May 02 | 10:13:59.09 | 9336.9252 | 12.86 | 4x72 | 224 | 1.7
May 03 | 9:56:11.36 | 9337.9127 | 12.93 | 4x72 | 234 | 1.3
June 19 | 6:32:04.07 | 9384.7675 | 16.29 | 4x72 | 182 | 1.6
June 19 | 6:38:49.98 | 9384.7722 | 16.29 | 4x72 | 171 | 1.7
June 20 | 6:32:32.52 | 9385.7677 | 16.37 | 4x72 | 241 | 1.3
June 21 | 6:27:02.17 | 9386.7639 | 16.44 | 4x72 | 280 | 1.4
June 23 | 6:00:08.67 | 9388.7451 | 16.58 | 4x72 | 258 | 1.3
June 24 | 5:57:16.62 | 9389.7431 | 16.65 | 4x72 | 283 | 1.2
June 25 | 6:06:01.63 | 9390.7491 | 16.72 | 4x72 | 235 | 1.4
June 26 | 5:57:21.88 | 9391.7430 | 16.80 | 4x72 | 293 | 1.3
June 27 | 5:59:26.04 | 9392.7444 | 16.87 | 4x72 | 286 | 1.5
June 28 | 6:02:42.25 | 9393.7467 | 16.94 | 4x72 | 295 | 1.4
June 29 | 5:56:45.44 | 9394.7425 | 17.01 | 4x72 | 242 | 1.6
June 30 | 5:58:09.69 | 9395.7435 | 17.08 | 4x72 | 220 | 1.6
July 01 | 5:58:11.25 | 9396.7434 | 17.16 | 4x72 | 241 | 1.6
July 02 | 5:57:49.39 | 9397.7432 | 17.23 | 4x72 | 195 | 2.0
July 17 | 5:56:11.39 | 9412.7417 | 18.31 | 4x72 | 271 | 1.3
July 19 | 5:46:33.47 | 9414.7350 | 18.45 | 4x72 | 239 | 1.4
November 16 | 15:08:48.28 | 9535.1336 | 27.10 | 4x72 | 273 | 1.3
November 17 | 15:38:53.47 | 9536.1546 | 27.18 | 4x72 | 236 | 1.3
November 18 | 15:01:36.89 | 9537.1288 | 27.25 | 4x72 | 280 | 1.3
November 19 | 15:34:03.76 | 9538.1514 | 27.32 | 4x72 | 302 | 1.1
November 20 | 15:17:58.29 | 9539.1403 | 27.39 | 4x72 | 260 | 1.3
November 20 | 15:55:15.68 | 9539.1662 | 27.39 | 4x72 | 273 | 1.4
November 21 | 15:10:49.20 | 9540.1354 | 27.46 | 4x72 | 278 | 1.4
November 22 | 14:53:39.28 | 9541.1236 | 27.53 | 4x72 | 293 | 1.2
December 09 | 15:17:03.16 | 9558.1411 | 28.76 | 4x72 | 284 | 1.3
December 10 | 14:49:25.95 | 9559.1219 | 28.83 | 4x72 | 233 | 1.5
December 11 | 15:49:49.61 | 9560.1639 | 28.90 | 4x72 | 217 | 1.4
December 13 | 14:59:58.41 | 9562.1294 | 29.05 | 4x72 | 287 | 1.1
December 15 | 14:37:49.46 | 9564.1142 | 29.19 | 4x72 | 256 | 1.4
December 17 | 15:55:58.97 | 9566.1686 | 29.34 | 4x72 | 203 | 2.0
December 18 | 15:05:25.50 | 9567.1335 | 29.40 | 4x72 | 248 | 1.5
2022 |
January 06 | 15:05:19.45 | 9586.1341 | 30.77 | 4x72 | 265 | 1.3
January 08 | 15:34:42.78 | 9588.1546 | 30.92 | 4x72 | 279 | 1.3
January 09 | 14:22:30.48 | 9589.1044 | 30.98 | 4x72 | 291 | 1.3
January 10 | 15:03:28.46 | 9590.1329 | 31.06 | 4x72 | 295 | 1.3
January 12 | 16:22:44.45 | 9592.1880 | 31.21 | 4x72 | 263 | 1.3
January 13 | 14:39:32.63 | 9593.1163 | 31.27 | 4x72 | 250 | 1.4
January 14 | 14:23:12.69 | 9594.1050 | 31.34 | 4x72 | 285 | 1.1
January 15 | 16:16:39.50 | 9595.1838 | 31.42 | 4x72 | 261 | 1.3
January 16 | 14:47:37.30 | 9596.1219 | 31.49 | 4x72 | 163 | 1.9
January 17 | 15:00:18.40 | 9597.1307 | 31.56 | 4x72 | 263 | 1.2
January 18 | 16:06:47.82 | 9598.1769 | 31.64 | 4x72 | 288 | 1.4
January 19 | 16:21:07.15 | 9599.1869 | 31.71 | 4x72 | 277 | 1.2
January 20 | 16:11:30.66 | 9600.1802 | 31.78 | 4x72 | 277 | 1.4
January 22 | 15:03:23.65 | 9602.1329 | 31.92 | 4x72 | 193 | 1.7
January 24 | 15:01:19.06 | 9604.1314 | 32.06 | 4x72 | 295 | 1.2
January 25 | 15:30:10.81 | 9605.1515 | 32.14 | 4x72 | 266 | 1.2
January 26 | 14:25:07.09 | 9606.1063 | 32.21 | 4x72 | 277 | 1.4
January 31 | 14:50:19.86 | 9611.1237 | 32.57 | 4x72 | 265 | 1.3
February 01 | 15:01:09.68 | 9612.1312 | 32.64 | 4x72 | 150 | 2.1
February 01 | 15:08:29.53 | 9612.1363 | 32.64 | 4x72 | 162 | 1.8
March 11 | 8:48:01.68 | 9649.8702 | 35.35 | 4x72 | 292 | 1.1
March 12 | 10:53:43.46 | 9650.9574 | 35.43 | 4x72 | 256 | 1.2
March 13 | 11:22:40.32 | 9651.9774 | 35.50 | 4x72 | 288 | 1.3
March 14 | 8:28:17.21 | 9652.8562 | 35.57 | 4x72 | 295 | 1.2
March 15 | 12:22:54.90 | 9654.0191 | 35.65 | 4x72 | 286 | 1.1
March 16 | 13:16:17.50 | 9655.0560 | 35.73 | 4x72 | 251 | 1.3
March 17 | 11:09:30.70 | 9655.9679 | 35.79 | 4x72 | 267 | 1.5
March 18 | 13:20:13.72 | 9657.0586 | 35.87 | 4x72 | 218 | 1.6
March 19 | 13:06:09.07 | 9658.0488 | 35.94 | 4x72 | 270 | 1.3
March 20 | 11:47:25.79 | 9658.9940 | 36.01 | 4x72 | 293 | 1.2
March 22 | 11:42:33.77 | 9660.9905 | 36.15 | 4x72 | 244 | 1.5
March 23 | 11:05:20.41 | 9661.9645 | 36.22 | 4x72 | 102 | 1.2
March 23 | 11:55:50.39 | 9661.9996 | 36.22 | 4x72 | 275 | 1.2
April 09 | 11:37:18.90 | 9678.9852 | 37.45 | 4x72 | 271 | 1.4
April 12 | 11:20:02.08 | 9681.9729 | 37.66 | 4x72 | 265 | 1.4
April 13 | 11:46:22.06 | 9682.9911 | 37.73 | 4x72 | 267 | 1.3
April 14 | 12:06:31.61 | 9684.0050 | 37.81 | 4x72 | 172 | 2.1
April 15 | 11:32:16.57 | 9684.9812 | 37.88 | 4x72 | 266 | 1.5
April 18 | 11:33:35.81 | 9687.9818 | 38.09 | 4x72 | 262 | 1.4
April 19 | 11:06:39.63 | 9688.9630 | 38.16 | 4x72 | 262 | 1.2
April 20 | 10:47:12.72 | 9689.9494 | 38.23 | 4x72 | 281 | 1.3
April 21 | 10:45:48.61 | 9690.9483 | 38.31 | 4x72 | 289 | 1.3
May 11 | 9:35:49.87 | 9710.8979 | 39.74 | 4x72 | 247 | 1.4
May 12 | 9:47:34.23 | 9711.9059 | 39.81 | 4x72 | 245 | 1.4
May 13 | 9:11:53.37 | 9712.8811 | 39.88 | 4x72 | 235 | 1.4
May 16 | 8:31:03.28 | 9715.8525 | 40.10 | 4x72 | 267 | 1.5
May 17 | 8:38:05.11 | 9716.8573 | 40.17 | 4x72 | 285 | 1.4
June 02 | 6:33:44.50 | 9732.7697 | 41.31 | 4x72 | 280 | 1.3
June 03 | 6:29:45.48 | 9733.7668 | 41.38 | 4x72 | 273 | 1.3
June 05 | 6:38:13.26 | 9735.7726 | 41.53 | 4x72 | 277 | 1.2
June 06 | 8:16:18.89 | 9736.8406 | 41.61 | 4x72 | 262 | 1.4
June 08 | 6:00:13.71 | 9738.7460 | 41.74 | 4x72 | 266 | 1.4
June 09 | 6:01:26.80 | 9739.7468 | 41.81 | 4x72 | 281 | 1.2
June 10 | 7:06:39.01 | 9740.7920 | 41.89 | 4x72 | 278 | 1.2
Table 8: Continued. Table 9:  CN Leo observations collected with SPIRou. The columns are the same as in Table LABEL:tab:log_evlac. Date | UT | HJD | $n_{\mathrm{cyc}}$ | $t_{exp}$ | S/N | $\sigma_{\mathrm{LSD}}$
---|---|---|---|---|---|---
| [hh:mm:ss] | $[-2450000]$ |  | [s] |  | [$10^{-4}I_{c}$]
2019 |
April 16 | 12:13:42.22 | 8590.0095 | 22.56 | 4x128 | 176 | 2.5
April 18 | 9:09:19.88 | 8591.8815 | 23.25 | 4x117 | 171 | 2.8
April 19 | 10:44:28.81 | 8592.9476 | 23.65 | 4x128 | 163 | 2.8
April 20 | 9:35:28.57 | 8593.8996 | 24.00 | 4x128 | 171 | 2.7
April 21 | 11:35:37.16 | 8594.9831 | 24.40 | 4x128 | 170 | 2.8
April 22 | 9:19:34.48 | 8595.8886 | 24.74 | 4x111 | 172 | 2.6
April 23 | 9:33:44.59 | 8596.8984 | 25.11 | 4x111 | 172 | 2.7
April 24 | 8:19:40.68 | 8597.8470 | 25.46 | 4x111 | 174 | 2.7
April 25 | 9:49:38.88 | 8598.9095 | 25.86 | 4x105 | 173 | 2.9
April 26 | 8:22:17.22 | 8599.8488 | 26.20 | 4x128 | 192 | 2.5
April 27 | 8:25:34.09 | 8600.8511 | 26.58 | 4x128 | 175 | 2.4
May 01 | 9:09:09.90 | 8604.8814 | 28.07 | 4x117 | 171 | 2.7
May 15 | 6:56:10.26 | 8618.7890 | 33.22 | 4x83 | 172 | 2.7
June 13 | 6:21:17.73 | 8647.7648 | 43.95 | 4x72 | 170 | 2.6
June 14 | 5:53:42.23 | 8648.7456 | 44.31 | 4x66 | 173 | 2.6
June 16 | 6:04:10.29 | 8650.7529 | 45.06 | 4x83 | 179 | 2.7
June 17 | 5:47:17.17 | 8651.7412 | 45.42 | 4x89 | 173 | 2.4
June 19 | 6:03:42.60 | 8653.7526 | 46.17 | 4x89 | 178 | 2.8
June 21 | 6:26:25.16 | 8655.7683 | 46.92 | 4x105 | 175 | 2.4
October 31 | 15:11:53.11 | 8788.1333 | 95.94 | 4x128 | 214 | 1.9
November 01 | 15:29:15.11 | 8789.1453 | 96.31 | 4x128 | 184 | 2.2
November 02 | 15:44:00.94 | 8790.1556 | 96.69 | 4x128 | 214 | 2.2
November 03 | 15:03:29.07 | 8791.1274 | 97.05 | 4x128 | 204 | 2.2
November 07 | 15:23:03.22 | 8795.1410 | 98.53 | 4x128 | 175 | 2.5
November 09 | 14:49:54.88 | 8797.1180 | 99.27 | 4x128 | 155 | 2.7
November 11 | 15:43:55.63 | 8799.1555 | 100.02 | 4x128 | 171 | 2.9
November 13 | 14:54:25.58 | 8801.1211 | 100.75 | 4x105 | 233 | 2.1
November 14 | 14:48:53.61 | 8802.1173 | 101.12 | 4x105 | 232 | 2.4
December 07 | 15:35:36.87 | 8825.1497 | 109.65 | 4x128 | 196 | 2.2
December 08 | 14:55:34.82 | 8826.1219 | 110.01 | 4x117 | 228 | 2.2
December 09 | 14:45:15.59 | 8827.1148 | 110.38 | 4x117 | 229 | 2.0
December 10 | 13:54:07.22 | 8828.0792 | 110.73 | 4x117 | 235 | 2.4
December 11 | 15:20:36.69 | 8829.1393 | 111.13 | 4x117 | 229 | 2.1
December 12 | 14:56:48.41 | 8830.1228 | 111.49 | 4x128 | 229 | 2.1
2020 |
January 26 | 12:28:07.86 | 8875.0195 | 128.12 | 4x111 | 233 | 2.3
January 27 | 12:23:43.97 | 8876.0165 | 128.49 | 4x117 | 230 | 2.2
January 28 | 13:14:47.18 | 8877.0519 | 128.87 | 4x128 | 219 | 2.0
February 05 | 8:54:48.40 | 8884.8714 | 131.77 | 4x111 | 228 | 2.1
February 09 | 8:19:52.83 | 8888.8471 | 133.24 | 4x128 | 182 | 2.5
February 16 | 7:47:00.30 | 8895.8243 | 135.82 | 4x111 | 227 | 2.1
February 17 | 9:28:00.93 | 8896.8945 | 136.22 | 4x128 | 206 | 2.2
February 18 | 7:46:31.47 | 8897.8240 | 136.57 | 4x128 | 229 | 2.0
February 19 | 10:11:34.85 | 8898.9247 | 136.97 | 4x105 | 233 | 2.2
March 11 | 12:52:11.14 | 8920.0362 | 144.79 | 4x111 | 233 | 1.8
March 12 | 11:12:15.25 | 8920.9668 | 145.14 | 4x100 | 236 | 2.2
May 07 | 10:50:12.93 | 8976.9515 | 165.87 | 4x128 | 218 | 2.2
May 09 | 9:49:53.90 | 8978.9097 | 166.60 | 4x111 | 231 | 2.0
May 12 | 9:47:44.18 | 8981.9082 | 167.71 | 4x128 | 222 | 2.1
May 13 | 9:52:29.17 | 8982.9114 | 168.08 | 4x100 | 231 | 2.2
May 14 | 7:54:40.23 | 8983.8296 | 168.42 | 4x94 | 229 | 1.9
May 15 | 9:57:54.46 | 8984.9152 | 168.82 | 4x128 | 196 | 2.3
June 05 | 6:51:51.44 | 9005.7860 | 176.55 | 4x128 | 197 | 2.1
June 07 | 7:06:16.26 | 9007.7960 | 177.30 | 4x128 | 175 | 2.7
June 08 | 6:17:14.27 | 9008.7620 | 177.65 | 4x128 | 179 | 3.2
June 09 | 6:43:04.47 | 9009.7799 | 178.03 | 4x128 | 174 | 2.6
June 10 | 7:04:25.72 | 9010.7947 | 178.41 | 4x128 | 230 | 1.9
October 31 | 15:20:34.06 | 9154.1393 | 231.50 | 4x111 | 226 | 2.1
November 02 | 15:34:13.01 | 9156.1488 | 232.24 | 4x128 | 211 | 2.1
November 03 | 15:43:02.82 | 9157.1549 | 232.61 | 4x89 | 225 | 2.1
November 04 | 14:48:35.82 | 9158.1171 | 232.97 | 4x83 | 222 | 2.3
November 05 | 14:39:48.09 | 9159.1110 | 233.34 | 4x78 | 220 | 2.3
December 24 | 13:17:50.80 | 9208.0541 | 251.47 | 4x78 | 228 | 2.2
December 25 | 14:34:01.34 | 9209.1070 | 251.86 | 4x83 | 225 | 2.4
December 26 | 13:02:53.07 | 9210.0437 | 252.20 | 4x117 | 223 | 2.4
December 29 | 14:57:23.81 | 9213.1232 | 253.34 | 4x128 | 158 | 2.5
December 30 | 14:59:32.48 | 9214.1247 | 253.71 | 4x128 | 223 | 2.4
December 31 | 14:19:08.41 | 9215.0966 | 254.07 | 4x100 | 226 | 2.4
2021 |
January 02 | 15:33:42.40 | 9217.1484 | 254.83 | 4x111 | 222 | 2.3
January 03 | 13:49:08.56 | 9218.0758 | 255.18 | 4x78 | 223 | 2.2
January 04 | 13:17:16.19 | 9219.0537 | 255.54 | 4x78 | 226 | 2.2
January 05 | 13:07:42.22 | 9220.0470 | 255.91 | 4x83 | 220 | 2.3
January 06 | 14:05:29.70 | 9221.0871 | 256.29 | 4x89 | 222 | 2.1
January 07 | 13:34:08.29 | 9222.0654 | 256.65 | 4x94 | 225 | 2.2
January 07 | 16:17:06.35 | 9222.1785 | 256.70 | 4x100 | 220 | 2.5
January 08 | 13:41:17.96 | 9223.0703 | 257.03 | 4x100 | 225 | 2.6
January 08 | 16:03:57.25 | 9223.1694 | 257.06 | 4x105 | 222 | 2.3
February 21 | 10:53:47.13 | 9266.9540 | 273.28 | 4x111 | 165 | 2.8
February 21 | 11:04:02.34 | 9266.9611 | 273.28 | 4x111 | 188 | 2.5
February 22 | 10:05:46.91 | 9267.9207 | 273.64 | 4x100 | 226 | 2.3
February 23 | 12:41:09.81 | 9269.0286 | 274.05 | 4x111 | 203 | 2.5
February 26 | 8:09:09.92 | 9271.8397 | 275.09 | 4x111 | 245 | 2.4
February 28 | 14:12:09.33 | 9274.0918 | 275.92 | 4x111 | 275 | 2.1
March 02 | 13:44:02.27 | 9276.0722 | 276.66 | 4x111 | 238 | 2.4
March 03 | 11:50:58.30 | 9276.9937 | 277.00 | 4x111 | 254 | 2.1
March 04 | 13:32:04.78 | 9278.0639 | 277.40 | 4x111 | 276 | 2.0
March 20 | 12:37:22.22 | 9294.0260 | 283.31 | 4x111 | 269 | 1.9
March 21 | 11:38:33.18 | 9294.9851 | 283.66 | 4x111 | 240 | 2.2
March 23 | 11:30:39.62 | 9296.9796 | 284.40 | 4x111 | 261 | 2.1
March 24 | 12:44:32.17 | 9298.0309 | 284.79 | 4x111 | 263 | 2.1
March 26 | 11:52:50.95 | 9299.9950 | 285.52 | 4x111 | 247 | 2.3
March 27 | 6:45:37.02 | 9300.7817 | 285.81 | 4x111 | 248 | 2.1
March 31 | 10:54:04.65 | 9304.9542 | 287.35 | 4x111 | 219 | 2.4
April 01 | 12:02:33.87 | 9306.0018 | 287.74 | 4x111 | 254 | 2.1
April 22 | 10:28:06.67 | 9326.9362 | 295.50 | 4x111 | 187 | 2.3
April 23 | 7:56:25.52 | 9327.8309 | 295.83 | 4x111 | 264 | 2.2
April 24 | 10:38:50.47 | 9328.9436 | 296.24 | 4x111 | 220 | 2.6
April 25 | 9:34:48.36 | 9329.8992 | 296.59 | 4x111 | 218 | 2.7
April 26 | 9:44:56.28 | 9330.9062 | 296.97 | 4x111 | 204 | 2.9
April 27 | 10:43:10.75 | 9331.9467 | 297.35 | 4x111 | 242 | 2.2
April 28 | 10:40:26.84 | 9332.9448 | 297.72 | 4x111 | 257 | 2.1
April 30 | 8:57:35.34 | 9334.8733 | 298.44 | 4x111 | 269 | 2.0
May 01 | 9:25:34.23 | 9335.8928 | 298.81 | 4x111 | 266 | 2.1
May 02 | 9:55:38.98 | 9336.9136 | 299.19 | 4x111 | 201 | 2.5
May 03 | 9:26:51.13 | 9337.8936 | 299.55 | 4x111 | 136 | 2.5
May 03 | 9:36:37.40 | 9337.9004 | 299.56 | 4x111 | 207 | 2.5
June 19 | 6:03:13.60 | 9384.7522 | 316.91 | 4x111 | 141 | 4.2
June 19 | 6:12:45.43 | 9384.7589 | 316.91 | 4x111 | 132 | 3.0
June 20 | 6:15:05.26 | 9385.7605 | 317.28 | 4x111 | 245 | 2.2
June 21 | 6:09:04.48 | 9386.7563 | 317.65 | 4x111 | 277 | 2.1
June 23 | 5:41:33.33 | 9388.7372 | 318.39 | 4x111 | 269 | 2.1
June 24 | 5:38:03.01 | 9389.7348 | 318.75 | 4x111 | 249 | 2.2
June 25 | 5:49:08.47 | 9390.7425 | 319.13 | 4x111 | 241 | 2.3
June 26 | 5:46:41.96 | 9391.7408 | 319.50 | 4x111 | 247 | 2.4
June 27 | 5:46:00.34 | 9392.7403 | 319.87 | 4x111 | 284 | 2.1
June 28 | 5:44:57.25 | 9393.7396 | 320.24 | 4x111 | 287 | 2.2
June 29 | 5:38:28.52 | 9394.7351 | 320.61 | 4x111 | 244 | 2.7
June 30 | 5:39:01.55 | 9395.7354 | 320.98 | 4x111 | 207 | 2.8
July 01 | 5:39:05.12 | 9396.7355 | 321.35 | 4x111 | 239 | 2.2
July 02 | 5:39:06.12 | 9397.7355 | 321.72 | 4x111 | 156 | 3.2
November 16 | 14:51:17.17 | 9535.1189 | 372.60 | 4x111 | 258 | 1.9
November 18 | 15:09:21.21 | 9537.1315 | 373.35 | 4x111 | 252 | 2.1
November 19 | 15:41:46.14 | 9538.1540 | 373.72 | 4x111 | 292 | 2.0
November 20 | 15:25:28.03 | 9539.1427 | 374.09 | 4x111 | 269 | 2.2
November 21 | 15:18:29.94 | 9540.1378 | 374.46 | 4x111 | 270 | 1.9
November 22 | 15:01:19.45 | 9541.1259 | 374.83 | 4x111 | 251 | 2.0
December 09 | 14:43:38.22 | 9558.1136 | 381.12 | 4x111 | 169 | 2.2
December 10 | 14:16:08.87 | 9559.0945 | 381.48 | 4x111 | 197 | 2.4
December 11 | 15:16:43.27 | 9560.1366 | 381.87 | 4x111 | 239 | 2.6
December 12 | 14:49:26.03 | 9561.1177 | 382.23 | 4x111 | 270 | 2.0
December 13 | 14:19:35.10 | 9562.0969 | 382.59 | 4x111 | 281 | 2.1
December 17 | 15:21:34.66 | 9566.1400 | 384.09 | 4x111 | 221 | 2.2
2022 |
January 08 | 16:08:05.82 | 9588.1723 | 392.25 | 4x111 | 237 | 2.1
January 09 | 14:54:37.52 | 9589.1213 | 392.60 | 4x111 | 290 | 2.2
January 10 | 15:11:57.75 | 9590.1333 | 392.98 | 4x111 | 268 | 2.0
January 11 | 15:15:49.27 | 9591.1360 | 393.35 | 4x111 | 228 | 1.9
January 13 | 15:15:45.97 | 9593.1359 | 394.09 | 4x111 | 273 | 2.0
January 14 | 14:57:46.42 | 9594.1235 | 394.45 | 4x111 | 275 | 1.9
January 15 | 14:30:58.09 | 9595.1048 | 394.82 | 4x111 | 217 | 2.5
January 15 | 16:24:18.06 | 9595.1835 | 394.85 | 4x111 | 248 | 2.0
January 16 | 16:17:06.12 | 9596.1785 | 395.22 | 4x111 | 208 | 2.1
January 17 | 14:48:47.16 | 9597.1172 | 395.56 | 4x111 | 259 | 1.9
January 18 | 16:14:14.92 | 9598.1766 | 395.96 | 4x111 | 281 | 1.9
January 19 | 16:03:54.92 | 9599.1694 | 396.32 | 4x111 | 242 | 2.0
January 20 | 16:19:00.43 | 9600.1799 | 396.70 | 4x111 | 275 | 2.0
January 22 | 14:51:44.64 | 9602.1193 | 397.42 | 4x111 | 197 | 2.1
January 23 | 12:33:17.06 | 9603.0231 | 397.75 | 4x111 | 253 | 1.8
January 24 | 15:39:39.54 | 9604.1525 | 398.17 | 4x111 | 278 | 1.9
January 25 | 14:54:38.57 | 9605.1213 | 398.53 | 4x111 | 268 | 2.1
January 26 | 15:22:06.80 | 9606.1404 | 398.90 | 4x111 | 258 | 2.1
January 27 | 15:04:51.26 | 9607.1284 | 399.27 | 4x111 | 251 | 1.6
January 31 | 15:25:45.41 | 9611.1429 | 400.76 | 4x111 | 264 | 1.8
February 01 | 15:16:42.73 | 9612.1366 | 401.13 | 4x111 | 224 | 2.4
March 11 | 8:37:55.01 | 9649.8597 | 415.10 | 4x111 | 287 | 1.8
March 12 | 8:40:28.21 | 9650.8614 | 415.47 | 4x111 | 289 | 2.0
March 13 | 8:01:40.95 | 9651.8345 | 415.83 | 4x111 | 292 | 1.8
March 15 | 10:37:12.00 | 9653.9425 | 416.61 | 4x111 | 281 | 2.1
March 17 | 8:44:34.57 | 9655.8643 | 417.32 | 4x111 | 244 | 2.1
March 20 | 9:52:07.34 | 9658.9112 | 418.45 | 4x111 | 270 | 1.8
March 22 | 9:46:22.44 | 9660.9072 | 419.19 | 4x111 | 226 | 2.4
April 12 | 11:02:15.07 | 9681.9599 | 426.99 | 4x111 | 246 | 2.2
April 18 | 11:23:26.56 | 9687.9746 | 429.21 | 4x111 | 251 | 2.1
May 11 | 9:17:57.19 | 9710.8875 | 437.70 | 4x111 | 230 | 2.4
May 14 | 9:36:00.15 | 9713.9000 | 438.82 | 4x111 | 198 | 2.4
June 10 | 6:03:21.25 | 9740.7523 | 448.76 | 4x111 | 281 | 2.2
Table 9: Continued. Table 10:  Optical and near-infrared measurements of longitudinal magnetic field for EV Lac. The columns are: (1) heliocentric Julian date of the observation, (2) Bl with formal error bar (see Eq. 4), and (3) instrument used. HJD | Bl | Instrument
---|---|---
$[-2450000]$ | [G] |
3569.1119 | 144.4$\pm$8.3 | ESPaDOnS
3606.1478 |  $-$305.9$\pm$20.4 | ESPaDOnS
3631.9065 | 23.9$\pm$9.4 | ESPaDOnS
3631.9447 |  $-$106.2$\pm$11.0 | ESPaDOnS
3631.9828 |  $-$82.9$\pm$9.6 | ESPaDOnS
3953.0731 |  $-$529.5$\pm$20.6 | ESPaDOnS
3955.0663 | 319.8$\pm$15.1 | ESPaDOnS
3956.0600 |  $-$354.3$\pm$13.8 | ESPaDOnS
3957.0597 |  $-$422.4$\pm$18.1 | ESPaDOnS
3958.0723 |  $-$282.6$\pm$15.9 | ESPaDOnS
3959.0728 | 290.1$\pm$15.0 | ESPaDOnS
3960.0761 |  $-$161.0$\pm$15.2 | ESPaDOnS
4986.5995 | 264.8$\pm$18.4 | NARVAL
4994.6321 | 0.5$\pm$20.9 | NARVAL
4995.5842 | 251.6$\pm$38.6 | NARVAL
5059.3898 | 48.0$\pm$18.3 | NARVAL
5370.5520 | 92.9$\pm$16.7 | NARVAL
5384.6114 |  $-$49.5$\pm$13.4 | NARVAL
5385.5601 |  $-$176.2$\pm$15.1 | NARVAL
5388.5822 | 23.4$\pm$14.6 | NARVAL
5390.5888 |  $-$203.6$\pm$25.4 | NARVAL
5392.5938 |  $-$73.6$\pm$14.2 | NARVAL
5396.5771 | 22.4$\pm$13.2 | NARVAL
5401.5894 | 76.7$\pm$14.0 | NARVAL
5402.5222 |  $-$87.1$\pm$14.5 | NARVAL
5403.5208 |  $-$270.5$\pm$18.7 | NARVAL
5411.5386 |  $-$194.8$\pm$15.6 | NARVAL
5414.5135 | 93.9$\pm$14.1 | NARVAL
5415.5838 |  $-$105.8$\pm$16.3 | NARVAL
5416.5916 |  $-$297.6$\pm$16.2 | NARVAL
5419.5954 |  $-$58.9$\pm$14.3 | NARVAL
5420.5813 |  $-$306.3$\pm$19.1 | NARVAL
5425.6210 |  $-$231.6$\pm$14.9 | NARVAL
5426.5908 |  $-$143.6$\pm$14.6 | NARVAL
5429.5743 |  $-$355.4$\pm$21.8 | NARVAL
5430.6048 |  $-$168.5$\pm$16.1 | NARVAL
7584.5642 |  $-$97.8$\pm$15.2 | NARVAL
7585.5742 |  $-$107.6$\pm$20.7 | NARVAL
7588.5683 | 5.1$\pm$13.9 | NARVAL
7595.5856 |  $-$93.8$\pm$15.0 | NARVAL
7608.4675 |  $-$61.2$\pm$15.2 | NARVAL
7612.5894 |  $-$50.4$\pm$17.5 | NARVAL
8738.0805 | 120.8$\pm$26.4 | SPIRou
8744.8848 |  $-$151.5$\pm$22.6 | SPIRou
8745.8461 | 47.1$\pm$22.6 | SPIRou
8746.9965 | 57.1$\pm$28.9 | SPIRou
8750.8405 | 130.8$\pm$21.0 | SPIRou
8751.9348 |  $-$14.5$\pm$22.0 | SPIRou
8752.8467 |  $-$82.0$\pm$21.9 | SPIRou
8758.8646 |  $-$42.6$\pm$34.0 | SPIRou
8761.8343 |  $-$71.1$\pm$22.1 | SPIRou
8762.8207 |  $-$114.5$\pm$22.0 | SPIRou
8763.9244 | 201.6$\pm$26.9 | SPIRou
8765.9023 |  $-$23.3$\pm$22.3 | SPIRou
8768.9492 | 64.0$\pm$24.4 | SPIRou
8770.8726 |  $-$160.7$\pm$24.0 | SPIRou
8772.8746 | 113.7$\pm$25.4 | SPIRou
8787.8492 |  $-$43.4$\pm$28.4 | SPIRou
8788.8650 |  $-$171.5$\pm$22.8 | SPIRou
8789.8035 | 121.3$\pm$21.6 | SPIRou
8790.8372 | 82.6$\pm$23.3 | SPIRou
8791.8001 | 18.3$\pm$29.9 | SPIRou
8792.9242 |  $-$136.4$\pm$25.1 | SPIRou
8794.8605 | 149.6$\pm$31.5 | SPIRou
8796.8614 |  $-$126.4$\pm$34.2 | SPIRou
8797.8884 |  $-$66.1$\pm$37.9 | SPIRou
8798.8829 | 182.5$\pm$42.1 | SPIRou
8799.8233 |  $-$57.3$\pm$29.1 | SPIRou
8800.8467 |  $-$53.9$\pm$31.4 | SPIRou
8801.8228 |  $-$101.7$\pm$31.7 | SPIRou
8822.7855 |  $-$164.5$\pm$33.4 | SPIRou
8825.8037 | 6.6$\pm$28.6 | SPIRou
8826.8194 | 9.1$\pm$32.7 | SPIRou
8827.7660 |  $-$205.7$\pm$29.1 | SPIRou
8828.7799 |  $-$5.1$\pm$25.3 | SPIRou
8829.7371 | 201.0$\pm$24.4 | SPIRou
8830.7822 |  $-$30.8$\pm$28.0 | SPIRou
9058.1250 |  $-$173.8$\pm$20.7 | SPIRou
9059.9728 | 54.1$\pm$17.8 | SPIRou
9060.9682 | 222.0$\pm$18.8 | SPIRou
9062.1226 |  $-$45.8$\pm$18.8 | SPIRou
9062.9968 |  $-$355.7$\pm$19.6 | SPIRou
9063.9722 |  $-$119.3$\pm$18.8 | SPIRou
9065.0092 | 175.9$\pm$18.9 | SPIRou
9065.9459 | 119.8$\pm$18.6 | SPIRou
9068.0267 |  $-$329.2$\pm$43.3 | SPIRou
9068.0312 |  $-$293.3$\pm$42.1 | SPIRou
9070.0256 | 206.8$\pm$30.3 | SPIRou
9071.0446 |  $-$137.7$\pm$21.0 | SPIRou
9072.0181 |  $-$381.7$\pm$19.1 | SPIRou
9073.0616 | 21.9$\pm$16.9 | SPIRou
9087.9929 |  $-$13.6$\pm$24.4 | SPIRou
9088.9728 |  $-$315.0$\pm$19.7 | SPIRou
9089.9193 |  $-$304.9$\pm$17.8 | SPIRou
9091.0053 | 116.5$\pm$17.4 | SPIRou
9092.0072 | 160.0$\pm$19.9 | SPIRou
9092.9975 |  $-$237.8$\pm$17.2 | SPIRou
9093.9327 |  $-$305.1$\pm$18.7 | SPIRou
9094.9279 | 12.8$\pm$16.2 | SPIRou
9096.0192 | 217.5$\pm$18.1 | SPIRou
9097.0196 |  $-$95.4$\pm$18.3 | SPIRou
9098.0086 |  $-$387.9$\pm$22.0 | SPIRou
9098.9935 |  $-$92.1$\pm$18.3 | SPIRou
9100.9554 | 71.0$\pm$17.0 | SPIRou
9101.9244 |  $-$258.0$\pm$20.4 | SPIRou
9102.9258 |  $-$273.3$\pm$18.2 | SPIRou
9110.8924 |  $-$327.2$\pm$21.0 | SPIRou
9111.9207 |  $-$166.5$\pm$26.8 | SPIRou
9112.9920 | 137.9$\pm$18.2 | SPIRou
9113.8478 | 184.2$\pm$25.4 | SPIRou
9114.8580 |  $-$227.5$\pm$18.5 | SPIRou
9115.8426 |  $-$338.7$\pm$19.5 | SPIRou
9117.8855 | 254.0$\pm$18.3 | SPIRou
9118.8149 |  $-$30.6$\pm$21.3 | SPIRou
9119.8209 |  $-$469.1$\pm$19.2 | SPIRou
9120.8204 |  $-$94.8$\pm$18.3 | SPIRou
9121.8184 | 191.1$\pm$18.3 | SPIRou
9122.8613 | 82.0$\pm$17.2 | SPIRou
9123.9069 |  $-$364.0$\pm$20.7 | SPIRou
9125.8420 | 120.7$\pm$18.7 | SPIRou
9126.8730 | 250.2$\pm$20.7 | SPIRou
9127.8539 |  $-$223.5$\pm$18.6 | SPIRou
9128.9388 |  $-$320.2$\pm$19.2 | SPIRou
9129.8487 |  $-$8.8$\pm$17.2 | SPIRou
9154.8442 |  $-$388.4$\pm$17.2 | SPIRou
9156.8789 | 216.9$\pm$16.9 | SPIRou
9157.8683 | 43.0$\pm$18.3 | SPIRou
9158.7828 |  $-$323.0$\pm$17.3 | SPIRou
9207.7586 |  $-$227.0$\pm$18.1 | SPIRou
9208.7037 | 23.9$\pm$17.9 | SPIRou
9209.7317 | 153.5$\pm$17.7 | SPIRou
9212.7597 |  $-$15.4$\pm$20.3 | SPIRou
9213.6973 | 136.5$\pm$22.7 | SPIRou
9214.7422 |  $-$29.1$\pm$21.0 | SPIRou
9215.6925 |  $-$355.8$\pm$21.3 | SPIRou
9216.7152 |  $-$173.8$\pm$18.7 | SPIRou
9217.7413 | 78.8$\pm$17.1 | SPIRou
9218.7086 | 130.8$\pm$17.2 | SPIRou
9219.7204 |  $-$293.7$\pm$19.6 | SPIRou
9220.7093 |  $-$357.7$\pm$18.6 | SPIRou
9221.7231 | 12.6$\pm$19.5 | SPIRou
9221.7556 | 36.1$\pm$19.4 | SPIRou
9222.7286 | 140.4$\pm$18.7 | SPIRou
9222.7623 | 163.7$\pm$18.7 | SPIRou
9386.0143 |  $-$370.1$\pm$17.9 | SPIRou
9387.0063 |  $-$83.9$\pm$18.7 | SPIRou
9387.9405 | 143.6$\pm$55.6 | SPIRou
9387.9446 | 120.0$\pm$42.8 | SPIRou
9389.0138 | 155.7$\pm$16.8 | SPIRou
9390.0506 |  $-$314.4$\pm$19.2 | SPIRou
9391.0437 |  $-$240.4$\pm$19.7 | SPIRou
9392.0613 | 194.9$\pm$18.0 | SPIRou
9393.0468 | 86.5$\pm$18.1 | SPIRou
9394.0751 |  $-$180.9$\pm$17.8 | SPIRou
9395.0160 |  $-$360.0$\pm$19.2 | SPIRou
9396.1236 | 174.1$\pm$19.3 | SPIRou
9397.0375 | 72.0$\pm$20.9 | SPIRou
9398.0271 | 105.6$\pm$20.4 | SPIRou
9413.0426 |  $-$174.6$\pm$20.2 | SPIRou
9414.0396 | 216.1$\pm$24.3 | SPIRou
9414.9480 | 171.8$\pm$28.5 | SPIRou
9416.0206 |  $-$264.2$\pm$17.3 | SPIRou
9417.0727 |  $-$321.1$\pm$19.7 | SPIRou
9421.0846 |  $-$442.5$\pm$28.9 | SPIRou
9423.0645 | 177.7$\pm$17.9 | SPIRou
9440.0290 | 249.7$\pm$18.8 | SPIRou
9440.8931 | 104.1$\pm$17.2 | SPIRou
9441.8620 |  $-$22.3$\pm$16.3 | SPIRou
9442.8526 |  $-$455.8$\pm$20.0 | SPIRou
9443.9164 | 82.2$\pm$18.8 | SPIRou
9445.0174 | 112.8$\pm$18.5 | SPIRou
9445.8257 | 167.7$\pm$17.7 | SPIRou
9446.9087 |  $-$405.0$\pm$32.7 | SPIRou
9447.9479 |  $-$164.1$\pm$24.4 | SPIRou
9448.9333 | 259.2$\pm$19.3 | SPIRou
9449.9149 | 111.8$\pm$18.2 | SPIRou
9451.0469 |  $-$294.0$\pm$23.0 | SPIRou
9451.8786 |  $-$367.3$\pm$19.4 | SPIRou
9452.8702 | 194.3$\pm$16.7 | SPIRou
9472.9042 |  $-$272.5$\pm$19.6 | SPIRou
9474.9013 | 272.7$\pm$17.4 | SPIRou
9475.8372 | 137.8$\pm$16.7 | SPIRou
9476.8210 |  $-$9.3$\pm$16.3 | SPIRou
9477.8592 |  $-$394.8$\pm$18.4 | SPIRou
9478.8431 | 17.4$\pm$22.1 | SPIRou
9479.8398 | 173.0$\pm$17.4 | SPIRou
9480.8265 | 154.8$\pm$17.5 | SPIRou
9481.8316 |  $-$340.5$\pm$18.2 | SPIRou
9502.7917 | 104.2$\pm$16.0 | SPIRou
9503.8325 |  $-$371.1$\pm$18.2 | SPIRou
9504.8194 |  $-$106.2$\pm$18.7 | SPIRou
9506.9874 | 85.4$\pm$16.6 | SPIRou
9507.8015 |  $-$239.0$\pm$17.9 | SPIRou
9508.8368 |  $-$242.1$\pm$19.3 | SPIRou
9509.7826 | 226.4$\pm$21.8 | SPIRou
9510.8080 | 87.4$\pm$17.0 | SPIRou
9511.8528 |  $-$92.6$\pm$20.2 | SPIRou
9512.7546 |  $-$363.7$\pm$17.6 | SPIRou
9513.7934 | 28.4$\pm$16.6 | SPIRou
9514.9074 | 129.1$\pm$17.7 | SPIRou
9537.7819 |  $-$25.7$\pm$17.8 | SPIRou
9538.7691 |  $-$330.5$\pm$18.8 | SPIRou
9539.7679 |  $-$43.1$\pm$20.1 | SPIRou
9540.7944 | 152.8$\pm$35.0 | SPIRou
Table 10: Continued. Table 11:  Optical and near-infrared measurements of longitudinal magnetic field for DS Leo. The columns are the same as in Table LABEL:tab:Bl_log_evlac. HJD | Bl | Instrument
---|---|---
$[-2450000]$ | [G] |
3748.1023 | 4.7$\pm$5.9 | ESPaDOnS
3748.9053 | 23.6$\pm$13.1 | ESPaDOnS
3774.0359 | 9.8$\pm$6.0 | ESPaDOnS
3781.1526 |  $-$7.9$\pm$9.1 | ESPaDOnS
4126.6502 |  $-$16.5$\pm$8.9 | NARVAL
4127.6349 |  $-$4.5$\pm$7.3 | NARVAL
4128.6469 | 1.7$\pm$5.9 | NARVAL
4129.6107 | 22.4$\pm$5.7 | NARVAL
4130.6536 | 23.1$\pm$5.4 | NARVAL
4133.6704 | 20.2$\pm$5.5 | NARVAL
4134.6568 | 14.0$\pm$5.5 | NARVAL
4135.6695 | 12.6$\pm$5.3 | NARVAL
4136.6303 | 30.1$\pm$10.0 | NARVAL
4462.6760 | 5.0$\pm$6.0 | NARVAL
4463.7049 | 7.8$\pm$5.3 | NARVAL
4465.7095 | 27.0$\pm$7.2 | NARVAL
4466.7091 | 25.0$\pm$5.4 | NARVAL
4468.7075 | 74.6$\pm$11.7 | NARVAL
4472.6258 | 16.4$\pm$13.3 | NARVAL
4473.6232 | 1.5$\pm$7.2 | NARVAL
4484.5805 | 17.1$\pm$11.7 | NARVAL
4485.5564 |  $-$6.5$\pm$10.5 | NARVAL
4487.5839 |  $-$0.8$\pm$8.6 | NARVAL
4489.5328 | 19.8$\pm$7.6 | NARVAL
4491.5490 | 13.4$\pm$7.5 | NARVAL
4492.5717 | 2.9$\pm$5.8 | NARVAL
4493.5834 | 1.7$\pm$6.4 | NARVAL
4499.6016 |  $-$10.8$\pm$6.8 | NARVAL
4501.5834 | 6.5$\pm$7.7 | NARVAL
4502.5820 | 12.2$\pm$6.0 | NARVAL
4503.5804 | 4.3$\pm$6.2 | NARVAL
4506.5933 | 18.7$\pm$6.1 | NARVAL
4507.5570 | 6.3$\pm$5.7 | NARVAL
4508.5856 | 29.6$\pm$6.3 | NARVAL
4509.5907 | 28.6$\pm$6.7 | NARVAL
4510.5857 | 27.9$\pm$8.4 | NARVAL
4511.6026 |  $-$0.9$\pm$6.2 | NARVAL
4512.5883 |  $-$0.5$\pm$6.8 | NARVAL
4513.5876 |  $-$2.1$\pm$6.0 | NARVAL
4862.6428 | 46.2$\pm$14.1 | NARVAL
5266.4448 | 29.3$\pm$7.0 | NARVAL
5268.4569 | 55.1$\pm$6.5 | NARVAL
5269.4287 | 41.6$\pm$6.8 | NARVAL
5270.5521 | 22.8$\pm$7.5 | NARVAL
5271.5597 | 9.2$\pm$5.3 | NARVAL
5272.5173 |  $-$5.3$\pm$11.5 | NARVAL
5278.4970 |  $-$16.0$\pm$5.7 | NARVAL
5284.4554 | 61.1$\pm$9.6 | NARVAL
5292.4695 | 163.2$\pm$27.1 | NARVAL
5293.3366 | 22.5$\pm$6.2 | NARVAL
5295.5424 | 38.9$\pm$6.1 | NARVAL
5296.4579 | 44.9$\pm$5.7 | NARVAL
5297.3607 | 33.0$\pm$5.7 | NARVAL
5302.5069 |  $-$10.0$\pm$5.9 | NARVAL
5303.4522 |  $-$9.5$\pm$5.8 | NARVAL
5304.3913 |  $-$17.4$\pm$5.6 | NARVAL
5310.3818 | 19.8$\pm$6.6 | NARVAL
5311.4403 | 21.0$\pm$6.5 | NARVAL
5312.3920 | 35.1$\pm$5.9 | NARVAL
5586.5978 | 18.2$\pm$7.0 | NARVAL
5587.5790 | 18.5$\pm$6.0 | NARVAL
5588.5849 | 25.8$\pm$5.6 | NARVAL
5589.5985 | 44.9$\pm$7.0 | NARVAL
5593.5929 |  $-$3.9$\pm$6.1 | NARVAL
5596.5681 |  $-$0.3$\pm$6.5 | NARVAL
5597.5715 |  $-$0.8$\pm$5.4 | NARVAL
5598.5820 | 18.2$\pm$6.3 | NARVAL
6048.3494 |  $-$4.6$\pm$7.2 | NARVAL
6050.3464 | 34.7$\pm$13.0 | NARVAL
6084.3812 | 61.7$\pm$17.4 | NARVAL
6092.3807 | 12.1$\pm$13.0 | NARVAL
6093.3758 | 68.2$\pm$19.5 | NARVAL
6094.3724 | 46.5$\pm$19.9 | NARVAL
6095.3935 | 3.3$\pm$12.2 | NARVAL
6100.3997 | 32.7$\pm$10.3 | NARVAL
6101.3914 | 23.4$\pm$8.9 | NARVAL
6102.4207 | 16.7$\pm$10.5 | NARVAL
6103.4078 | 24.9$\pm$14.6 | NARVAL
6667.6945 | 0.1$\pm$5.9 | NARVAL
6669.6203 | 4.5$\pm$6.7 | NARVAL
6744.3769 | 31.2$\pm$14.3 | NARVAL
6754.4270 | 33.0$\pm$8.1 | NARVAL
6756.4720 | 24.4$\pm$5.7 | NARVAL
6757.4436 | 14.9$\pm$5.7 | NARVAL
6759.5429 | 7.3$\pm$7.0 | NARVAL
6760.4614 | 16.6$\pm$6.2 | NARVAL
6761.4555 | 10.6$\pm$5.9 | NARVAL
6762.4030 | 17.1$\pm$8.0 | NARVAL
6763.4675 | 13.5$\pm$5.9 | NARVAL
6764.4372 | 1.0$\pm$5.7 | NARVAL
6765.4610 | 4.6$\pm$7.1 | NARVAL
6786.3669 | 18.8$\pm$6.7 | NARVAL
6787.3714 | 3.9$\pm$6.0 | NARVAL
9158.1132 |  $-$23.1$\pm$7.1 | SPIRou
9159.1076 | 0.1$\pm$7.1 | SPIRou
9208.0545 | 1.3$\pm$7.5 | SPIRou
9209.1069 |  $-$3.3$\pm$7.3 | SPIRou
9210.0431 |  $-$17.6$\pm$7.6 | SPIRou
9213.1208 |  $-$10.8$\pm$10.0 | SPIRou
9214.1222 |  $-$10.7$\pm$8.1 | SPIRou
9215.0966 | 8.5$\pm$7.1 | SPIRou
9217.1479 | 50.3$\pm$7.2 | SPIRou
9218.0762 | 30.9$\pm$7.3 | SPIRou
9219.0543 | 25.0$\pm$7.2 | SPIRou
9220.0473 | 7.9$\pm$7.4 | SPIRou
9221.0873 | 19.4$\pm$7.4 | SPIRou
9222.0928 | 0.5$\pm$7.2 | SPIRou
9223.0996 |  $-$25.3$\pm$7.2 | SPIRou
9266.9790 |  $-$26.9$\pm$8.0 | SPIRou
9268.0356 |  $-$34.3$\pm$6.9 | SPIRou
9269.0974 |  $-$10.1$\pm$21.6 | SPIRou
9269.1272 |  $-$10.5$\pm$9.9 | SPIRou
9271.9320 | 24.7$\pm$7.1 | SPIRou
9274.1081 | 28.4$\pm$6.2 | SPIRou
9276.0892 | 8.3$\pm$8.1 | SPIRou
9277.0101 | 19.6$\pm$7.6 | SPIRou
9278.0748 | 5.7$\pm$6.0 | SPIRou
9294.0409 |  $-$22.1$\pm$6.3 | SPIRou
9295.0002 |  $-$28.7$\pm$7.6 | SPIRou
9296.0629 |  $-$16.8$\pm$8.1 | SPIRou
9296.9974 |  $-$17.8$\pm$6.2 | SPIRou
9298.0459 |  $-$15.9$\pm$6.3 | SPIRou
9300.0194 | 13.8$\pm$10.0 | SPIRou
9300.7965 | 17.5$\pm$7.2 | SPIRou
9304.9681 | 24.8$\pm$7.1 | SPIRou
9306.0155 | 3.7$\pm$6.5 | SPIRou
9326.9507 | 21.8$\pm$7.5 | SPIRou
9327.8698 | 24.1$\pm$7.1 | SPIRou
9328.9553 | 24.1$\pm$7.5 | SPIRou
9329.9107 | 8.5$\pm$10.4 | SPIRou
9330.9185 | 5.0$\pm$14.0 | SPIRou
9330.9527 | 56.5$\pm$16.7 | SPIRou
9331.9585 | 31.4$\pm$7.2 | SPIRou
9332.9564 | 20.3$\pm$6.7 | SPIRou
9334.9279 | 4.2$\pm$6.4 | SPIRou
9335.9337 |  $-$22.0$\pm$7.1 | SPIRou
9336.9252 |  $-$34.0$\pm$8.0 | SPIRou
9337.9127 |  $-$26.9$\pm$7.2 | SPIRou
9384.7675 |  $-$16.4$\pm$10.6 | SPIRou
9384.7722 |  $-$8.1$\pm$10.1 | SPIRou
9385.7677 |  $-$4.6$\pm$7.5 | SPIRou
9386.7639 | 39.9$\pm$6.2 | SPIRou
9388.7451 | 27.9$\pm$6.9 | SPIRou
9389.7431 | 17.5$\pm$6.3 | SPIRou
9390.7491 |  $-$16.2$\pm$7.2 | SPIRou
9391.7430 |  $-$36.3$\pm$6.2 | SPIRou
9392.7444 |  $-$48.3$\pm$6.7 | SPIRou
9393.7467 |  $-$53.5$\pm$6.2 | SPIRou
9394.7425 |  $-$33.7$\pm$7.6 | SPIRou
9395.7435 |  $-$19.6$\pm$8.5 | SPIRou
9396.7434 |  $-$15.0$\pm$7.9 | SPIRou
9397.7432 | 38.1$\pm$10.3 | SPIRou
9412.7417 | 2.6$\pm$6.8 | SPIRou
9414.7350 | 30.3$\pm$7.6 | SPIRou
9535.1336 | 8.1$\pm$6.7 | SPIRou
9536.1546 | 0.6$\pm$7.2 | SPIRou
9537.1288 | 17.3$\pm$6.7 | SPIRou
9538.1514 | 3.0$\pm$5.9 | SPIRou
9539.1403 |  $-$3.0$\pm$7.2 | SPIRou
9539.1662 | 11.7$\pm$6.8 | SPIRou
9540.1354 |  $-$7.2$\pm$6.7 | SPIRou
9541.1236 |  $-$2.4$\pm$6.4 | SPIRou
9558.1411 |  $-$20.4$\pm$6.7 | SPIRou
9559.1219 |  $-$5.6$\pm$8.1 | SPIRou
9560.1639 |  $-$10.5$\pm$8.4 | SPIRou
9562.1294 |  $-$2.2$\pm$6.1 | SPIRou
9564.1142 | 23.9$\pm$7.1 | SPIRou
9566.1686 | 26.4$\pm$11.7 | SPIRou
9567.1335 | 2.1$\pm$7.6 | SPIRou
9586.1341 |  $-$15.5$\pm$6.4 | SPIRou
9588.1546 |  $-$16.4$\pm$6.2 | SPIRou
9589.1044 |  $-$25.0$\pm$6.1 | SPIRou
9590.1329 |  $-$30.8$\pm$5.9 | SPIRou
9592.1880 | 31.6$\pm$6.6 | SPIRou
9593.1163 | 36.9$\pm$6.9 | SPIRou
9594.1050 | 33.5$\pm$6.0 | SPIRou
9595.1838 | 29.1$\pm$6.4 | SPIRou
9596.1219 | 19.4$\pm$9.9 | SPIRou
9597.1307 |  $-$4.8$\pm$6.3 | SPIRou
9598.1769 |  $-$1.7$\pm$6.2 | SPIRou
9599.1869 |  $-$22.7$\pm$6.2 | SPIRou
9600.1802 |  $-$22.5$\pm$6.5 | SPIRou
9602.1329 | 11.4$\pm$10.1 | SPIRou
9604.1314 |  $-$14.0$\pm$5.8 | SPIRou
9605.1515 |  $-$5.9$\pm$6.5 | SPIRou
9606.1063 | 10.3$\pm$6.2 | SPIRou
9611.1237 | 14.3$\pm$6.7 | SPIRou
9612.1312 |  $-$15.2$\pm$10.5 | SPIRou
9612.1363 | 2.1$\pm$9.6 | SPIRou
9649.8702 | 28.7$\pm$5.8 | SPIRou
9650.9574 | 15.3$\pm$6.9 | SPIRou
9651.9774 | 28.7$\pm$6.2 | SPIRou
9652.8562 | 23.9$\pm$6.0 | SPIRou
9654.0191 |  $-$7.1$\pm$6.1 | SPIRou
9655.0560 |  $-$3.6$\pm$7.0 | SPIRou
9655.9679 |  $-$14.7$\pm$6.7 | SPIRou
9657.0586 |  $-$14.8$\pm$8.5 | SPIRou
9658.0488 |  $-$20.6$\pm$6.7 | SPIRou
9658.9940 |  $-$19.4$\pm$6.0 | SPIRou
9660.9905 | 5.4$\pm$7.6 | SPIRou
9661.9645 | 9.8$\pm$6.3 | SPIRou
9661.9996 | 9.8$\pm$6.3 | SPIRou
9678.9852 | 12.5$\pm$6.7 | SPIRou
9681.9729 |  $-$10.0$\pm$7.1 | SPIRou
9682.9911 |  $-$34.1$\pm$6.7 | SPIRou
9684.0050 |  $-$18.1$\pm$11.7 | SPIRou
9684.9812 |  $-$15.7$\pm$7.0 | SPIRou
9687.9818 |  $-$5.8$\pm$7.6 | SPIRou
9688.9630 | 8.6$\pm$6.9 | SPIRou
9689.9494 | 7.2$\pm$6.5 | SPIRou
9690.9483 | 2.5$\pm$6.6 | SPIRou
9710.8979 |  $-$5.0$\pm$7.4 | SPIRou
9711.9059 |  $-$22.3$\pm$7.1 | SPIRou
9712.8811 |  $-$13.4$\pm$7.7 | SPIRou
9715.8525 |  $-$20.7$\pm$6.9 | SPIRou
9716.8573 | 12.9$\pm$6.7 | SPIRou
9732.7697 | 3.7$\pm$6.6 | SPIRou
9733.7668 | 9.3$\pm$6.8 | SPIRou
9735.7726 |  $-$1.8$\pm$6.7 | SPIRou
9736.8406 |  $-$15.7$\pm$6.9 | SPIRou
9738.7460 |  $-$4.2$\pm$6.9 | SPIRou
9739.7468 |  $-$2.2$\pm$6.5 | SPIRou
9740.7920 | 6.2$\pm$6.5 | SPIRou
Table 11: Continued. Table 12:  Optical and near-infrared measurements of longitudinal magnetic field for CN Leo. The columns are the same as in Table LABEL:tab:Bl_log_evlac. HJD | Bl | Instrument
---|---|---
$[-2450000]$ | [G] |
4546.9935 |  $-$633.8$\pm$52.7 | ESPaDOnS
4547.9162 |  $-$669.7$\pm$61.8 | ESPaDOnS
4547.9584 |  $-$779.1$\pm$64.6 | ESPaDOnS
4548.8749 |  $-$825.1$\pm$64.4 | ESPaDOnS
8590.0095 |  $-$576.1$\pm$73.6 | SPIRou
8591.8815 |  $-$632.6$\pm$70.9 | SPIRou
8592.9476 |  $-$592.8$\pm$77.2 | SPIRou
8593.8996 |  $-$577.7$\pm$71.6 | SPIRou
8594.9831 |  $-$596.9$\pm$76.2 | SPIRou
8595.8886 |  $-$551.8$\pm$69.6 | SPIRou
8596.8984 |  $-$617.6$\pm$72.0 | SPIRou
8597.8470 |  $-$554.8$\pm$69.2 | SPIRou
8598.9095 |  $-$490.6$\pm$74.2 | SPIRou
8599.8488 |  $-$651.8$\pm$67.0 | SPIRou
8600.8511 |  $-$577.6$\pm$65.9 | SPIRou
8604.8814 |  $-$579.3$\pm$73.0 | SPIRou
8618.7890 |  $-$588.8$\pm$75.9 | SPIRou
8647.7648 |  $-$645.6$\pm$77.5 | SPIRou
8648.7456 |  $-$682.4$\pm$83.9 | SPIRou
8650.7529 |  $-$529.2$\pm$76.1 | SPIRou
8651.7412 |  $-$413.7$\pm$76.7 | SPIRou
8653.7526 |  $-$656.2$\pm$79.1 | SPIRou
8655.7683 |  $-$444.1$\pm$75.2 | SPIRou
8788.1333 |  $-$441.9$\pm$51.6 | SPIRou
8789.1453 |  $-$469.5$\pm$62.6 | SPIRou
8790.1556 |  $-$460.3$\pm$54.0 | SPIRou
8791.1274 |  $-$534.4$\pm$57.9 | SPIRou
8795.1410 |  $-$437.4$\pm$73.6 | SPIRou
8797.1180 |  $-$547.4$\pm$108.2 | SPIRou
8799.1555 |  $-$493.0$\pm$91.1 | SPIRou
8801.1211 |  $-$257.9$\pm$61.9 | SPIRou
8802.1173 |  $-$492.1$\pm$66.4 | SPIRou
8825.1497 |  $-$425.0$\pm$67.3 | SPIRou
8826.1219 |  $-$512.7$\pm$58.9 | SPIRou
8827.1148 |  $-$521.8$\pm$57.0 | SPIRou
8828.0792 |  $-$401.8$\pm$56.5 | SPIRou
8829.1393 |  $-$514.5$\pm$58.2 | SPIRou
8830.1228 |  $-$440.7$\pm$56.5 | SPIRou
8875.0195 |  $-$543.8$\pm$54.7 | SPIRou
8876.0165 |  $-$533.8$\pm$52.2 | SPIRou
8877.0519 |  $-$493.1$\pm$54.0 | SPIRou
8884.8714 |  $-$509.5$\pm$51.8 | SPIRou
8888.8471 |  $-$585.1$\pm$61.4 | SPIRou
8895.8243 |  $-$512.7$\pm$51.0 | SPIRou
8896.8945 |  $-$536.2$\pm$60.0 | SPIRou
8897.8240 |  $-$423.4$\pm$52.0 | SPIRou
8898.9247 |  $-$589.4$\pm$55.4 | SPIRou
8920.0362 |  $-$461.1$\pm$51.3 | SPIRou
8920.9668 |  $-$476.0$\pm$51.2 | SPIRou
8976.9515 |  $-$568.9$\pm$60.1 | SPIRou
8978.9097 |  $-$417.4$\pm$50.9 | SPIRou
8981.9082 |  $-$535.9$\pm$52.8 | SPIRou
8982.9114 |  $-$564.9$\pm$54.3 | SPIRou
8983.8296 |  $-$382.6$\pm$49.4 | SPIRou
8984.9152 |  $-$470.6$\pm$68.4 | SPIRou
9005.7860 |  $-$451.6$\pm$56.9 | SPIRou
9007.7960 |  $-$555.0$\pm$70.4 | SPIRou
9008.7620 |  $-$542.4$\pm$88.7 | SPIRou
9009.7799 |  $-$623.4$\pm$86.5 | SPIRou
9010.7947 |  $-$516.0$\pm$47.5 | SPIRou
9154.1393 |  $-$492.8$\pm$49.1 | SPIRou
9156.1488 |  $-$477.1$\pm$47.5 | SPIRou
9157.1549 |  $-$508.2$\pm$51.2 | SPIRou
9158.1171 |  $-$648.4$\pm$56.5 | SPIRou
9159.1110 |  $-$438.6$\pm$53.6 | SPIRou
9208.0541 |  $-$518.3$\pm$54.8 | SPIRou
9209.1070 |  $-$662.0$\pm$56.2 | SPIRou
9210.0437 |  $-$636.0$\pm$56.9 | SPIRou
9213.1232 |  $-$470.8$\pm$61.8 | SPIRou
9214.1247 |  $-$652.1$\pm$55.3 | SPIRou
9215.0966 |  $-$665.6$\pm$57.3 | SPIRou
9217.1484 |  $-$675.6$\pm$54.9 | SPIRou
9218.0758 |  $-$504.9$\pm$54.8 | SPIRou
9219.0537 |  $-$532.3$\pm$53.9 | SPIRou
9220.0470 |  $-$585.9$\pm$57.6 | SPIRou
9221.0871 |  $-$533.6$\pm$52.4 | SPIRou
9222.0654 |  $-$561.2$\pm$54.3 | SPIRou
9222.1785 |  $-$616.2$\pm$54.4 | SPIRou
9223.0703 |  $-$585.6$\pm$55.9 | SPIRou
9223.1694 |  $-$599.8$\pm$55.7 | SPIRou
9266.9540 |  $-$531.2$\pm$66.8 | SPIRou
9266.9611 |  $-$615.6$\pm$66.1 | SPIRou
9267.9207 |  $-$609.4$\pm$56.4 | SPIRou
9269.0286 |  $-$673.1$\pm$68.8 | SPIRou
9271.8397 |  $-$581.6$\pm$56.3 | SPIRou
9274.0918 |  $-$638.0$\pm$51.0 | SPIRou
9276.0722 |  $-$563.4$\pm$54.4 | SPIRou
9276.9937 |  $-$660.1$\pm$52.1 | SPIRou
9278.0639 |  $-$568.2$\pm$43.7 | SPIRou
9294.0260 |  $-$577.6$\pm$47.4 | SPIRou
9294.9851 |  $-$576.7$\pm$48.1 | SPIRou
9296.9796 |  $-$513.5$\pm$46.1 | SPIRou
9298.0309 |  $-$632.2$\pm$48.3 | SPIRou
9299.9950 |  $-$609.5$\pm$47.1 | SPIRou
9300.7817 |  $-$634.4$\pm$53.6 | SPIRou
9304.9542 |  $-$572.0$\pm$50.7 | SPIRou
9306.0018 |  $-$610.6$\pm$47.3 | SPIRou
9326.9362 |  $-$578.2$\pm$56.4 | SPIRou
9327.8309 |  $-$703.8$\pm$52.3 | SPIRou
9328.9436 |  $-$734.4$\pm$60.3 | SPIRou
9329.8992 |  $-$666.4$\pm$58.8 | SPIRou
9330.9062 |  $-$679.2$\pm$64.9 | SPIRou
9331.9467 |  $-$644.3$\pm$51.2 | SPIRou
9332.9448 |  $-$662.3$\pm$53.8 | SPIRou
9334.8733 |  $-$607.8$\pm$48.0 | SPIRou
9335.8928 |  $-$627.8$\pm$49.3 | SPIRou
9336.9136 |  $-$643.4$\pm$67.4 | SPIRou
9337.8936 |  $-$620.4$\pm$54.8 | SPIRou
9337.9004 |  $-$620.4$\pm$54.8 | SPIRou
9384.7522 |  $-$597.8$\pm$127.1 | SPIRou
9384.7589 |  $-$592.4$\pm$86.9 | SPIRou
9385.7605 |  $-$570.4$\pm$51.2 | SPIRou
9386.7563 |  $-$629.1$\pm$48.7 | SPIRou
9388.7372 |  $-$544.6$\pm$44.0 | SPIRou
9389.7348 |  $-$668.5$\pm$50.6 | SPIRou
9390.7425 |  $-$663.1$\pm$54.5 | SPIRou
9391.7408 |  $-$640.6$\pm$53.5 | SPIRou
9392.7403 |  $-$654.5$\pm$48.1 | SPIRou
9393.7396 |  $-$598.9$\pm$50.6 | SPIRou
9394.7351 |  $-$688.8$\pm$54.2 | SPIRou
9395.7354 |  $-$690.6$\pm$63.2 | SPIRou
9396.7355 |  $-$649.1$\pm$54.8 | SPIRou
9397.7355 |  $-$674.6$\pm$77.6 | SPIRou
9535.1189 |  $-$525.7$\pm$50.9 | SPIRou
9537.1315 |  $-$618.9$\pm$50.8 | SPIRou
9538.1540 |  $-$665.1$\pm$50.5 | SPIRou
9539.1427 |  $-$605.4$\pm$47.5 | SPIRou
9540.1378 |  $-$584.7$\pm$46.5 | SPIRou
9541.1259 |  $-$512.2$\pm$47.7 | SPIRou
9558.1136 |  $-$555.7$\pm$64.6 | SPIRou
9559.0945 |  $-$558.5$\pm$58.7 | SPIRou
9560.1366 |  $-$618.6$\pm$55.3 | SPIRou
9561.1177 |  $-$510.0$\pm$47.5 | SPIRou
9562.0969 |  $-$508.2$\pm$45.8 | SPIRou
9566.1400 |  $-$462.7$\pm$60.3 | SPIRou
9588.1723 |  $-$477.5$\pm$48.5 | SPIRou
9589.1213 |  $-$520.8$\pm$45.6 | SPIRou
9590.1333 |  $-$528.1$\pm$44.8 | SPIRou
9591.1360 |  $-$493.6$\pm$51.3 | SPIRou
9593.1359 |  $-$566.9$\pm$46.5 | SPIRou
9594.1235 |  $-$479.8$\pm$43.5 | SPIRou
9595.1048 |  $-$387.1$\pm$69.0 | SPIRou
9595.1835 |  $-$511.1$\pm$48.4 | SPIRou
9596.1785 |  $-$516.0$\pm$53.0 | SPIRou
9597.1172 |  $-$546.3$\pm$47.2 | SPIRou
9598.1766 |  $-$480.6$\pm$43.1 | SPIRou
9599.1694 |  $-$528.8$\pm$48.5 | SPIRou
9600.1799 |  $-$539.3$\pm$46.3 | SPIRou
9602.1193 |  $-$516.0$\pm$56.8 | SPIRou
9603.0231 |  $-$470.1$\pm$48.3 | SPIRou
9604.1525 |  $-$467.0$\pm$44.8 | SPIRou
9605.1213 |  $-$542.9$\pm$47.7 | SPIRou
9606.1404 |  $-$512.9$\pm$47.7 | SPIRou
9607.1284 |  $-$478.2$\pm$45.9 | SPIRou
9611.1429 |  $-$497.2$\pm$46.0 | SPIRou
9612.1366 |  $-$575.6$\pm$56.6 | SPIRou
9649.8597 |  $-$577.9$\pm$45.3 | SPIRou
9650.8614 |  $-$556.2$\pm$46.2 | SPIRou
9651.8345 |  $-$516.9$\pm$44.7 | SPIRou
9653.9425 |  $-$492.8$\pm$43.1 | SPIRou
9655.8643 |  $-$519.3$\pm$49.3 | SPIRou
9658.9112 |  $-$536.1$\pm$43.8 | SPIRou
9660.9072 |  $-$486.9$\pm$57.1 | SPIRou
9681.9599 |  $-$566.0$\pm$50.8 | SPIRou
9687.9746 |  $-$448.5$\pm$49.7 | SPIRou
9710.8875 |  $-$624.3$\pm$55.3 | SPIRou
9713.9000 |  $-$607.5$\pm$61.9 | SPIRou
9740.7523 |  $-$496.4$\pm$43.6 | SPIRou
Table 12: Continued.


# Dataset Distillation via Adversarial Prediction Matching

Mingyang Chen1,2 Bo Huang1,2 Junda Lu3 Bing Li4 Yi Wang5 Minhao Cheng2 Wei
Wang1
1The Hong Kong University of Science and Technology (Guangzhou)
2The Hong Kong University of Science and Technology 3CSIRO Data61, Australia
4University of Electronic Science and Technology of China 5Dongguan University
of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Dataset distillation is the technique of synthesizing smaller condensed
datasets from large original datasets while retaining necessary information to
persist the effect. In this paper, we approach the dataset distillation
problem from a novel perspective: we regard minimizing the prediction
discrepancy on the real data distribution between models, which are
respectively trained on the large original dataset and on the small distilled
dataset, as a conduit for condensing information from the raw data into the
distilled version. An adversarial framework is proposed to solve the problem
efficiently. In contrast to existing distillation methods involving nested
optimization or long-range gradient unrolling, our approach hinges on single-
level optimization. This ensures the memory efficiency of our method and
provides a flexible tradeoff between time and memory budgets, allowing us to
distil ImageNet-1K using a minimum of only 6.5GB of GPU memory. Under the
optimal tradeoff strategy, it requires only 2.5$\times$ less memory and
5$\times$ less runtime compared to the state-of-the-art. Empirically, our
method can produce synthetic datasets just 10% the size of the original, yet
achieve, on average, 94% of the test accuracy of models trained on the full
original datasets including ImageNet-1K, significantly surpassing state-of-
the-art. Additionally, extensive tests reveal that our distilled datasets
excel in cross-architecture generalization capabilities. Our code is available
at https://github.com/mchen725/DD_APM/.

##  1 Introduction

Cutting-edge machine learning models across diverse domains are progressively
dependent on extensive datasets. Nevertheless, the enormity of these datasets
introduces formidable challenges in aspects such as data storage,
preprocessing, model training, continual learning (Aljundi et al., 2019a; Zhao
& Bilen, 2021a), and other auxiliary tasks including hyperparameter tuning
(Bergstra et al., 2011; Baik et al., 2020) and architecture search (Pham et
al., 2018; Jin et al., 2019). Exploring surrogate datasets emerges as a viable
solution, offering comparable efficacy while significantly reducing scale.
While existing research on core-set selection has underscored the feasibility
of employing representative subsets from the original datasets (Chen et al.,
2010; Rebuffi et al., 2017; Aljundi et al., 2019b; Toneva et al., 2019),
Dataset Distillation (DD) is a nascent approach that generate new data
instances not present in the original sets. This methodology notably expands
the scope for optimization, potentially retaining optimal training signals
from the principal dataset, and has demonstrated remarkable efficacy and
effectiveness 111https://github.com/Guang000/Awesome-Dataset-Distillation.

Dataset distillation via imitating the behaviour of “teacher models” acquired
through training on original datasets, by the intermediary “student models”
trained on distilled datasets is perceived as an effective solution (Zhao &
Bilen, 2021a; Cazenavette et al., 2022). These methods utilize the teacher-
student paradigm as a medium to condense knowledge from raw data into a more
succinct form. Particularly, Matching Training Trajectory (MTT) frameworks (Du
et al., 2023; Cui et al., 2023) exhibit superior performance over alternate
distillation paradigms. Compared to its predecessor, i.e., the gradient
matching framework (Zhao et al., 2021), its superior performance can be
attributed to the strategy of matching long-term training trajectories of
teachers rather than short-term gradients. However, over-fitting on the
synthetic dataset can cause a remarkable disparity between the obtained and
real trajectories, deteriorating the effectiveness and incurring considerable
memory burdens during gradient derivation w.r.t synthetic samples. Hence, such
methodologies remain confined to a localized imitation of behaviours elicited
by teachers.

Figure 1: We contrast our adversarial prediction matching framework with the
training trajectory matching (MTT) framework (Cazenavette et al., 2022). Our
method has three primary advantages: 1. It imitates the converged teachers’
prediction, avoiding short-sightedness on local trajectories; 2. Synthetic
samples $u$ are updated by a single-level loss function, significantly
enhancing memory complexity; 3. It requires only one well-trained teacher,
markedly diminishing storage overhead.

Prompted by these insights, we formally formulate a novel prediction-matching-
based objective. This objective motivates synthetic data to enable student
models to directly emulate the predictions of teacher models on the authentic
data distributions of the original datasets, circumventing the risk of the
distilled features becoming shortsighted. To effectively optimize the
objective and bypass the nested optimization challenges in various prevailing
methods (Du et al., 2023; Loo et al., 2023; Cui et al., 2023), we formulate an
adversarial framework. A depiction of the basic proposition of our adversarial
framework and its superiorities over the trajectory matching paradigm are
provided in Figure 1. Intuitively, we obtain training snapshots of students by
aligning predictions with teachers’ on synthetic samples, then adversarially
refine these samples to approach “critical points” in the real data
distribution, which can cause substantial prediction disagreement between
teachers and the aggregated snapshots with high probabilities in the original
distribution. This strategy adeptly enables the distilled data to encapsulate
crucial features from the original, motivating consistent alignment between
intermediary students and teachers throughout the training phase.

Notably, although the utilized teacher-student paradigm in DD has a similar
form to Knowledge Distillation (KD) (Hinton et al., 2015; Fang et al., 2019),
especially our method hinges a prediction-matching paradigm, the research
focal points are orthogonal. Instead of matching predictions between two
designated models, the goal of DD is to create synthetic datasets endowed with
prominent generalization ability that can off-the-shelf train unseen models.
To this end, we devote ourselves to harnessing adequate information for the
distilled data from the original, employing the prediction-matching paradigm
as a conduit.

The contributions of this paper can be summarized as:

  * •

A novel imitation-based prediction-matching notation is introduced for
effective knowledge condensation from original datasets to distilled
counterparts by aligning predictions of models, trained respectively on
original and distilled datasets, over the authentic data distribution. An
adversarial framework is consequently proposed to formulate the problem as a
min-max game, allowing for trivial single-level optimization of distilled
samples.

  * •

Our method is proven to exhibit low memory complexity, offering a flexible
tradeoff between memory consumption and runtime. Empirically, it enables the
distillation of the ImageNet-1K dataset using a mere 6.5GB of GPU memory,
achieving superior outcomes with only 2.5$\times$ less memory and 5$\times$
reduced runtime compared to the foremost approach.

  * •

Experimental results illustrate that our distilled datasets, with just 10% the
size of the original, can averagely attain 94% of the test accuracies of
models than training on the original benchmark datasets, significantly
surpassing existing solutions. Additionally, the distilled dataset
demonstrates superior performance in cross-architecture generalization and
excels in the Neural Architecture Search (NAS) task.

##  2 Related Work of Dataset Distillation

Solving Bi-Level Optimizations. A common approach in dataset distillation is
to model the task as a bi-level optimization problem, aiming to minimize the
generalization error on the original data caused by models trained on the
distilled data (Wang et al., 2018; Bohdal et al., 2020; Zhou et al., 2022;
Nguyen et al., 2021; Zhao & Bilen, 2021a; Loo et al., 2023). However, solving
this problem is non-trivial, especially when the employed proxy models are
optimized by gradient descent because it requires unrolling a complicated
computational graph. To tackle this challenge, several recent works propose to
approximate the model training with Kernel Ridge Regression (KRR) to get a
closed-form solution for the optimum weights (Zhou et al., 2022; Nguyen et
al., 2021) or directly solving the problem by the Implicit Gradient (IG)
through convexifying the model training by the Neural Tangent Kernel (NTK)
(Loo et al., 2023). Nonetheless, these methods either require extensive
computations or are compromised by the looseness in the convex relaxation.

Imitation-based Approaches. Another widespread notion in dataset distillation
involves mimicking specific behaviours elicited by the original dataset with
those of the distilled dataset. Following this concept, the gradient matching
frameworks (Zhao et al., 2021; Zhao & Bilen, 2021a; Kim et al., 2022) and
training trajectory matching frameworks (Cazenavette et al., 2022; Du et al.,
2023) propose to emulate the training dynamics of models experienced on the
original dataset, making models trained on the distilled data converge to a
neighbourhood in the parameter space. Although a significant improvement has
been achieved by state-of-the-art methods, the update of synthetic data is
typically memory-intensive as it also requires unrolling the computation graph
of multistep gradient descent. While Cui et al. (2023) addresses this issue by
allowing constant memory usage during gradient derivation, aligning
trajectories obtained from extensive steps over synthetic data with long-term
real trajectories becomes challenging. Consequent tradeoff also leads to
inevitable truncation bias (Zhou et al., 2022). Additionally, distribution
matching methods (Zhao & Bilen, 2021b; Wang et al., 2022) heuristically
propose to align the feature distribution of original data with that of
distilled data output by several proxy extractors, exhibiting inferior
effectiveness.

##  3 Method

###  3.1 Dataset Distillation via Matching Prediction on Real Data
Distribution

Suppose a large labelled training dataset
$\mathcal{T}=\left\\{(x_{i},y_{i})\right\\}_{i=1}^{|\mathcal{T}|}$, where
$x_{i}\in\mathbb{R}^{d}$ and $y_{i}\in\\{1,2,\ldots,C\\}$, is given to be
distilled into a smaller distilled dataset
$\mathcal{S}=\left\\{(u_{i},v_{i})\right\\}_{i=1}^{|\mathcal{S}|}$, with
$u_{i}\in\mathbb{R}^{d}$ and $v_{i}\in\mathbb{R}^{C}$, such that
$|\mathcal{S}|\ll|\mathcal{T}|$. Our objective is to construct $\mathcal{S}$
such that, for a given model architecture parameterized by $\theta\sim
p(\theta)$, models trained respectively on $\mathcal{S}$ and $\mathcal{T}$
have the minimum discrepancy when predicting over the original data
distribution with $p(x)=\sum_{y}p(x|y)p(y)$:

$\min_{\mathcal{S}}\sum_{y=1}^{C}\int p(x|y)\cdot
d\left(f_{\theta^{\mathcal{T}}}(x),f_{\theta^{\mathcal{S}}}(x)\right)dx$ (1)

We omit $p\left(y\right)$ as we assume it follows a uniform distribution.
Where, $d\left(.,.\right)$ refers to a distance function and
$f_{\theta}\left(.\right)$ refers to the logtis output by the penultimate
layer of models before the softmax. We employ logits for the prediction
matching instead of the confidences normalized by the softmax function because
logits contain more information for knowledge transfer (Yuan et al., 2020; Fu
et al., 2020; Yin et al., 2020). $\theta^{\mathcal{T}}$ and
$\theta^{\mathcal{S}}$ refer to the optimized parameters obtained by
minimizing empirical loss over $\mathcal{T}$ and $\mathcal{S}$ respectively.
Typically, this process is performed by an iterative optimization algorithm,
denoted as $Alg$, such as gradient descent. For instance,
$\theta^{\mathcal{S}}=Alg(\ell(\theta,\mathcal{S}),E)$, where $\ell(.,.)$ is
any suitable loss function for model training, and $E$ denotes the number of
training epochs.

We emphasize that minimizing the objective in Equation (1) is non-trivial as
two reasons: i) calculating the gradient w.r.t $\mathcal{S}$ requires
unrolling the recursive computation graph through the entire training process
of $\theta^{\mathcal{S}}$, which is computationally expensive and memory-
intensive, even $|\mathcal{S}|$ is small; ii) Directly minimizing the
prediction discrepancy across the entire distribution is computationally
infeasible due to the curse of dimensionality and the continuous nature of the
variables.

###  3.2 Adversarial Prediction Matching Framework

To tackle the above challenges, we first posit the existence of a subset
$\mathcal{R}_{e}\subseteq\mathbb{R}^{d}$ within the feasible domain of $x$,
where every element $x_{e}\in\mathcal{R}_{e}$ satisfies the following
conditions: i) it is associated with a high conditional probability
$p(x_{e}|y)$ for its corresponding class $y$; and ii) it induces substantial
logits differences between the $f_{\theta^{\mathcal{T}}}$ and
$f_{\theta_{e}^{\mathcal{S}}}$, where $e\in\\{0,\ldots,E-1\\}$ refers to the
index of student checkpoints collected at each epoch of running $Alg$. We
conjecture that such elements can “dominate” the expectation of logits
difference over $p(x)$, denoted as:

$\mathbb{E}_{x_{e}\in\mathcal{R}_{e}}[d\left(f_{\theta^{\mathcal{T}}}(x_{e}),f_{\theta_{e}^{\mathcal{S}}}(x_{e})\right)]\geq\mathbb{E}_{x\sim
p(x)}[d\left(f_{\theta^{\mathcal{T}}}(x),f_{\theta_{e}^{\mathcal{S}}}(x)\right)]$
(2)

Figure 2: Trends of $L_{2}$ norm of $\nabla_{\theta}\ell(\theta,\mathcal{S})$
with $\mathcal{S}$ derived by Algorithm 1, utilizing checkpoints from a
specified epoch (left three figures) and assorted epochs (the far right
figure) for CIFAR-10, where $|\mathcal{S}|=10\times 50$. For example,
“ckpt@100” denotes $\mathcal{S}$ is synthesized by referencing student models
solely at epoch 100; “integration” utilizes models from epochs 100, 200, 300,
400 and 500.

Namely, directly minimizing the expectation of logits difference over $x_{e}$
can facilitate the prediction alignment of each $\theta_{e}^{\mathcal{S}}$ to
the target $\theta^{\mathcal{T}}$ over the original distribution. Based on
this assumption, we propose to optimize the original objective by a
curriculum-based adversarial framework. In each round, we first train a
student model from $\theta_{0}^{\mathcal{S}}$ by the optimizer $Alg$ on
$\mathcal{S}=\\{(u_{i},f_{\theta^{\mathcal{T}}}(u_{i}))\\}$ for $E$ epochs
with $d(.,.)$ as the loss function, then alternately force synthetic samples
$\\{u_{i}\\}_{i=1}^{|\mathcal{S}|}$ to approach “hard samples”
$x_{e}\in\mathcal{R}_{e},\forall e\in\\{0,\ldots,E-1\\}$ with the following
target:

$\max_{u}\frac{1}{|\mathcal{S}|}\sum_{i=1}^{|\mathcal{S}|}\sum_{e=0}^{E-1}\log[p(u_{i}|y_{u_{i}})d\left(f_{\theta^{\mathcal{T}}}\left(u_{i}\right),f_{\theta_{e}^{\mathcal{S}}}\left(u_{i}\right)\right)]$
(3)

Where, $y_{u_{i}}$ refers to the class in which $u_{i}$ is associated for
distilling. However, the inherent complexity in estimating $p(u|y_{u})$ due to
the high-dimensional nature of data still hinders the optimization of Equation
(3). To address this issue, we advocate for substituting $p(u|y_{u})$ with
$p(y_{u}|u)$ for the maximization of the stated target. According to Bayes’
theorem, we have $p(u|y_{u})\propto p(y_{u}|u)p(u)$, presuming a uniform
distribution for $p(y)$. Given the stark contrast between the high-dimensional
continuum inherent in $u$ and the finite discreteness in $y$, we posit,
heuristically, that prioritizing the maximization of $p(y_{u}|u)$ is a
plausible strategy for the overall maximization of $p(y_{u}|u)p(u)$. Regarding
the teacher model derived from the original dataset is reliable, we employ
$p(y_{u}|u;\theta^{\mathcal{T}})=\text{softmax}(f_{\theta^{\mathcal{T}}}(u))_{y_{u}}$
as an approximation for $p(y_{u}|u)$. Consequently, our loss function designed
for updating $\\{u_{i}\\}_{i=1}^{|\mathcal{S}|}$ to maximize the stated target
is defined as follows:

$\mathcal{L}_{u}=\frac{1}{|\mathcal{S}|}\sum_{i=1}^{|\mathcal{S}|}\sum_{e=0}^{E-1}-\log\left[d\left(f_{\theta^{\mathcal{T}}}(u_{i}),f_{\theta_{e}^{\mathcal{S}}}(u_{i})\right)\right]+\alpha
H\left(y_{u_{i}},p(y_{u_{i}}|u_{i};\theta^{\mathcal{T}})\right)$ (4)

Where,
$H\left(y_{u_{i}},p(y_{u_{i}}|u_{i};\theta^{\mathcal{T}})\right)=-\sum_{c=1}^{C}y_{c}\log\left(p(y_{c}|u_{i};\theta^{\mathcal{T}})\right)=-\log\left(p(y_{u_{i}}|u_{i};\theta^{\mathcal{T}})\right)$
denotes the cross-entropy loss of $\theta^{\mathcal{T}}$ on $u_{i}$ w.r.t the
associated distilling class $y_{u_{i}}$ and $\alpha$ refers to a hyper-
parameter for flexible tradeoff. Intuitively, this loss function encourages
the synthetic samples to evolve into valuable, hard points, thus contributing
beneficially to the overall training process of $\theta^{\mathcal{S}}$.

Practical Implementation. We outline the overall procedure of our proposed
Adversarial Prediction Matching (APM) framework in Algorithm 1. We initialize
$\mathcal{S}$ by sampling real data from $\mathcal{T}$, ensuring each $u_{i}$
has a high initial probability of $p(y_{u_{i}}|u_{i};\theta^{\mathcal{T}})$.
Notably, to fulfil the inherent generalization requisite, we adopt the
strategy from MTT frameworks (Cazenavette et al., 2022; Du et al., 2023; Cui
et al., 2023), obtaining a set of teacher models,
$\\{\theta^{\mathcal{T}}\\}$, prior to commencement. Subsequently, in each
iteration, a $\theta^{\mathcal{T}}$ is sampled from the set to be aligned with
a newly initialized $\theta_{0}^{\mathcal{S}}\sim p(\theta)$. To constrain the
memory overhead, we compute the loss in Equation (4) with $K$ select
checkpoints, collected at regular intervals during training. We empirically
validate in Section 4.4 that this does not compromise the efficacy of our
method.

Necessity of Involving Training Snapshots in $\mathcal{L}_{u}$. Ideally, a
distilled dataset should remain informative throughout the entire model
training process, avoiding limitations to specific stages. Nonetheless, our
observations indicate that relying on proxy student models from a singular
stage can introduce a significant inductive bias. To clarify this, we generate
synthetic samples using Algorithm 1 but with checkpoints from a specified
epoch to calculate $\mathcal{L}_{u}$. Then, the resultant dataset is used to
train newly initialized models, and variations in gradient norm relative to
model parameters are observed. Figure 2 reveals a noteworthy trend where peaks
in gradient norms are prominent and only appear near the epochs of the
selected checkpoints. Conversely, jointly utilizing these checkpoints for
synthetic sample generation exhibits a stable pattern throughout training.

Input: An architecture parameterized by $\theta$; A set of well-trained
teacher weights $\\{\theta^{\mathcal{T}}\\}$; Rounds $R$; Epochs $E$; Number
of select checkpoints $K$; Model learning rate $\eta$; Synthetic samples
learning rate $\gamma$; Mini-batch size $B$

Output: Distilled synthetic dataset $\mathcal{S}$

1 Initialize synthetic samples by randomly sampling
$\\{u_{i}\\}_{i=1}^{|\mathcal{S}|}\subset\\{x_{i}\\}_{i=1}^{|\mathcal{T}|}$;

2 for  _$r=1$ to $R$_ do

3 Initialize $\theta^{\mathcal{T}}\sim\\{\theta^{\mathcal{T}}\\}$,
$\theta^{\mathcal{S}}_{0}\sim p(\theta)$, and an empty checkpoint list;

4 Randomly sample a mini-batch
$\\{u^{r}_{i}\\}_{i=1}^{B}\subseteq\\{u_{i}\\}_{i=1}^{|\mathcal{S}|}$;

5 for  _$e=1$ to $E$_ do

6 Calculate the prediction matching loss:
$\mathcal{L}_{\theta}=\frac{1}{B}\sum_{i=1}^{B}d(f_{\theta^{\mathcal{T}}}(u^{r}_{i}),f_{\theta_{e-1}^{\mathcal{S}}}(u^{r}_{i}))$;

7 Update
$\theta^{\mathcal{S}}_{e}=\theta^{\mathcal{S}}_{e-1}-\eta\cdot\nabla_{\theta^{\mathcal{S}}_{e-1}}\mathcal{L}_{\theta}$;

8 if  _$e\mod\lfloor E/K\rfloor=0$_ then

9 Store $\theta^{\mathcal{S}}_{e}$ in the checkpoint list;

10

11

12 Calculate $\mathcal{L}_{u}$ in Equation (4) with $K$ checkpoints;

13 Update $u^{r}=u^{r}-\gamma\cdot\nabla_{u^{r}}\mathcal{L}_{u}$;

14

15Calculate $v_{i}=f_{\theta^{\mathcal{T}}}(u_{i}),\forall u_{i}$ and set
$\mathcal{S}=\\{(u_{i},v_{i})\\}_{i=1}^{|\mathcal{S}|}$;

Algorithm 1 Dataset distillation via adversarial prediction matching

###  3.3 Superiority Analysis

Low Memory Complexity. Based on the definition of the loss function proposed
for updating synthetic samples in Equation (4), all computational operations
conducted on each synthetic sample $u_{i}$, e.g., the computation of each
$d(f_{\theta^{\mathcal{T}}}(u_{i}),f_{\theta_{e}^{\mathcal{S}}}(u_{i}))$ and
$H\left(y_{u_{i}},p(y_{u_{i}}|u_{i};\theta^{\mathcal{T}})\right)$ (abbreviated
as $d^{e}_{i}$ and $H_{i}$, respectively), are integrated by addition. Thanks
to this characteristic, two inherent advantages in memory efficiency emerge:
1. the gradient of $\mathcal{L}_{u}$ w.r.t each $u_{i}$ can be independently
derived without interference from other samples of $u_{j\neq i}$; 2. the
partial derivative of each computation term, e.g., $\frac{\partial
d^{e}_{i}}{\partial u_{i}}$ or $\frac{\partial H_{i}}{\partial u_{i}}$, can be
independently calculated, and finally updating $u_{i}$ based on the
accumulated gradients of each term. Thus, the minimal memory complexity of our
method can be expressed as $\mathcal{O}(\mathcal{G}_{\text{ours}})$, where
$\mathcal{G}_{ours}$ refers to the computation graph size for a single input
sample, and it is dominated by $\frac{\partial d^{e}_{i}}{\partial u_{i}}$ or
$\frac{\partial H_{i}}{\partial u_{i}}$ which only involves a trivial
backpropagation through models. In contrast, TESLA (Cui et al., 2023) achieves
the optimal memory efficiency among MMT frameworks by reducing the memory
complexity of the original (Cazenavette et al., 2022) from
$\mathcal{O}\left(TB\mathcal{G}_{MTT}\right)$ to
$\mathcal{O}\left(B\mathcal{G}_{MTT}\right)$, where $T$ and $B$ denote
specified updating steps of deriving trajectories and the mini-batch updating
size for synthetic data, respectively. Compared with ours, except the mini-
batch size $B$, the computation graph size $\mathcal{G}_{MTT}$ is dominated by
a complicated Jacobian matrix $\frac{\partial}{\partial
u_{i}}\nabla_{\theta}\ell(\theta,\mathcal{S})\in\mathbb{R}^{|\theta|\times
d}$. The remarkable memory efficiency of our method enables the distillation
of various-size synthetic datasets for large-scale datasets, such as
ImageNet-1K, even under memory-constrained circumstances.

Customized Tradeoff between Time and Memory. We show that
$\nabla_{u_{i}}\mathcal{L}_{u}$ for each $u_{i}$ can be calculated
independently and further decomposed into partial derivatives of each
computational term. In practice, this implies multiple model invocations for
separate calculations, elongating the runtime. Thus, our algorithm optionally
allows computation of the gradients w.r.t multiple $u$ simultaneously,
offering a flexible tradeoff between memory and runtime. Further, Section 4.3
empirically validates our capability to distribute distillation tasks to
independent workers (e.g., generating $N$ subsets of size $|\mathcal{S}|/N$ in
parallel and aggregating), a feature absent in prior dataset distillation
methods.

Significantly Reduced Storage vs. MTT. Our approach only requires matching
predictions of the converged teacher, requiring one checkpoint per teacher.
However, MTT frameworks necessitate saving checkpoints from every epoch of
training a teacher. Therefore, even utilizing the same setup to acquire an
equal number of teachers, the storage usage of our method is only a fraction
(i.e., 1/epoch) of that of MTT frameworks. Moreover, an ablation study in
Section 4.4 demonstrates that the performance of our method obtained with 10
teachers is comparable to that with 100 teachers.

##  4 Experiments

###  4.1 Experimental Setups

Datasets. We evaluate the performance of our method on distilling four
benchmark datasets, including CIFAR-10/100 (Krizhevsky & Hinton, 2009), Tiny
ImageNet (Le & Yang, 2015) and ImageNet-1K (Russakovsky et al., 2015).
Notably, following the protocol of Zhou et al. (2022) and Cui et al. (2023),
we resize the resolution of ImageNet-1K images to $64\times 64$ in our
evaluations for fair comparison. We provide further detailed information of
benchmark datasets in Appendix A.1.

Evaluation Setup and Baselines. We employ several recent distillation
frameworks as baselines including a short-term gradient matching method Data
Siamese Augmentation (DSA) (Zhao & Bilen, 2021a), a feature distribution
matching method Distribution Matching (DM) (Zhao & Bilen, 2021b), two bi-
level-optimization-based methods which are FRePo (Zhou et al., 2022) and RCIG
(Loo et al., 2023), and three training trajectory matching methods which are
Matching Training Trajectory (MTT) (Cazenavette et al., 2022) and its two
variations TESLA (Cui et al., 2023) and Flat Trajectory Distillation (FTD) (Du
et al., 2023).

Evaluation Metrics. To evaluate compared methods, we adhere to the standard
protocol prevalent in the existing literature. We train 5 randomly initialized
networks from scratch over the distilled dataset and evaluate their
performances by reporting the mean and standard deviation of their accuracy on
the test set. We use the terminology “IPC” (an acronym for “images per class”)
in dataset distillation to indicate the size of distilled datasets.
Specifically, we examine an IPC of 50/500/1000 for CIFAR-10 and an IPC of
10/50/500 for CIFAR-100, Tiny ImageNet, and ImageNet-1K.

Implementation Details. We implement our method based on the procedure of
Algorithm 1. Specifically, we set the teacher number of
$\\{\theta^{{\bm{T}}}\\}$ as 100 to be fairly compared with MTT and its
variations TESLA and FTD. We set the distance metric $d$ by $L_{1}$ distance,
the running round $R=2,500$, the number of checkpoints $K=5$ and the
coefficient of cross-entropy regularizer $\alpha=0.1$ by default. Further
details of our hyper-parameter setup and guidelines are in Appendix A.7. For a
fair comparison, we stay with all the compared precedents to adopt the 3-layer
ConvNet architecture (Gidaris & Komodakis, 2018) for distilling CIFAR-10 and
CIFAR-100, and extend to the 4-layer ConvNet for Tiny ImageNet and
ImageNet-1K. Moreover, we also employ ZCA whitening for CIFAR-10 and CIFAR-100
as done by previous works (Zhou et al., 2022; Loo et al., 2023; Cazenavette et
al., 2022; Du et al., 2023; Cui et al., 2023). All the experimental results of
our method can be obtained on one RTX4090 GPU, including the extensive
ImageNet-1K dataset.

###  4.2 Comparison to State-of-the-art

Evaluation on ConvNets. As illustrated by Table 1, our method demonstrates
superior performance than all the baselines among all the evaluation
scenarios. Notably, ConvNet models trained on our distilled datasets, which
are just $10\%$ the size of the original, manage to achieve, on average,
$96\%$ of the test accuracies of models trained on the full CIFAR-10,
CIFAR-100, and Tiny ImageNet. Conversely, models trained on equivalent-sized
random subsets can only reach $58\%$, while the state-of-the-art trajectory
matching framework, FTD, achieves a maximum of $87\%$. Specifically, the
average test accuracies of our method surpass those of FTD by $3.1\%$ on
CIFAR-10, $2.1\%$ on CIFAR-100, and $5.8\%$ on Tiny ImageNet. Compared to the
state-of-the-art bi-level optimization framework RCIG, our framework not only
exhibits superior performances at a smaller IPC but also demonstrates
feasibility and effectiveness when generating synthetic sets with a larger
IPC. Furthermore, thanks to the memory efficiency of our adversarial
prediction matching framework, our method stands as one of the only two
capable of successfully distilling the extensive ImageNet-1K dataset, and it
additionally exceeds the competitor, TESLA, by $4.4\%$ on average. A
visualization study of our synthetic data is shown in Appendix A.8.

Cross-Architecture Generalization. In Table 2, we assess the generalization
performance of datasets distilled by our method and by several representative
baselines, including DSA, DM, RCIG, and FTD, on unseen model architectures
over CIFAR and Tiny ImageNet with an IPC of 50. We utilize three different
neural network architectures for evaluation: VGG11 (Simonyan & Zisserman,
2015), ResNet18 (He et al., 2016) and Vision Transformer (ViT) (Dosovitskiy et
al., 2021). Those models are widely employed for cross-architecture
evaluations of recent dataset distillation works (Cui et al., 2022; 2023;
Cazenavette et al., 2023). The results demonstrate that models learned with
our synthetic data have superior performances than models trained on datasets
synthesized by other baselines. Additionally, compared to the results on
ConvNet shown in Table 1, we surprisingly noticed larger performance
improvements of our method to the state-of-the-art FTD method across multiple
unseen models. Moreover, our results obtained on ResNet18 even exceed the
results obtained on ConvNet when testing on CIFAR-100 and Tiny ImageNet.The
remarkable transferability demonstrates that our method is more capable of
capturing the essential features for learning. We also report the
generalization result of our distilled dataset of ImageNet-1K in Appendix A.3.

Table 1: Comparison of the test accuracy (%) between ours and other distillation methods on CIFAR10/100, Tiny ImageNet and ImageNet-1K datasets. Entries marked as absent are due to scalability issues. See Appendix A.2 for detailed reasons. Dataset |  IPC | Random | DSA | DM | FRePo | RCIG | MTT | TESLA | FTD | Ours |  Whole
---|---|---|---|---|---|---|---|---|---|---|---
CIFAR-10 |  50 | 50.2±0.5 | 60.6±0.5 | 63.0±0.4 | 71.7±0.2 | 73.5±0.3 | 71.6±0.2 | 72.6±0.7 | 73.8±0.2 | 75.0±0.2 |  85.6±0.2
|  500 | 71.9±0.3 | - | 74.3±0.2 | - | - | 78.6±0.2 | 78.4±0.5 | 78.7±0.2 | 83.4±0.3 |
|  1000 | 78.2±0.3 | - | 79.2±0.2 | - | - | 79.7±0.2 | 80.3±0.3 | 81.2±0.2 | 84.6±0.2 |
CIFAR-100 |  10 | 14.4±0.5 | 32.3±0.3 | 29.7±0.3 | 42.5±0.2 | 44.1±0.4 | 40.1±0.4 | 41.7±0.3 | 43.4±0.3 | 44.6±0.3 |  56.8±0.2
|  50 | 30.5±0.3 | 42.8±0.4 | 43.6±0.4 | 44.3±0.2 | 46.7±0.3 | 47.7±0.2 | 47.9±0.3 | 50.7±0.3 | 53.3±0.2 |
|  100 | 43.1±0.2 | 44.7±0.2 | 47.1±0.4 | - | 47.4±0.3 | 50.3±0.1 | 51.1±0.3 | 52.8±0.3 | 55.2±0.2 |
Tiny ImageNet |  10 | 4.8±0.3 | 15.9±0.2 | 12.9±0.4 | 25.4±0.2 | 29.4±0.2 | 23.2±0.2 | 14.1±0.3 | 24.5±0.2 | 30.0±0.3 |  40.0±0.4
|  50 | 15.1±0.3 | 21.6±0.3 | 21.2±0.3 | - | - | 28.0±0.3 | 33.4±0.5 | 31.5±0.3 | 38.2±0.4 |
|  100 | 24.3±0.3 | - | 29.4±0.2 | - | - | 33.7±0.6 | 34.7±0.2 | 34.5±0.4 | 39.6±0.2 |
ImageNet-1K |  10 | 4.1±0.1 | - | - | - | - | - | 17.8±1.3 | - | 24.8±0.2 |  37.2±0.5
|  50 | 16.2±0.8 | - | - | - | - | - | 27.9±1.2 | - | 30.7±0.3 |
|  100 | 19.5±0.5 | - | - | - | - | - | 29.2±1.0 | - | 32.6±0.3 |

###  4.3 Memory and Time Efficiency Analysis

High Efficiency for Distilling ImageNet-1K. To empirically evident the
superiorities of the memory complexity and the flexible tradeoff between
runtime and memory usage we discussed in Section 3.3, we compare our method
with TESLA, which exhibits the optimal memory complexity among trajectory
matching frameworks, for distilling ImageNet-1K with an IPC of $50$. We
conduct tests on a Tesla A100 GPU (80GB) for a fair comparison. Figure 4
represents the averaged peak memory usage and runtime per iteration over 50
rounds collected by running algorithms with the reported default hyper-
parameters for obtaining the results shown in Table 1. Both methods set the
mini-batch size for updating synthetic samples in each iteration as $500$.
However, our method allows for an equivalent update by distributing the
updates of 500 samples across multiple segments. For instance, “Segment-100”
implies dividing 500 samples into five parts and updating sequentially.
Through this approach, our method can flexibly tradeoff memory usage and
runtime, enabling the distillation of ImageNet-1K with merely 6.5GB of memory
and achieving superior distillation performance. Additionally, we observe that
even when updating 500 samples at once, our method consumes $2.5\times$ less
memory than Tesla. Given that the default running iteration of Tesla is double
ours, our overall runtime is only one-fifth of theirs.

Parallelizability of Dataset Distillation. To further enhance productivity, a
logical approach is to divide the generation of a synthetic dataset and
allocate it to multiple workers for parallel execution without compromising
effectiveness. We refer to this feature as the parallelizability of dataset
distillation methods. We evaluate this feature of our method and several
representative frameworks including DM, FRePo and MTT in Figure 4.
Specifically, we separately craft 10 groups of synthetic data with an IPC of
50 for CIFAR-10 and 5 groups with an IPC of 10 for Tiny ImageNet. We can see
from the figure that the performances of models attained on our synthetic
datasets persistently increase with the growing number of merged groups.
Moreover, we are the only framework that can achieve equivalent performances
as the original results shown in Table 1. This indicates that, unlike other
frameworks, running our method in parallel does not capture repetitive or
redundant features.

Table 2: Cross-architecture generalization performance in test accuracy (%) of synthetic datasets distilled for CIFAR-10/100 and Tiny ImageNet at an IPC of 50. Dataset | Model | Random | DSA | DM | RCIG | FTD | Ours
---|---|---|---|---|---|---|---
CIFAR-10 | VGG11 | 46.7±0.6 | 51.0±1.1 | 59.2±0.8 | 47.9±0.3 | 59.1±0.2 | 65.8±1.0
ResNet18 | 48.1±0.3 | 47.3±1.0 | 57.5±0.9 | 59.5±0.2 | 64.7±0.3 | 69.2±0.8
ViT | 38.9±0.6 | 22.6±1.0 | 27.2±1.6 | 27.7±0.8 | 38.7±0.9 | 42.2±1.1
CIFAR-100 | VGG11 | 28.4±0.3 | 29.8±0.1 | 34.6±0.4 | 36.7±0.1 | 42.5±0.2 | 46.6±0.2
ResNet18 | 41.0±0.8 | 41.7±0.7 | 37.7±0.5 | 36.5±0.4 | 48.4±0.1 | 55.2±0.3
ViT | 25.5±0.2 | 27.4±0.3 | 28.2±0.6 | 15.7±0.6 | 30.2±0.5 | 35.2±0.6
Tiny ImageNet | VGG11 | 21.1±0.3 | 21.5±0.3 | 22.0±0.4 | - | 27.2±0.3 | 35.1±0.1
ResNet18 | 30.7±0.8 | 30.9±1.1 | 20.3±0.4 | - | 35.7±0.6 | 40.2±0.2
ViT | 17.4±0.6 | 18.6±0.3 | 18.1±0.4 | - | 21.5±0.1 | 24.5±0.8

Figure 3: Flexible tradeoff between memory usage and runtime per iteration of
our method compared with TESLA’s for distilling ImageNet-1K with an IPC of 50.

Figure 4: Test accuracy (%) varying with merging different numbers of
distributively generated sub-batches. IPC of each sub-batch is 50 for CIFAR-10
and 10 for Tiny ImageNet.

###  4.4 Ablation Study

We study the effectiveness or the influence of the following four components
of our method. Number of Pre-trained Teachers: To enhance the generalization
ability, we propose to match prediction with different teachers sampled from a
pre-gathered pool $\\{\theta^{T}\\}$. Figure 5(a) reveals a consistent
performance increase with the size of $\\{\theta^{T}\\}$, emphasizing the
importance of dynamically altering the teacher model to circumvent overfitting
to a singular model. Notably, the result with $10$ teachers is nearly
equivalent to that with $100$ teachers, highlighting the potential for
computation and storage resource conservation. Number of Involved Checkpoints:
As illustrated by Figure 5(b), utilizing $K$ checkpoints in Equation (4)
improves the performance of only using the last checkpoint. Besides, we
observe that setting $K$ larger than 5 will not cause significant changes in
the performance, allowing equivalent effectiveness with reduced resources.
Mini-Batch Size: In Sec 3.3, we emphasize that the loss function for updating
one synthetic sample is computationally independent of others. Fig 5(c) shows
the invariance of performances of synthetic datasets distilled with different
mini-batch sizes $B$. Thus, both memory consumption and runtime of each round
can be effectively reduced when the mini-batch size is small. Number of Epoch
for the Model Update: We examine the effect of the epoch number $E$ for
training student models on the synthetic dataset in Figure 5(d). We can see a
significant decrease when $E$ is set higher than $300$. This is because a
large $E$ will cause most of the $K$ collected checkpoints to be nearly
converged. Thus, condensed features derived by using them as proxies lack
sufficient knowledge beneficial for the early stages of model training.
Moreover, we comprehensively study the influence of varying the coefficient of
cross-entropy regularizer, $\alpha$, over the four benchmark datasets in
Appendix A.5.

Table 3: Comparison of Spearman’s ranking correlation of NAS on CIFAR-10. Results are obtained from models trained on various distilled datasets with an IPC of 50 (i.e., 1% of the original dataset) over 200 epochs. For reference, the correlation achieved by training models on the full original dataset for 20 epochs stands at 0.52, albeit requiring $10\times$ the search time than ours. Random | DSA | DM | MTT | FRePo | Ours
---|---|---|---|---|---
0.18±0.06 | 0.12±0.07 | 0.13±0.06 | 0.31±0.01 | -0.07±0.07 | 0.47±0.03

(a) Teacher number

(b) Checkpoint number

(c) Mini-batch size

(d) Updating epoch of $\theta^{{\bm{S}}}$

Figure 5: Ablation studies of our method conducted on CIFAR-10 with an IPC of
50.

###  4.5 Application: Neural Architecture Search

To explore the practicality of our method, we utilize the distilled datasets
as a proxy for the intricate Neural Architecture Search (NAS) task. For this,
we adopt the same NAS-Bench-201 task (Dong & Yang, 2020), previously employed
by a recent dataset distillation benchmark (Cui et al., 2022). Specifically,
we randomly sample 100 networks from NAS-Bench-201, in which the search space
is comprised of 15,625 networks with ground-truth performances collected by
training on the entire training dataset of CIFAR-10 for 50 epochs under 5
random seeds, and ranked according to their average accuracy on a held-out
validation set of 10k images. We compare our methods with various baselines
including Random (a randomly selected subset), DSA, DM, MTT and FRePo by
training the 100 different architectural models over synthetic datasets with
an IPC of 50 for 200 epochs. The Spearman’s correlation between the rankings
derived from the synthetic datasets and the ground-truth ranking is used as
the evaluation metric. Comparison results are reported in Table 3. Notably,
due to the inherent variability in training, an average ranking correlation of
0.76 is achieved locally even when adhering to the ground-truth protocol. We
can clearly see from the table that the ranking correlation attained by our
method markedly surpasses other baselines. Especially, although MTT and
FRePo’s test accuracies in Table 1 are on par with ours for an IPC of 50 on
CIFAR-10, the efficiency of our method for this NAS task is evident. Moreover,
the correlation achieved by training models on the entire original training
dataset of CIFAR-10 for 20 epochs is 0.52. However, our distilled datasets
deliver comparable performance in just one-tenth of the search time.

##  5 Conclusions

In this work, we introduce an innovative formulation for dataset distillation
via prediction matching, realized with an adversarial framework, devised for
effective problem resolution. Comprehensive experiments validate that our
distilled datasets surpass contemporary benchmarks, showcasing exceptional
cross-architecture capabilities. Furthermore, we illustrate the superior
memory efficiency and practical viability of large-scale dataset distillation
of our method. For future research, we intend to explore advancements in
efficiency by diminishing its reliance on the number of teacher models.

## References

  * Aljundi et al. (2019a) Rahaf Aljundi, Eugene Belilovsky, Tinne Tuytelaars, Laurent Charlin, Massimo Caccia, Min Lin, and Lucas Page-Caccia.  Online continual learning with maximal interfered retrieval.  In _NeurIPS_ , pp. 11849–11860, 2019a.
  * Aljundi et al. (2019b) Rahaf Aljundi, Min Lin, Baptiste Goujaud, and Yoshua Bengio.  Gradient based sample selection for online continual learning.  In _NeurIPS_ , pp. 11816–11825, 2019b.
  * Baik et al. (2020) Sungyong Baik, Myungsub Choi, Janghoon Choi, Heewon Kim, and Kyoung Mu Lee.  Meta-learning with adaptive hyperparameters.  In _NeurIPS_ , 2020.
  * Bergstra et al. (2011) James Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl.  Algorithms for hyper-parameter optimization.  In _NIPS_ , pp. 2546–2554, 2011.
  * Bohdal et al. (2020) Ondrej Bohdal, Yongxin Yang, and Timothy M. Hospedales.  Flexible dataset distillation: Learn labels instead of images.  _CoRR_ , abs/2006.08572, 2020.
  * Cazenavette et al. (2022) George Cazenavette, Tongzhou Wang, Antonio Torralba, Alexei A. Efros, and Jun-Yan Zhu.  Dataset distillation by matching training trajectories.  In _CVPR_ , pp. 10708–10717. IEEE, 2022.
  * Cazenavette et al. (2023) George Cazenavette, Tongzhou Wang, Antonio Torralba, Alexei A. Efros, and Jun-Yan Zhu.  Generalizing dataset distillation via deep generative prior.  In _CVPR_ , pp. 3739–3748. IEEE, 2023.
  * Chen et al. (2010) Yutian Chen, Max Welling, and Alexander J. Smola.  Super-samples from kernel herding.  In _UAI_ , pp. 109–116. AUAI Press, 2010.
  * Cui et al. (2022) Justin Cui, Ruochen Wang, Si Si, and Cho-Jui Hsieh.  DC-BENCH: dataset condensation benchmark.  In _NeurIPS_ , 2022.
  * Cui et al. (2023) Justin Cui, Ruochen Wang, Si Si, and Cho-Jui Hsieh.  Scaling up dataset distillation to imagenet-1k with constant memory.  In _ICML_ , volume 202 of _Proceedings of Machine Learning Research_ , pp. 6565–6590. PMLR, 2023.
  * Dong & Yang (2020) Xuanyi Dong and Yi Yang.  Nas-bench-201: Extending the scope of reproducible neural architecture search.  In _ICLR_. OpenReview.net, 2020.
  * Dosovitskiy et al. (2021) Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby.  An image is worth 16x16 words: Transformers for image recognition at scale.  In _ICLR_. OpenReview.net, 2021.
  * Du et al. (2023) Jiawei Du, Yidi Jiang, Vincent Y. F. Tan, Joey Tianyi Zhou, and Haizhou Li.  Minimizing the accumulated trajectory error to improve dataset distillation.  In _CVPR_ , pp. 3749–3758. IEEE, 2023.
  * Fang et al. (2019) Gongfan Fang, Jie Song, Chengchao Shen, Xinchao Wang, Da Chen, and Mingli Song.  Data-free adversarial distillation.  _CoRR_ , abs/1912.11006, 2019.
  * Fu et al. (2020) Jie Fu, Xue Geng, Zhijian Duan, Bohan Zhuang, Xingdi Yuan, Adam Trischler, Jie Lin, Chris Pal, and Hao Dong.  Role-wise data augmentation for knowledge distillation.  2020\.
  * Gidaris & Komodakis (2018) Spyros Gidaris and Nikos Komodakis.  Dynamic few-shot visual learning without forgetting.  In _CVPR_ , pp. 4367–4375. Computer Vision Foundation / IEEE Computer Society, 2018.
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _CVPR_ , pp. 770–778. IEEE Computer Society, 2016.
  * Hinton et al. (2015) Geoffrey E. Hinton, Oriol Vinyals, and Jeffrey Dean.  Distilling the knowledge in a neural network.  _CoRR_ , abs/1503.02531, 2015.
  * Jin et al. (2019) Haifeng Jin, Qingquan Song, and Xia Hu.  Auto-keras: An efficient neural architecture search system.  In _KDD_ , pp. 1946–1956. ACM, 2019.
  * Kim et al. (2022) Jang-Hyun Kim, Jinuk Kim, Seong Joon Oh, Sangdoo Yun, Hwanjun Song, Joonhyun Jeong, Jung-Woo Ha, and Hyun Oh Song.  Dataset condensation via efficient synthetic-data parameterization.  In _ICML_ , volume 162 of _Proceedings of Machine Learning Research_ , pp. 11102–11118. PMLR, 2022.
  * Krizhevsky & Hinton (2009) Alex Krizhevsky and Geoffrey Hinton.  Learning multiple layers of features from tiny images.  https://www.cs.toronto.edu/~kriz/cifar.html, 2009.  Accessed: March 1, 2023.
  * Le & Yang (2015) Ya Le and Xuan Yang.  Tiny imagenet visual recognition challenge.  _CS 231N_ , 7(7):3, 2015.
  * Loo et al. (2023) Noel Loo, Ramin M. Hasani, Mathias Lechner, and Daniela Rus.  Dataset distillation with convexified implicit gradients.  In _ICML_ , volume 202 of _Proceedings of Machine Learning Research_ , pp. 22649–22674. PMLR, 2023.
  * Nguyen et al. (2021) Timothy Nguyen, Zhourong Chen, and Jaehoon Lee.  Dataset meta-learning from kernel ridge-regression.  In _ICLR_. OpenReview.net, 2021.
  * Pham et al. (2018) Hieu Pham, Melody Y. Guan, Barret Zoph, Quoc V. Le, and Jeff Dean.  Efficient neural architecture search via parameter sharing.  _CoRR_ , abs/1802.03268, 2018.
  * Rebuffi et al. (2017) Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H. Lampert.  icarl: Incremental classifier and representation learning.  In _CVPR_ , pp. 5533–5542. IEEE Computer Society, 2017.
  * Russakovsky et al. (2015) Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael S. Bernstein, Alexander C. Berg, and Li Fei-Fei.  Imagenet large scale visual recognition challenge.  _Int. J. Comput. Vis._ , 115(3):211–252, 2015\.
  * Simonyan & Zisserman (2015) Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  In _ICLR_ , 2015.
  * Toneva et al. (2019) Mariya Toneva, Alessandro Sordoni, Remi Tachet des Combes, Adam Trischler, Yoshua Bengio, and Geoffrey J. Gordon.  An empirical study of example forgetting during deep neural network learning.  In _ICLR (Poster)_. OpenReview.net, 2019.
  * Wang et al. (2022) Kai Wang, Bo Zhao, Xiangyu Peng, Zheng Zhu, Shuo Yang, Shuo Wang, Guan Huang, Hakan Bilen, Xinchao Wang, and Yang You.  CAFE: learning to condense dataset by aligning features.  In _CVPR_ , pp. 12186–12195. IEEE, 2022.
  * Wang et al. (2018) Tongzhou Wang, Jun-Yan Zhu, Antonio Torralba, and Alexei A. Efros.  Dataset distillation.  _CoRR_ , abs/1811.10959, 2018.
  * Yin et al. (2020) Hongxu Yin, Pavlo Molchanov, Jose M. Alvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K Jha, and Jan Kautz.  Dreaming to distill: Data-free knowledge transfer via deepinversion.  In _The IEEE/CVF Conf. Computer Vision and Pattern Recognition (CVPR)_ , 2020.
  * Yin et al. (2023) Zeyuan Yin, Eric P. Xing, and Zhiqiang Shen.  Squeeze, recover and relabel: Dataset condensation at imagenet scale from A new perspective.  _CoRR_ , abs/2306.13092, 2023.
  * Yuan et al. (2020) Li Yuan, Francis EH Tay, Guilin Li, Tao Wang, and Jiashi Feng.  Revisiting knowledge distillation via label smoothing regularization.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 3903–3911, 2020.
  * Zhao & Bilen (2021a) Bo Zhao and Hakan Bilen.  Dataset condensation with differentiable siamese augmentation.  In _ICML_ , volume 139 of _Proceedings of Machine Learning Research_ , pp. 12674–12685. PMLR, 2021a.
  * Zhao & Bilen (2021b) Bo Zhao and Hakan Bilen.  Dataset condensation with distribution matching.  _CoRR_ , abs/2110.04181, 2021b.
  * Zhao et al. (2021) Bo Zhao, Konda Reddy Mopuri, and Hakan Bilen.  Dataset condensation with gradient matching.  In _ICLR_. OpenReview.net, 2021.
  * Zhou et al. (2022) Yongchao Zhou, Ehsan Nezhadarya, and Jimmy Ba.  Dataset distillation using neural feature regression.  In _NeurIPS_ , 2022.

##  Appendix A Appendix

###  A.1 Benchmark Datasets

We provide further details of four employed benchmark datasets including
CIFAR-10/100 (Krizhevsky & Hinton, 2009), Tiny ImageNet (Le & Yang, 2015) and
ImageNet-1K (Russakovsky et al., 2015). For the low-resolution datasets
CIFAR-10 and CIFAR-100, there are $5000\times 10$ and $500\times 100$ training
samples, and $1000\times 10$ and $100\times 100$ testing samples that are
under $32\times 32$ resolution in 3-channels RGB format, respectively. For
high-resolution datasets, Tiny ImageNet comprises $500\times 200$ training
samples and $50\times 200$ testing samples under $64\times 64$ resolution with
3 RGB channels in total. Besides, ImageNet-1K is a widely employed large-scale
dataset that includes 1,000 classes, encompassing 1,281,167 training images
and 50,000 testing images with 3 RGB channels. Following the protocol of Zhou
et al. (2022) and Cui et al. (2023), we resize the resolution of ImageNet-1K
images to $64\times 64$.

###  A.2 Scalability Issues of Competitors

In this section, we clarify the reasons behind the absent entries for various
baseline methods in Table 1. For DSA, achieving results for distillation tasks
with an IPC exceeding 50 is unattainable due to the lack of official
implementation support in such scenarios, where each iteration’s runtime
becomes prohibitively long (e.g., exceeding 5 minutes). Distilling ImageNet-1K
with an IPC of 10 induces an Out-Of-Memory (OOM) error, even on a Tesla A100
GPU (80GB). The absence of DM entries for distilling ImageNet-1K is also
attributed to OOM errors, resulting from the extensive number of classes. For
FRePo, there are two constraining issues: first, its official implementation
does not support an IPC above 50; second, the computation of the Gram matrix
of features from proxy models on real and synthetic samples demands
significant memory, constraining its application on large-scale datasets like
Tiny ImageNet or ImageNet-1K. Concerning RCIG, it manages to evade OOM issues
by minimizing the mini-batch size for updating synthetic samples, but our
empirical results indicate that this compromises the method’s efficacy. The
model trained with such a distilled dataset yields inferior performance
compared to even smaller synthetic datasets. Lastly, for MTT and FTD, the
memory consumption escalates notably with the stipulated updating steps for
aligning trajectories of teacher models, making the acquisition of reasonable
results challenging with limited memory resources.

###  A.3 Cross-Architecture Generalization on ImageNet-1K

We examine and contrast the cross-architecture generalization capabilities of
our synthetic datasets, distilled for ImageNet-1K with an IPC of 50, against
the sole valid competitor, TESLA (Cui et al., 2023), as detailed in Table A4.
The outcomes indicate that our approach significantly elevates the performance
on randomly sampled sub-batches of equivalent size from the original dataset.
Notably, the achievements on VGG11 and ResNet18 even exceed those obtained on
the proxy architecture (i.e., ConvNet), evidencing reliable applicability
across various unseen convolutional models. The modest advancement on ViT can
likely be ascribed to the disparities in receptive fields between ViT’s
segmented patches and ConvNet’s convolutional windows. Addressing this
deficiency is considered a pivotal avenue for our future endeavours.

Table A4: Cross-architecture generalization performance in test accuracy (%) of synthetic datasets distilled for ImageNet-1K at an IPC of 50. |  | Random | TESLA | Ours
---|---|---|---|---
ImageNet-1K | ConvNetD4 | 16.2±0.8 | 26.7±1.2 | 30.7±0.3
VGG11 | 15.3±0.2 | 24.6±0.8 | 31.1±0.1
ResNet18 | 18.7±1.2 | 28.7±0.5 | 33.2±0.5
ViT | 9.5±0.3 | 14.3±0.5 | 11.7±1.2

###  A.4 Comparison of Generalization Among Different Proxy Model
Architectures

In our formal implementation, we stay with precedents Cazenavette et al.
(2022); Zhao & Bilen (2021a); Zhao et al. (2021); Wang et al. (2022) and adopt
the commonly used ConvNet architecture as the proxy model for both teacher
models and student models in the dataset distillation tasks. Here, we also
investigate the feasibility of using some other CNN-based architectures as the
proxy model to generate distilled datasets and compare their inter-model
generalization. We employ another two CNN-based network architectures for
comparison including VGG11 (Simonyan & Zisserman, 2015) and ResNet18 (He et
al., 2016) and compare the generalization of the distilled dataset generated
respectively on different architectures on CIFAR-10 and Tiny ImageNet with an
IPC of 50. For reference, the test accuracies (%) of ConvNet, VGG11 and
ResNet18 training on the whole original datasets are $85.6\pm 0.2$, $88.6\pm
0.6$ and $93.4\pm 0.5$ over CIFAR-10, and are $40.0\pm 0.4$, $46.8\pm 0.6$ and
$62.4\pm 1.2$ over Tiny ImageNet, respectively. In Table A5, the distilled
datasets are generated with one architecture (denoted as “proxy”) and then
evaluated on another architecture (denoted as “evaluation”) by training a
model from scratch and testing on real testing data. The findings from these
experiments unveil several intriguing, and at times, unexpected insights.
First, a model trained using data derived from ResNet18 and evaluated on the
same architecture, remarkably achieves a test accuracy of $54.8\pm 0.6$,
considerably surpassing results obtained from datasets generated by the other
two models. Importantly, this outcome also significantly surpasses the result
(i.e., $41.1\pm 0.4$) reported in Yin et al. (2023), a recent study dedicated
to generating distilled datasets specifically for models equipped with batch
normalization (e.g., ResNets) on Tiny ImageNet and ImageNet-1K. However,
second, we also identified relatively poorer generalization performances of
these synthetic datasets on other models, a phenomenon that aligns with the
observations made in Yin et al. (2023). Third, it is observed that models of
relative complexity, such as VGG11 and ResNet18, demonstrate superior
generalization on Tiny ImageNet as compared to their performance on CIFAR-10.
We hypothesize this may attributed to the potential overfitting of complex
models on simpler datasets, leading to the injection of obscure patterns
during the synthesis of samples via backpropagation. The information contained
by patterns may not be recognizable by other models. Similar findings have
been noted in preceding studies on dataset distillation (Zhao et al., 2021;
Zhao & Bilen, 2021a; b). Such observations prompt us to delve deeper into the
underlying causes of these phenomena and to explore methodologies to more
effectively integrate sophisticated models with advanced predictive
capabilities in dataset distillation processes. We consider the exploration of
these notions as pivotal focal points for our impending research endeavours in
the dataset distillation.

Table A5: Comparison of cross-architecture generalization performances (%) of synthetic datasets generated by different proxy architectures for CIFAR-10 and Tiny ImageNet with an IPC of 50. | CIFAR-10 | Tiny ImageNet
---|---|---
Proxy\Evaluation | ConvNet | VGG11 | ResNet18 | ConvNet | VGG11 | ResNet18
ConvNet | 75.0±0.2 | 65.8±1.0 | 69.2±0.8 | 30.0±0.3 | 35.1±0.1 | 40.2±0.2
VGG11 | 55.2±0.3 | 54.1±0.8 | 54.3±0.4 | 32.8±0.1 | 36.6±0.6 | 41.8±0.4
ResNet18 | 61.7±0.3 | 59.7±0.8 | 63.2±0.9 | 26.9±0.3 | 32.3±0.4 | 54.8±0.6

###  A.5 Ablation Study on the Coefficient of Cross-Entropy Regularizer

We carry out an exhaustive ablation study concerning the coefficient of the
cross-entropy regularizer, $\alpha$, as defined in Equation (4). In Table A6,
we examine the performance of ConvNet models in response to variations in
$\alpha$ across four benchmark datasets: CIFAR-10 with an IPC of 50 and
CIFAR-100, Tiny ImageNet, and ImageNet-1K, each with an IPC of 10. Our
findings reveal that increasing $\alpha$ above $0$ yields modest improvements,
attesting to the efficacy of constraining synthetic samples to maintain a high
probability of $p(x|y;\theta^{T})$. Moreover, the performance appears to
remain robust against alterations in $\alpha$, indicating the stability of
this parameter. This stability proves beneficial during the hyperparameter
tuning phase, especially when adapting our approach to diverse distillation
tasks.

Table A6: Ablation study on the coefficient of cross-entropy regularizer $\alpha$ over the four benchmark datasets with an IPC of 50 for CIFAR-10 and an IPC of 10 for CIFAR-100, Tiny ImageNet and ImageNet-1K. $\alpha$ | CIFAR-10 | CIFAR-100 | Tiny ImageNet | ImageNet-1K
---|---|---|---|---
0 | 74.2±0.3 | 44.0±0.2 | 29.1±0.5 | 23.9±0.7
0.1 | 75.0±0.2 | 44.5±0.3 | 30.0±0.3 | 24.8±0.2
1 | 75.0±0.2 | 44.6±0.4 | 30.0±0.2 | 24.4±0.3
10 | 74.8±0.3 | 44.6±0.3 | 29.9±0.2 | 24.3±0.2

###  A.6 Comparison of Distance Functions

In this work, we utilize the _Manhattan distance_ (also known as $L_{1}$
distance) as the default distance function for assessing the logits
disagreement between teachers and students. To demonstrate its efficacy, we
compare it with two alternative distance metrics, namely, the Euclidean
distance and the cosine similarity, across three experimental scenarios
including CIFAR-10 with an IPC of 50, CIFAR-100 and Tiny ImageNet with an IPC
of 10. The comparison results are presented in Table A7. The results
demonstrate that utilizing the _Manhattan distance_ for evaluating logits
disagreement results in superior performance compared to the other two
distance metrics across all three experimental cases, thereby validating our
selection of the distance function for our formal implementation.

Table A7: Comparison of test accuracies (%) between Manhattan distance and two candidates including Euclidean distance and the cosine similarity on CIFAR-10 with an IPC of 50, CIFAR-100 and Tiny ImageNet with an IPC of 10.  | CIFAR-10 | CIFAR-100 | Tiny ImageNet
---|---|---|---
Cosine | 68.8±0.3 | 36.2±0.4 | 23.8±0.3
Euclidean | 74.1±0.3 | 43.9±0.2 | 29.2±0.4
Manhattan | 75.0±0.2 | 44.5±0.3 | 30.0±0.3

###  A.7 Hyper-Parameter Setup and Guidelines

In Table A8, we present detailed information regarding the hyper-parameter
configuration used to obtain the ConvNet results (i.e., the proxy model
utilized for distillation) depicted in Table 1. In accordance with the
ablation study outlined in Section 4.4, we recommend meticulous tuning of the
student model learning rate, $\eta$, and the student model update epoch, $E$,
when adapting our method to new datasets or models. This is to avoid an
overreliance on distilled features of knowledge from nearly converged
checkpoint models. Additionally, considering the stability of tuning the
coefficient of the cross-entropy regularizer shown in Appendix A.5, it is
advisable to initiate the tuning process with a smaller value of $\alpha$
(e.g., $0.1$).

Table A8: Hyper-parameters setup for obtaining the results on ConvNet shown in Table 1. Reminder: $\gamma$ \- synthetic samples learning rate; $\eta$ \- student model learning rate; $E$ \- proxy student model update epoch number; $B$ \- mini-batch size; $\alpha$ \- cross-entropy regularizer coefficient. | IPC | Model | $\gamma$ | $\eta$ | $E$ | Evaluation Epoch | Iteration | ZCA | $B$ | $\alpha$
---|---|---|---|---|---|---|---|---|---|---
CIFAR-10 | 50 | ConvNet | 1 | 0.02 | 250 | 1000 | 2500 | TRUE | full | 0.10
500 | ConvNet | 1 | 0.02 | 250 | 1000 | 2500 | TRUE | 2500 | 0.10
1000 | ConvNet | 1 | 0.02 | 250 | 600 | 2500 | TRUE | 2500 | 0.10
CIFAR-100 | 10 | ConvNet | 1 | 0.02 | 250 | 1000 | 2500 | TRUE | full | 0.10
50 | ConvNet | 1 | 0.04 | 250 | 1000 | 2500 | TRUE | 2500 | 0.10
100 | ConvNet | 1 | 0.04 | 250 | 600 | 2500 | TRUE | 2500 | 0.10
Tiny ImageNet | 10 | ConvNetD4 | 20 | 0.10 | 250 | 1000 | 2500 | FALSE | 500 | 0.10
50 | ConvNetD4 | 40 | 0.10 | 250 | 1000 | 2500 | FALSE | 500 | 0.10
100 | ConvNetD4 | 40 | 0.10 | 250 | 600 | 2500 | FALSE | 500 | 0.10
ImageNet-1K | 10 | ConvNetD4 | 20 | 0.10 | 250 | 1000 | 2500 | FALSE | 500 | 0.10
50 | ConvNetD4 | 20 | 0.10 | 250 | 600 | 2500 | FALSE | 500 | 0.10
100 | ConvNetD4 | 20 | 0.10 | 250 | 600 | 2500 | FALSE | 500 | 0.10

###  A.8 Visualization Study

In this section, we visualise our synthetic samples for CIFAR-10, CIFAR-100,
Tiny ImageNet and ImageNet-1K. Due to the large class numbers of CIFAR-100,
Tiny ImageNet and ImageNet-1k, we only exhibit the generated synthetic samples
for the former 50 classes.

Figure 6: Synthetic image visualization for CIFAR10.

Figure 7: CIFAR100 - [1,25].

Figure 8: CIFAR100 - [26,50].

Figure 9: Tiny ImageNet - [1,25].

Figure 10: Tiny ImageNet - [26,50].

Figure 11: ImageNet-1K - [1,25].

Figure 12: ImageNet-1K - [26,50].