$NN\to W^{\pm}\gamma$ | p p > ww a QED=2 QCD=0 [QCD] | $1.519\times 10^{-7}$ |  ${}^{+15\%}_{-12\%}$ ${}^{+4\%}_{-4\%}$ | 3.37 | $2.302\times 10^{-5}$ |  ${}^{+14\%}_{-13\%}$ ${}^{+4\%}_{-4\%}$ | 9.71
$NN\to W^{-}\gamma$ | p p > w- a QED=2 QCD=0 [QCD] | $7.928\times 10^{-8}$ |  ${}^{+15\%}_{-12\%}$ ${}^{+4\%}_{-4\%}$ | 3.48 | $1.193\times 10^{-5}$ |  ${}^{+14\%}_{-13\%}$ ${}^{+4\%}_{-4\%}$ | 9.00
$NN\to W^{+}\gamma$ | p p > w+ a QED=2 QCD=0 [QCD] | $7.314\times 10^{-8}$ |  ${}^{+15\%}_{-12\%}$ ${}^{+4\%}_{-4\%}$ | 3.30 | $1.102\times 10^{-5}$ |  ${}^{+14\%}_{-13\%}$ ${}^{+4\%}_{-4\%}$ | 10.48
$NN\to Z\gamma$ | p p > z a QED=2 QCD=0 [QCD] | $9.157\times 10^{-8}$ |  ${}^{+7\%}_{-6\%}$ ${}^{+3\%}_{-3\%}$ | 1.60 | $6.991\times 10^{-6}$ |  ${}^{+11\%}_{-11\%}$ ${}^{+5\%}_{-5\%}$ | 2.90
$NN\to W^{\pm}H$ | p p > ww h QED=2 QCD=0 [QCD] | $3.478\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+2\%}_{-2\%}$ | 1.18 | $1.386\times 10^{-5}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+7\%}_{-7\%}$ | 1.27
$NN\to ZH$ | p p > z h QED=2 QCD=0 [QCD] | $1.867\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+2\%}_{-2\%}$ | 1.17 | $7.630\times 10^{-6}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+7\%}_{-7\%}$ | 1.26
$NN\to W^{-}H$ | p p > w- h QED=2 QCD=0 [QCD] | $1.741\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+2\%}_{-2\%}$ | 1.18 | $6.955\times 10^{-6}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+7\%}_{-7\%}$ | 1.27
$NN\to W^{+}H$ | p p > w+ h QED=2 QCD=0 [QCD] | $1.740\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+2\%}_{-2\%}$ | 1.18 | $6.958\times 10^{-6}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+7\%}_{-7\%}$ | 1.27
$NN\to W^{+}W^{-}\gamma$ | p p > w+ w- a QED=3 QCD=0 [QCD] | $1.931\times 10^{-9}$ |  ${}^{+9\%}_{-7\%}$ ${}^{+5\%}_{-5\%}$ | 1.68 | $5.277\times 10^{-7}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 4.09
$NN\to W^{+}W^{-}H$ | p p > w+ w- h QED=3 QCD=0 [QCD] | $1.798\times 10^{-9}$ |  ${}^{+3\%}_{-3\%}$ ${}^{+3\%}_{-3\%}$ | 1.25 | $1.588\times 10^{-7}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+5\%}_{-5\%}$ | 1.44
$NN\to W^{+}W^{-}W^{\pm}$ | p p > w+ w- ww QED=3 QCD=0 [QCD] | $3.222\times 10^{-8}$ |  ${}^{+6\%}_{-4\%}$ ${}^{+3\%}_{-3\%}$ | 1.44 | $4.032\times 10^{-6}$ |  ${}^{+8\%}_{-9\%}$ ${}^{+4\%}_{-4\%}$ | 2.41
$NN\to W^{+}W^{-}Z$ | p p > w+ w- z QED=3 QCD=0 [QCD] | $2.250\times 10^{-8}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+4\%}_{-4\%}$ | 1.47 | $3.876\times 10^{-6}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 2.63
$NN\to ZZ\gamma$ | p p > z z a QED=3 QCD=0 [QCD] | $2.057\times 10^{-10}$ |  ${}^{+5\%}_{-5\%}$ ${}^{+5\%}_{-5\%}$ | 1.38 | $3.147\times 10^{-8}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+3\%}_{-3\%}$ | 2.00
$NN\to ZZH$ | p p > z z h QED=3 QCD=0 [QCD] | $4.663\times 10^{-10}$ |  ${}^{+3\%}_{-3\%}$ ${}^{+4\%}_{-4\%}$ | 1.23 | $3.874\times 10^{-8}$ |  ${}^{+4\%}_{-5\%}$ ${}^{+5\%}_{-5\%}$ | 1.28
$NN\to ZZW^{\pm}$ | p p > z z ww QED=3 QCD=0 [QCD] | $7.247\times 10^{-9}$ |  ${}^{+7\%}_{-6\%}$ ${}^{+4\%}_{-4\%}$ | 1.54 | $1.311\times 10^{-6}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 2.96
$NN\to ZZZ$ | p p > z z z QED=3 QCD=0 [QCD] | $2.632\times 10^{-9}$ |  ${}^{+4\%}_{-3\%}$ ${}^{+4\%}_{-4\%}$ | 1.29 | $2.385\times 10^{-7}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+5\%}_{-5\%}$ | 1.50
$NN\to\gamma\gamma W^{\pm}$ | p p > a a ww QED=3 QCD=0 [QCD] | $1.090\times 10^{-10}$ |  ${}^{+11\%}_{-9\%}$ ${}^{+5\%}_{-5\%}$ | 1.95 | $3.347\times 10^{-8}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 5.02
$NN\to HHW^{\pm}$ | p p > h h ww QED=3 QCD=0 [QCD] | $7.769\times 10^{-11}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+4\%}_{-4\%}$ | 1.17 | $7.398\times 10^{-9}$ |  ${}^{+2\%}_{-4\%}$ ${}^{+5\%}_{-5\%}$ | 1.20
$NN\to\gamma\gamma Z$ | p p > a a z QED=3 QCD=0 [QCD] | $4.376\times 10^{-11}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+5\%}_{-5\%}$ | 1.38 | $7.060\times 10^{-9}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+3\%}_{-3\%}$ | 2.18
$NN\to HHZ$ | p p > h h z QED=3 QCD=0 [QCD] | $4.948\times 10^{-11}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+4\%}_{-4\%}$ | 1.17 | $4.855\times 10^{-9}$ |  ${}^{+3\%}_{-4\%}$ ${}^{+5\%}_{-5\%}$ | 1.20
$NN\to W^{\pm}\gamma H$ | p p > ww a h QED=3 QCD=0 [QCD] | $1.906\times 10^{-11}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+5\%}_{-5\%}$ | 1.41 | $3.739\times 10^{-9}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+3\%}_{-3\%}$ | 2.12
$NN\to W^{\pm}Z\gamma$ | p p > ww z a QED=3 QCD=0 [QCD] | $8.528\times 10^{-10}$ |  ${}^{+12\%}_{-10\%}$ ${}^{+5\%}_{-5\%}$ | 2.08 | $3.060\times 10^{-7}$ |  ${}^{+9\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 5.48
$NN\to W^{\pm}ZH$ | p p > ww z h QED=3 QCD=0 [QCD] | $7.545\times 10^{-10}$ |  ${}^{+4\%}_{-4\%}$ ${}^{+4\%}_{-4\%}$ | 1.34 | $9.329\times 10^{-8}$ |  ${}^{+6\%}_{-7\%}$ ${}^{+4\%}_{-4\%}$ | 1.73
$NN\to Z\gamma H$ | p p > z a h QED=3 QCD=0 [QCD] | $1.510\times 10^{-11}$ |  ${}^{+4\%}_{-4\%}$ ${}^{+5\%}_{-5\%}$ | 1.27 | $1.809\times 10^{-9}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+4\%}_{-4\%}$ | 1.58
$NN\to\gamma j$ | p p > a j QED=1 QCD=1 [QCD] | $1.951\times 10^{-5}$ |  ${}^{+3\%}_{-4\%}$ ${}^{+4\%}_{-4\%}$ | 1.19 | $1.625\times 10^{-3}$ |  ${}^{+3\%}_{-6\%}$ ${}^{+6\%}_{-6\%}$ | 1.37
$NN\to\gamma\gamma$ | p p > a a QED=2 QCD=0 [QCD] | $3.593\times 10^{-8}$ |  ${}^{+4\%}_{-4\%}$ ${}^{+3\%}_{-3\%}$ | 1.29 | $1.914\times 10^{-6}$ |  ${}^{+9\%}_{-10\%}$ ${}^{+6\%}_{-6\%}$ | 2.28
$NN\to\gamma\gamma\gamma$ | p p > a a a QED=3 QCD=0 [QCD] | $3.615\times 10^{-12}$ |  ${}^{+8\%}_{-7\%}$ ${}^{+6\%}_{-6\%}$ | 1.51 | $1.007\times 10^{-9}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+3\%}_{-3\%}$ | 2.58
$NN\to t\bar{t}$ | p p > t t$\sim$ QED=0 QCD=2 [QCD] | $5.406\times 10^{-5}$ |  ${}^{+10\%}_{-12\%}$ ${}^{+8\%}_{-8\%}$ | 1.48 | $2.864\times 10^{-2}$ |  ${}^{+7\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 1.50
$NN\to t\bar{t}W^{-}$ | p p > t t$\sim$ w- QED=1 QCD=2 [QCD] | $3.202\times 10^{-8}$ |  ${}^{+8\%}_{-10\%}$ ${}^{+5\%}_{-5\%}$ | 1.46 | $8.594\times 10^{-6}$ |  ${}^{+15\%}_{-14\%}$ ${}^{+2\%}_{-2\%}$ | 2.71
$NN\to t\bar{t}W^{+}$ | p p > t t$\sim$ w+ QED=1 QCD=2 [QCD] | $3.199\times 10^{-8}$ |  ${}^{+8\%}_{-10\%}$ ${}^{+5\%}_{-5\%}$ | 1.46 | $8.578\times 10^{-6}$ |  ${}^{+15\%}_{-14\%}$ ${}^{+2\%}_{-2\%}$ | 2.71
$NN\to t\bar{t}Z$ | p p > t t$\sim$ z QED=1 QCD=2 [QCD] | $5.000\times 10^{-8}$ |  ${}^{+9\%}_{-11\%}$ ${}^{+7\%}_{-7\%}$ | 1.46 | $5.530\times 10^{-5}$ |  ${}^{+8\%}_{-7\%}$ ${}^{+2\%}_{-2\%}$ | 1.53
$NN\to t\bar{t}H$ | p p > t t$\sim$ h QED=1 QCD=2 [QCD] | $2.844\times 10^{-8}$ |  ${}^{+4\%}_{-9\%}$ ${}^{+7\%}_{-7\%}$ | 1.28 | $3.184\times 10^{-5}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+2\%}_{-2\%}$ | 1.42
$NN\to t\bar{t}\gamma$ | p p > t t$\sim$ a QED=1 QCD=2 [QCD] | $7.193\times 10^{-9}$ |  ${}^{+13\%}_{-14\%}$ ${}^{+9\%}_{-9\%}$ | 1.62 | $1.439\times 10^{-5}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 1.58

Table 3: Per nucleon NLO in QCD cross sections (in $\mu$b) for various
processes, for 12C collisions, and the representative collision energies
$\sqrt{s_{NN}}=5{\rm~{}TeV}$ and $100{\rm~{}TeV}$, the corresponding mg5amc
simulation syntax (with define ww = w+ w-), the associated scale and PDF
uncertainties [$\%$], and QCD $K$-factor.

Process | Syntax | $\sqrt{s_{NN}}=5{\rm~{}TeV}$ | $\sqrt{s_{NN}}=100{\rm~{}TeV}$
---|---|---|---
$\sigma^{\mathrm{NLO}}$ [$\mu$b] | $\delta_{\mathrm{scale}}\quad\delta_{\mathrm{PDF}}$ | $K^{\mathrm{NLO}}$ |  $\sigma^{\mathrm{NLO}}$ [$\mu$b] | $\delta_{\mathrm{scale}}\quad\delta_{\mathrm{PDF}}$ | $K^{\mathrm{NLO}}$
$pp\to W^{\pm}$ | p p > ww QED=1 QCD=0 [QCD] | $6.419\times 10^{-2}$ |  ${}^{+4\%}_{-6\%}$ ${}^{+4\%}_{-5\%}$ | 1.24 | $1.238\times 10^{+0}$ |  ${}^{+9\%}_{-17\%}$ ${}^{+10\%}_{-6\%}$ | 1.40
$pp\to W^{-}$ | p p > w- QED=1 QCD=0 [QCD] | $2.541\times 10^{-2}$ |  ${}^{+4\%}_{-5\%}$ ${}^{+3\%}_{-5\%}$ | 1.23 | $5.673\times 10^{-1}$ |  ${}^{+9\%}_{-17\%}$ ${}^{+9\%}_{-6\%}$ | 1.40
$pp\to W^{+}$ | p p > w+ QED=1 QCD=0 [QCD] | $3.862\times 10^{-2}$ |  ${}^{+4\%}_{-6\%}$ ${}^{+4\%}_{-5\%}$ | 1.24 | $6.708\times 10^{-1}$ |  ${}^{+9\%}_{-17\%}$ ${}^{+10\%}_{-6\%}$ | 1.40
$pp\to Z$ | p p > z QED=1 QCD=0 [QCD] | $1.834\times 10^{-2}$ |  ${}^{+3\%}_{-5\%}$ ${}^{+3\%}_{-4\%}$ | 1.23 | $3.834\times 10^{-1}$ |  ${}^{+8\%}_{-15\%}$ ${}^{+9\%}_{-6\%}$ | 1.38
$pp\to W^{+}W^{-}$ | p p > w+ w- QED=2 QCD=0 [QCD] | $2.626\times 10^{-5}$ |  ${}^{+3\%}_{-2\%}$ ${}^{+4\%}_{-5\%}$ | 1.32 | $1.131\times 10^{-3}$ |  ${}^{+9\%}_{-11\%}$ ${}^{+4\%}_{-4\%}$ | 1.64
$pp\to W^{\pm}Z$ | p p > ww z QED=2 QCD=0 [QCD] | $1.030\times 10^{-5}$ |  ${}^{+5\%}_{-4\%}$ ${}^{+4\%}_{-5\%}$ | 1.44 | $5.728\times 10^{-4}$ |  ${}^{+11\%}_{-12\%}$ ${}^{+3\%}_{-4\%}$ | 2.06
$pp\to W^{-}Z$ | p p > w- z QED=2 QCD=0 [QCD] | $3.480\times 10^{-6}$ |  ${}^{+5\%}_{-4\%}$ ${}^{+5\%}_{-7\%}$ | 1.45 | $2.551\times 10^{-4}$ |  ${}^{+11\%}_{-12\%}$ ${}^{+3\%}_{-4\%}$ | 2.08
$pp\to W^{+}Z$ | p p > w+ z QED=2 QCD=0 [QCD] | $6.838\times 10^{-6}$ |  ${}^{+5\%}_{-4\%}$ ${}^{+4\%}_{-5\%}$ | 1.44 | $3.227\times 10^{-4}$ |  ${}^{+11\%}_{-12\%}$ ${}^{+3\%}_{-3\%}$ | 2.08
$pp\to ZZ$ | p p > z z QED=2 QCD=0 [QCD] | $3.445\times 10^{-6}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+4\%}_{-6\%}$ | 1.26 | $1.522\times 10^{-4}$ |  ${}^{+6\%}_{-9\%}$ ${}^{+5\%}_{-5\%}$ | 1.40
$pp\to W^{\pm}\gamma$ | p p > ww a QED=2 QCD=0 [QCD] | $1.485\times 10^{-7}$ |  ${}^{+15\%}_{-12\%}$ ${}^{+3\%}_{-4\%}$ | 3.27 | $2.346\times 10^{-5}$ |  ${}^{+15\%}_{-13\%}$ ${}^{+2\%}_{-3\%}$ | 9.43
$pp\to W^{-}\gamma$ | p p > w- a QED=2 QCD=0 [QCD] | $5.000\times 10^{-8}$ |  ${}^{+16\%}_{-12\%}$ ${}^{+5\%}_{-6\%}$ | 3.67 | $1.141\times 10^{-5}$ |  ${}^{+15\%}_{-13\%}$ ${}^{+2\%}_{-3\%}$ | 8.98
$pp\to W^{+}\gamma$ | p p > w+ a QED=2 QCD=0 [QCD] | $9.943\times 10^{-8}$ |  ${}^{+14\%}_{-11\%}$ ${}^{+4\%}_{-3\%}$ | 3.12 | $1.213\times 10^{-5}$ |  ${}^{+15\%}_{-13\%}$ ${}^{+2\%}_{-3\%}$ | 9.89
$pp\to Z\gamma$ | p p > z a QED=2 QCD=0 [QCD] | $1.019\times 10^{-7}$ |  ${}^{+7\%}_{-6\%}$ ${}^{+4\%}_{-5\%}$ | 1.62 | $7.629\times 10^{-6}$ |  ${}^{+11\%}_{-11\%}$ ${}^{+2\%}_{-3\%}$ | 2.80
$pp\to W^{\pm}H$ | p p > ww h QED=2 QCD=0 [QCD] | $3.634\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+4\%}_{-6\%}$ | 1.18 | $1.524\times 10^{-5}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 1.25
$pp\to ZH$ | p p > z h QED=2 QCD=0 [QCD] | $1.837\times 10^{-7}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+4\%}_{-5\%}$ | 1.18 | $8.333\times 10^{-6}$ |  ${}^{+4\%}_{-7\%}$ ${}^{+4\%}_{-4\%}$ | 1.25
$pp\to W^{-}H$ | p p > w- h QED=2 QCD=0 [QCD] | $1.219\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+5\%}_{-7\%}$ | 1.17 | $6.738\times 10^{-6}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 1.25
$pp\to W^{+}H$ | p p > w+ h QED=2 QCD=0 [QCD] | $2.404\times 10^{-7}$ |  ${}^{+2\%}_{-1\%}$ ${}^{+4\%}_{-5\%}$ | 1.18 | $8.546\times 10^{-6}$ |  ${}^{+4\%}_{-8\%}$ ${}^{+4\%}_{-4\%}$ | 1.26
$pp\to W^{+}W^{-}\gamma$ | p p > w+ w- a QED=3 QCD=0 [QCD] | $1.998\times 10^{-9}$ |  ${}^{+9\%}_{-8\%}$ ${}^{+5\%}_{-6\%}$ | 1.69 | $5.466\times 10^{-7}$ |  ${}^{+9\%}_{-8\%}$ ${}^{+1\%}_{-2\%}$ | 3.83
$pp\to W^{+}W^{-}H$ | p p > w+ w- h QED=3 QCD=0 [QCD] | $1.752\times 10^{-9}$ |  ${}^{+3\%}_{-3\%}$ ${}^{+5\%}_{-7\%}$ | 1.25 | $1.694\times 10^{-7}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+2\%}_{-3\%}$ | 1.40
$pp\to W^{+}W^{-}W^{\pm}$ | p p > w+ w- ww QED=3 QCD=0 [QCD] | $3.299\times 10^{-8}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+5\%}_{-6\%}$ | 1.44 | $4.245\times 10^{-6}$ |  ${}^{+8\%}_{-9\%}$ ${}^{+2\%}_{-3\%}$ | 2.33
$pp\to W^{+}W^{-}Z$ | p p > w+ w- z QED=3 QCD=0 [QCD] | $2.116\times 10^{-8}$ |  ${}^{+7\%}_{-6\%}$ ${}^{+5\%}_{-6\%}$ | 1.48 | $4.013\times 10^{-6}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 2.56
$pp\to ZZ\gamma$ | p p > z z a QED=3 QCD=0 [QCD] | $2.138\times 10^{-10}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+6\%}_{-8\%}$ | 1.38 | $3.402\times 10^{-8}$ |  ${}^{+6\%}_{-7\%}$ ${}^{+2\%}_{-3\%}$ | 1.97
$pp\to ZZH$ | p p > z z h QED=3 QCD=0 [QCD] | $4.096\times 10^{-10}$ |  ${}^{+3\%}_{-3\%}$ ${}^{+6\%}_{-7\%}$ | 1.23 | $3.984\times 10^{-8}$ |  ${}^{+4\%}_{-5\%}$ ${}^{+3\%}_{-4\%}$ | 1.28
$pp\to ZZW^{\pm}$ | p p > z z ww QED=3 QCD=0 [QCD] | $7.329\times 10^{-9}$ |  ${}^{+7\%}_{-6\%}$ ${}^{+5\%}_{-6\%}$ | 1.53 | $1.362\times 10^{-6}$ |  ${}^{+9\%}_{-9\%}$ ${}^{+2\%}_{-2\%}$ | 2.83
$pp\to ZZZ$ | p p > z z z QED=3 QCD=0 [QCD] | $2.231\times 10^{-9}$ |  ${}^{+4\%}_{-4\%}$ ${}^{+6\%}_{-7\%}$ | 1.30 | $2.382\times 10^{-7}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+4\%}_{-4\%}$ | 1.46
$pp\to\gamma\gamma W^{\pm}$ | p p > a a ww QED=3 QCD=0 [QCD] | $1.043\times 10^{-10}$ |  ${}^{+11\%}_{-9\%}$ ${}^{+6\%}_{-6\%}$ | 1.83 | $3.342\times 10^{-8}$ |  ${}^{+9\%}_{-9\%}$ ${}^{+2\%}_{-3\%}$ | 4.85
$pp\to HHW^{\pm}$ | p p > h h ww QED=3 QCD=0 [QCD] | $7.913\times 10^{-11}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+6\%}_{-7\%}$ | 1.17 | $7.940\times 10^{-9}$ |  ${}^{+3\%}_{-4\%}$ ${}^{+2\%}_{-3\%}$ | 1.20
$pp\to\gamma\gamma Z$ | p p > a a z QED=3 QCD=0 [QCD] | $5.213\times 10^{-11}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+7\%}_{-8\%}$ | 1.38 | $7.776\times 10^{-9}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+2\%}_{-4\%}$ | 2.09
$pp\to HHZ$ | p p > h h z QED=3 QCD=0 [QCD] | $4.597\times 10^{-11}$ |  ${}^{+2\%}_{-2\%}$ ${}^{+6\%}_{-7\%}$ | 1.17 | $5.104\times 10^{-9}$ |  ${}^{+2\%}_{-4\%}$ ${}^{+3\%}_{-3\%}$ | 1.20
$pp\to W^{\pm}\gamma H$ | p p > ww a h QED=3 QCD=0 [QCD] | $1.901\times 10^{-11}$ |  ${}^{+6\%}_{-5\%}$ ${}^{+7\%}_{-8\%}$ | 1.39 | $3.790\times 10^{-9}$ |  ${}^{+6\%}_{-6\%}$ ${}^{+2\%}_{-3\%}$ | 2.06
$pp\to W^{\pm}Z\gamma$ | p p > ww z a QED=3 QCD=0 [QCD] | $8.288\times 10^{-10}$ |  ${}^{+12\%}_{-9\%}$ ${}^{+6\%}_{-6\%}$ | 1.99 | $3.052\times 10^{-7}$ |  ${}^{+10\%}_{-9\%}$ ${}^{+1\%}_{-2\%}$ | 5.25
$pp\to W^{\pm}ZH$ | p p > ww z h QED=3 QCD=0 [QCD] | $7.654\times 10^{-10}$ |  ${}^{+5\%}_{-4\%}$ ${}^{+6\%}_{-7\%}$ | 1.33 | $9.913\times 10^{-8}$ |  ${}^{+6\%}_{-7\%}$ ${}^{+2\%}_{-3\%}$ | 1.68
$pp\to Z\gamma H$ | p p > z a h QED=3 QCD=0 [QCD] | $1.642\times 10^{-11}$ |  ${}^{+4\%}_{-4\%}$ ${}^{+6\%}_{-8\%}$ | 1.26 | $2.006\times 10^{-9}$ |  ${}^{+5\%}_{-6\%}$ ${}^{+2\%}_{-3\%}$ | 1.55
$pp\to\gamma j$ | p p > a j QED=1 QCD=1 [QCD] | $2.200\times 10^{-5}$ |  ${}^{+3\%}_{-5\%}$ ${}^{+2\%}_{-2\%}$ | 1.18 | $1.717\times 10^{-3}$ |  ${}^{+4\%}_{-7\%}$ ${}^{+3\%}_{-3\%}$ | 1.37
$pp\to\gamma\gamma$ | p p > a a QED=2 QCD=0 [QCD] | $4.331\times 10^{-8}$ |  ${}^{+5\%}_{-4\%}$ ${}^{+4\%}_{-5\%}$ | 1.29 | $2.099\times 10^{-6}$ |  ${}^{+9\%}_{-10\%}$ ${}^{+3\%}_{-4\%}$ | 2.20
$pp\to\gamma\gamma\gamma$ | p p > a a a QED=3 QCD=0 [QCD] | $4.510\times 10^{-12}$ |  ${}^{+8\%}_{-7\%}$ ${}^{+8\%}_{-9\%}$ | 1.52 | $1.114\times 10^{-9}$ |  ${}^{+7\%}_{-7\%}$ ${}^{+2\%}_{-4\%}$ | 2.50
$pp\to t\bar{t}$ | p p > t t$\sim$ QED=0 QCD=2 [QCD] | $5.326\times 10^{-5}$ |  ${}^{+10\%}_{-13\%}$ ${}^{+6\%}_{-5\%}$ | 1.48 | $2.880\times 10^{-2}$ |  ${}^{+7\%}_{-10\%}$ ${}^{+3\%}_{-3\%}$ | 1.49
$pp\to t\bar{t}W^{-}$ | p p > t t$\sim$ w- QED=1 QCD=2 [QCD] | $1.736\times 10^{-8}$ |  ${}^{+10\%}_{-11\%}$ ${}^{+9\%}_{-11\%}$ | 1.48 | $7.197\times 10^{-6}$ |  ${}^{+17\%}_{-15\%}$ ${}^{+2\%}_{-3\%}$ | 2.57
$pp\to t\bar{t}W^{+}$ | p p > t t$\sim$ w+ QED=1 QCD=2 [QCD] | $4.676\times 10^{-8}$ |  ${}^{+9\%}_{-11\%}$ ${}^{+8\%}_{-8\%}$ | 1.46 | $1.012\times 10^{-5}$ |  ${}^{+17\%}_{-15\%}$ ${}^{+2\%}_{-2\%}$ | 2.61
$pp\to t\bar{t}Z$ | p p > t t$\sim$ z QED=1 QCD=2 [QCD] | $4.393\times 10^{-8}$ |  ${}^{+10\%}_{-12\%}$ ${}^{+7\%}_{-6\%}$ | 1.47 | $5.414\times 10^{-5}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-2\%}$ | 1.50
$pp\to t\bar{t}H$ | p p > t t$\sim$ h QED=1 QCD=2 [QCD] | $2.726\times 10^{-8}$ |  ${}^{+5\%}_{-10\%}$ ${}^{+7\%}_{-6\%}$ | 1.28 | $3.166\times 10^{-5}$ |  ${}^{+6\%}_{-7\%}$ ${}^{+2\%}_{-2\%}$ | 1.41
$pp\to t\bar{t}\gamma$ | p p > t t$\sim$ a QED=1 QCD=2 [QCD] | $7.386\times 10^{-9}$ |  ${}^{+14\%}_{-15\%}$ ${}^{+9\%}_{-8\%}$ | 1.62 | $1.418\times 10^{-5}$ |  ${}^{+8\%}_{-8\%}$ ${}^{+2\%}_{-3\%}$ | 1.55

Table 4: Per nucleon NLO in QCD cross sections (in $\mu$b) for various
processes, for proton collisions, and the representative collision energies
$\sqrt{s_{NN}}=5{\rm~{}TeV}$ and $100{\rm~{}TeV}$, the corresponding mg5amc
simulation syntax (with define ww = w+ w-), the associated scale and PDF
uncertainties [$\%$], and QCD $K$-factor.

##  6 Extrapolating Cross Sections from One Nucleus to Another

In Sec. 5 we surveyed the inclusive and fiducial cross sections for dozens of
high-energy processes as a function of collider energy. One observation (of
many) was that the ratios of the nucleon-level cross section for symmetric
208Pb, 131Xe, and 12C collisions with respect to symmetric proton collisions
tended to converge to some number for sufficiently high collider energies. In
many cases, these were near but not equal to unity. Despite different nuclei
exhibiting qualitatively different parton densities (Sec. 3) and parton
luminosities (Sec. 4), findings suggest that it may be possible to extrapolate
reliably a particular cross section for one pair of colliding nuclei from a
second pair at sufficiently high collider energies. In this section, we
explore quantitatively this possibility.

For the inclusive production of a particular final state $\mathcal{F}$,
possibly with phase space restrictions, we consider the cross section estimate
(est.) obtained by multiply a true nuclear-level cross section (true) by the
appropriate ratio of atomic numbers. Symbolically, this is given by

$\displaystyle\sigma^{\rm
est.}(\mathcal{A}_{1}\mathcal{A}_{2}\to\mathcal{F}+X)$ $\displaystyle=\
\left(\frac{A_{1}A_{2}}{A_{3}A_{4}}\right)\ \times\ \sigma^{\rm
true}(\mathcal{A}_{3}\mathcal{A}_{4}\to\mathcal{F}+X)$ (6.1a) $\displaystyle=\
\left(A_{1}A_{2}\right)\ \times\ \sigma^{\rm
true}_{NN}(N_{3}N_{4}\to\mathcal{F}+X)\ .$ (6.1b)

According to the definition in Eq. (5.2), this reduces to the product of (a)
the target atomic numbers $(A_{1}A_{2})$ and (b) the starting nucleon-level
cross section $\sigma_{N_{3}N_{4}}$. One can then define the following as the
relative difference [$\%$] between the estimated and true nucleus-level cross
sections:

$\displaystyle\frac{\delta\sigma^{\mathcal{A}_{1}\mathcal{A}_{2}}_{\mathcal{A}_{3}\mathcal{A}_{4}}}{\sigma^{\rm
true}_{\mathcal{A}_{1}\mathcal{A}_{2}}}$ $\displaystyle=\ \frac{\sigma^{\rm
est.}(\mathcal{A}_{1}\mathcal{A}_{2}\to\mathcal{F}+X)\ -\ \sigma^{\rm
true}(\mathcal{A}_{1}\mathcal{A}_{2}\to\mathcal{F}+X)}{\sigma^{\rm
true}_{\mathcal{A}_{1}\mathcal{A}_{2}}(\mathcal{A}_{1}\mathcal{A}_{2}\to\mathcal{F}+X)}$
(6.2a) $\displaystyle=\ \frac{\left(A_{1}A_{2}\right)\times\sigma^{\rm
true}_{NN}(N_{3}N_{4}\to\mathcal{F}+X)\ -\
\left(A_{1}A_{2}\right)\times\sigma^{\rm
true}_{NN}(N_{1}N_{2}\to\mathcal{F}+X)\
}{\left(A_{1}A_{2}\right)\times\sigma^{\rm
true}_{NN}(N_{1}N_{2}\to\mathcal{F}+X)}$ (6.2b) $\displaystyle=\
\frac{\sigma^{\rm true}_{NN}(N_{3}N_{4}\to\mathcal{F}+X)\ -\ \sigma^{\rm
true}_{NN}(N_{1}N_{2}\to\mathcal{F}+X)\ }{\sigma^{\rm
true}_{NN}(N_{1}N_{2}\to\mathcal{F}+X)}\ .$ (6.2c)

For the case of symmetric beams, we have $\mathcal{A}_{4}=\mathcal{A}_{3}$ and
$\mathcal{A}_{2}=\mathcal{A}_{1}$ as well as $N_{4}=N_{3}$ and $N_{2}=N_{1}$,
but otherwise the expression remains the same. Two observations can be drawn
from Eq. (6.2): First and most importantly, the expression quantifies the
error in using Eq. (6.1) to extrapolate cross sections across nuclei. Second
but more interesting, we find that an uncertainty at the nuclear level can be
expressed in terms of nucleon-level quantities. To guide further discussion,
we note that intuitively the closer this error is to zero, the better the
extrapolation. However, a nonzero number is also useful as one can extract an
“offset” factor $\Delta_{\rm offset}$. One can then, in principle, improve the
extrapolation formula of Eq. (6.1) by the following modification

$\displaystyle\sigma^{\rm
est.}(\mathcal{A}_{1}\mathcal{A}_{2}\to\mathcal{F}+X)$ $\displaystyle=\
\left(A_{1}A_{2}\right)\ \times\ \sigma^{\rm
true}_{NN}(N_{3}N_{4}\to\mathcal{F}+X)\ +\ \Delta_{\rm offset}\ .$ (6.3)

In Figs. 17 and 18, we show as a function of nucleon-nucleon collision energy
$\sqrt{s_{NN}}$ the extrapolation error, as defined in Eq. (6.2), in
estimating the scattering cross sections at NLO in QCD for our various
processes and isotopes from the true predictions for 208Pb. Also shown is the
uncertainty band stemming from the PDF uncertainty of 208Pb. In all plots, we
consider (top panel) 1H, (middle panel) 12C, and (bottom panel) 131Xe.

Figure 17: As a function of $\sqrt{s_{NN}}$, the extrapolation error [Eq.
(6.2)] in estimating the NLO in QCD cross section, with PDF uncertainty, for
(a) single weak boson, (b) diboson, (c) associated photon, and (d) associated
Higgs production in 1H (top panel), 12C (middle), and 131Xe (bottom)
collisions from 208Pb collisions.

Figure 18: Same as Fig. 17 but for (a) $W^{+}W^{-}X$, (b) $ZZX$, (c)
$W^{\pm}ZX$ and $\gamma HX$, (d) $\gamma\gamma X$ and $HHX$, (e) inclusive
photon, and (e) top quark processes.

We start our assessment with Fig. 17, where we have: (a) single weak boson
production, (b) diboson production, (c) associated photon-weak boson
production, and (d) associated Higgs-weak boson production. The first
remarkable finding is the extrapolation to 131Xe. For all processes and
collision energies, the extrapolation error sits at or just around zero, with
a PDF uncertainty in the range of $\mathcal{O}(\pm 5\%)-\mathcal{O}(\pm
20\%)$. For $\sqrt{s_{NN}}\gtrsim 5-10{\rm~{}TeV}$, a similar trend holds for
12C. While the error for carbon converges to $\delta\sigma/\sigma|_{\rm
est.}\sim 0\%$ to $-10\%$, this lies within the $\mathcal{O}(\pm
5\%)-\mathcal{O}(\pm 20\%)$ PDF uncertainty. For low collision energies, i.e.,
$\sqrt{s_{NN}}\lesssim 5-10{\rm~{}TeV}$, the error can grow to about
$\delta\sigma/\sigma|_{\rm est.}\sim-25\%$. We broadly attribute these
similarities and differences to two ingredients: (i) the relative nucleon
content of the isotopes, which are comparable and are approximately
$Z:(A-Z)\sim 40:60\ (40:60)\ [50:50]$ for 208Pb (131Xe) [12C], and (ii) the
growing important of valence quarks (gluons) at the lowest (highest) collision
energies that we consider. For example: As discussed in Sec. 5.3, $V\gamma$
production is sensitive to the nucleus’ isospin since this is related to the
net electric charge of its constituents.

When extrapolating predictions for proton collisions from lead collisions (in
practice, one will more likely do the reverse), we observe much larger
extrapolation errors and hence much worse agreement. At the largest collider
energies, cross sections can be off by as much as $\delta\sigma/\sigma|_{\rm
est.}\sim-25\%$, and over $\delta\sigma/\sigma|_{\rm est.}\sim+100\%$ at the
lowest energies. The largest disagreements are driven by the $W^{-}X$
channels, and ultimately reflects the difference in valance $d$ quark
distributions in lead and the proton. However, in charge-symmetric channels
such as $W^{\pm}+\gamma/Z/H$, $W^{+}W^{-}$, and $ZZ$ production, we find much
better agreement. With the exception of $V\gamma$, extrapolation errors for
these channels are small and comparable to those for 12C and 131Xe;
differences remain below the $\mathcal{O}(10\%)$ level.

Moving onto Fig. 18, we show the same information as in Fig. 17 but for (a)
$W^{+}W^{-}X$, (b) $ZZX$, (c) $W^{\pm}ZX$ and $\gamma HX$, (d) $\gamma\gamma
X$ and $HHX$, (e) inclusive photon, and (e) top quark processes. Globally, we
observe qualitatively and quantitatively similar behavior: For channels that
are charge- and flavor-symmetric, scattering rates for symmetric 208Pb
collisions can be used to reliably estimate scattering rates for symmetric 1H,
12C, and 131Xe rates to within $\mathcal{O}(10\%)$, particularly for collision
energies $\sqrt{s_{NN}}\gtrsim 5{\rm~{}TeV}-10{\rm~{}TeV}$. The exception are
channels with one or more high-energy photons in the final state; these retain
sizable sensitivity to the net isospin of the nucleus. For charge- and flavor-
asymmetric channels, e.g., $t\overline{t}W^{-}$ and $t\overline{t}W^{+}$, we
find that the extrapolation errors can quickly converse to nonzero values that
are process and isotope dependent, suggesting that the offset correction in
Eq. (6.3) is a viable remedy.

##  7 Discussion and Conclusions

The density and distribution of partons in nuclei differ both qualitatively
and quantitatively from protons. This stems from an interplay of nuclear,
hadronic, as well as partonic dynamics, and has a direct impact on parton
luminosities. (See Fig. 1 for illustrations of typical dynamics at these
levels.) Subsequently, scattering cross sections of high-mass particle physics
processes in high-energy ion collisions differ both qualitatively and
quantitatively from those in proton collisions. We explore a variety of such
high-energy processes, ranging from single $W^{\pm}$ and $Z$ production and
associated $W^{\pm}H$ and $ZH$ production to triple $W^{+}W^{-}H$ and $ZZH$
production as well as associated $t\overline{t}W^{\pm}$ and $t\overline{t}Z$
production.

In this work, we have attempted to map out the ion and collider energy
dependence by systematically cataloging the scattering rates and theoretical
uncertainties of 42 high-$p_{T}$ processes in 208Pb-208Pb, 131Xe-131Xe, and
12C-12C collisions over the energy range $\sqrt{s_{NN}}=1-100$ TeV. Our work
takes advantage of publicly available precision Monte Carlo simulation tools
developed initially for proton-proton collisions [26, 27]. Our work also
relies on advances in nuclear PDF fits, which have seen significant
improvements over the past decade [20]. We carry out this work up to NLO in
QCD in order to properly account for additional partonic and kinematic sub-
channels, which can be numerically important due to the presence of accidental
cancellations at lowest order in perturbation theory, large virtual
corrections at one loop, the largeness of the strong coupling constant, etc.

While our main results are shown in Sec. 5 (cross sections), Secs. 3 and 4
feature, respectively, a study of nPDFs and luminosities (and their ratios)
for various partons, nuclei, scales, and kinematic ranges. When taken
altogether, we are able to build an interesting narrative on how the interplay
between nuclear dynamics (e.g., meson exchange), hadronic dynamics (e.g.,
long-range gluon exchanges), and partonic dynamics (e.g., DGLAP evolution)
propagate into cross sections. In many cases, this interplay leads to
sufficiently large differences in cross sections with sufficiently small
uncertainties that they can be observed experimentally in ongoing and future
high-energy ion collisions. However, at the same time, we find consistently
the increasing role of partonic dynamics as the dominant driver of
phenomenology at increasing collider energies for many processes. Motivated by
this, we explored in Sec. 6 the error one makes when trying to extrapolate
cross sections for one nucleus from predictions for a second nucleus.

In the context of the inclusive production of high-energy systems in multi-TeV
collisions, more interesting findings include the following:

  * •

While nPDFs are known to exhibit large $A$ dependence for $x\gtrsim 0.1$, much
of this variation is softened at the level of parton luminosities. We
attribute this to an increased role of the gluon density and $g\to
q\overline{q}$ splittings (that are driven by perturbative dynamics), which
are flavor symmetric (for massless quarks) and therefore exhibits approximate
nuclear isospin symmetry, i.e., is approximately independent of $(A,Z)$
configurations. This is particularly true for kinematic thresholds below
$(\hat{s}/s_{NN})\lesssim 10^{-3}$.

  * •

Categorically, for processes that sum over all charge and flavor
configurations, we find that the ratio of nucleon-averaged cross sections
$\sigma_{NN}$ [as defined in Eq. (5.2)] with respect to the proton cross
section $\sigma_{pp}$ converges a flat value at or just below unity once the
nucleon-nucleon collision energy surpasses $\sqrt{s_{NN}}\gtrsim
5-10{\rm~{}TeV}$. Not only does this include simpler single- and diboson
processes, e.g., $Z$ and $W^{\pm}H$ production, but more complex triboson and
top quark processes, e.g., $W^{+}W^{-}Z$ and $t\overline{t}H$ production. This
follows from the increased importance of sea quarks and sea antiquarks (and
hence gluons) in the production of sub-TeV systems in multi-TeV collisions,
and the smaller sensitivity of these partons to an averaged nucleon’s $(A,Z)$
configuration. The exception are channels with multiple $Z$ bosons and photons
since these are indirectly sensitive to the net isospin of nucleons, but the
impact is small.

  * •

Categorically, for processes that are charge or flavor asymmetric, e.g.,
$W^{-}$, $W^{-}H$, $W^{-}\gamma$, and $t\overline{t}W^{-}$ production, the
aforementioned convergence still occurs when collision energies are
sufficiently large such that $(\sqrt{\hat{s}}/\sqrt{s_{NN}})\lesssim 4-6\times
10^{-3}$. This universal-like behavior across so many processes and ion
configurations suggests the onset of a common phenomenon, such as an increased
importance / dominance of sea quarks and hence gluons in many-TeV collisions.

  * •

In many cases, we found large QCD corrections at NLO (with $K^{\rm
NLO}>1.5-2$). Such large corrections, which are well-documented for proton
collisions, have various origins. These include radiation amplitude zeros
(accidental cancellations at lowest order), the opening of new partonic
channels, DGLAP evolution, and sizable virtual corrections. It is important to
stress that any ion collision prediction of high-energy systems, albeit a
precision measurement or not, should include QCD corrections if and when
available. Importantly, nPDF uncertainties in cross sections are comparable or
smaller than scale NLO in QCD uncertainties.

  * •

In practice (see Sec. 6), many nucleus-level cross sections and uncertainties
for many processes can be reliably extrapolated, even at NLO in QCD, by
scaling proton cross sections by the appropriate $A$ factor(s). For channels
that are charge- and flavor-symmetric, scattering rates for symmetric 208Pb
collisions can be used to reliably estimate scattering rates for symmetric 1H,
12C, and 131Xe collisions to within $\mathcal{O}(10\%)$, particularly for
collision energies $\sqrt{s_{NN}}\gtrsim 5{\rm~{}TeV}-10{\rm~{}TeV}$.
Exceptions consist of channels with one or more high-energy photons in the
final state; these retain sizable sensitivity to the net isospin of the
nucleus.

In conclusion, with the start of Run III activities at the LHC and ongoing
preparations for its high luminosity era, we look forward to the upcoming
transition to a precision particle physics program using high-energy ion-ion
and ion-proton collisions. And subsequently, through the combined use of
precision matrix elements, precision event generators / simulation tools,
precision detector systems, a precise determination of the nuclear structure
could be achieved through the extraction of nuclear PDFs from measurements of
hard, electroweak-scale processes in multi-TeV ion collisions.

## Acknowledgements

The authors thank Krzysztof Kutak, Aleksander Kusina, and Fred Olness for
enlightening discussions. BF acknowledges support from Grant ANR-21-CE31-0013,
Project DMwithLLPatLHC, from the _Agence Nationale de la Recherche_ (ANR),
France. RR acknowledges the support of Narodowe Centrum Nauki under Grant Nos.
2019/34/E/ST2/00186 and 2023/49/B/ST2/04330 (SNAIL). The authors acknowledge
support from the COMETA COST Action CA22130. This work was partially funded by
the Polish Academy of Sciences under the “Funding for the dissemination and
promotion of scientific activities” program (PAN.BFB.S.KA.140.022.2023).

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# Adaptive Explicit Kernel Minkowski Weighted K-means

Amir Aradnia
Department of Computer Engineering
Amirkabir University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
Maryam Amir Haeri
Learning, Data-Analytics and Technology Department
University of Twente
Enschede, Netherlands
<EMAIL_ADDRESS>
Mohammad Mehdi Ebadzadeh
Department of Computer Engineering
Amirkabir University of Technology
Tehran, Iran
<EMAIL_ADDRESS>

###### Abstract

The K-means algorithm is among the most commonly used data clustering methods.
However, the regular K-means can only be applied in the input space and it is
applicable when clusters are linearly separable. The kernel K-means, which
extends K-means into the kernel space, is able to capture nonlinear structures
and identify arbitrarily shaped clusters. However, kernel methods often
operate on the kernel matrix of the data, which scale poorly with the size of
the matrix or suffer from the high clustering cost due to the repetitive
calculations of kernel values. Another issue is that algorithms access the
data only through evaluations of $K(x_{i},x_{j})$, which limits many processes
that can be done on data through the clustering task. This paper proposes a
method to combine the advantages of the linear and nonlinear approaches by
using driven corresponding approximate finite-dimensional feature maps based
on spectral analysis. Applying approximate finite-dimensional feature maps
were only discussed in the Support Vector Machines (SVM) problems before. We
suggest using this method in kernel K-means era as alleviates storing huge
kernel matrix in memory, further calculating cluster centers more efficiently
and access the data explicitly in feature space. These explicit feature maps
enable us to access the data in the feature space explicitly and take
advantage of K-means extensions in that space. We demonstrate our Explicit
Kernel Minkowski Weighted K-mean (Explicit KMWK-mean) method is able to be
more adopted and find best-fitting values in new space by applying additional
Minkowski exponent and feature weights parameter. Moreover, it can reduce the
impact of concentration on nearest neighbour search by suggesting investigate
among other norms instead of Euclidean norm, includes Minkowski norms and
fractional norms (as an extension of the Minkowski norms with p<1). The
proposed method is evaluated by four benchmark data sets and compared the
performance with commonly used kernel clustering approach. Experiments show
the proposed method consistently achieves superiors clustering performances
while it can be used to reduce memory consumption as well.

_K_ eywords Kernel clustering $\cdot$ Minkowski metric $\cdot$ Features map
$\cdot$ K-means

##  1 Introduction

Clustering can be considered as the most important unsupervised learning
problem. Clustering methods are used to determine the intrinsic grouping in a
set of unlabeled data. It has been shown to be useful in many practical
domains such as web search, image segmentation, image compression, gene
expression analysis, recommendation systems and mining text data [4, 17, 8].
K-means clustering [20] is one of the most popular conventional clustering
algorithms, despite its age. It aims to partition sample of N observation into
K compact clusters in an iterative process. The K-means algorithm only works
reasonably well when 1) clusters can be separated by hyper-planes and 2) each
data point belongs to the closest cluster center. If one of these principles
does not hold, the standard K-means algorithm will likely not give a good
result. Kernel-based clustering methods overcome these limitations by using an
appropriate non-linear mapping to higher dimensional feature space. Thus, it
enables the K-means algorithm to partition data points by the linear separator
in the new space, that has non-linear projection back in the original space.
Various types of kernel-based methods such as the kernel version of the SOM
(Self-organizing map) algorithm [19, 16], kernel neural gas[21], one Class SVM
(Support Vector Machines) [2] and kernel fuzzy clustering [26, 25] have been
proposed by researchers. In this paper, we focus on kernel k-means clustering
because of its efficiency and simpleness. Furthermore, various studies [24, 7,
6] claim that different kernel-based clustering methods show similar result as
kernel K-means.

Although kernel-based methods have received considerable attention from the
machine learning community in recent years, they still suffer from the
following problems in real applications. First, the high clustering cost due
to either the repeated calculations of kernel values or requiring huge amounts
of memory to store the kernel matrix makes it unsuitable for large corpora.
Second, algorithms access the data only through evaluations of K(x, y).
Therefore, many processes on the data points like fighting the concentration
phenomenon, handling noise, etc. limited to work in the original space.

The aim of this paper is to develop a clustering method that can group data
points with both linear and non-linear structures while try to address the two
mentioned problems of kernel clustering methods. As the main contribution of
this paper, we address both the space complexity of storing kernel matrix and
lack of accessing to data in feature space by proposing new _Adaptive Explicit
Kernel Minkowski Weighted K-means (Explicit KMWK-mean)_ method. The proposed
method combines the advantages of the linear and nonlinear approaches by using
driven corresponding approximate finite-dimensional feature maps based on 1D
Fourier analysis [23] for a large family of additive kernels, known as
$\gamma$-homogeneous kernels. The proposed method, first map data to feature
space. This feature space is data independent approximation of
$\gamma$-homogeneous kernels. Then, in order to provide a better fit to
cluster structure, the weighted version of Minkowski K-means in low-
dimensional feature space would be applied. Especially, we analyze the
concentration of the Euclidean norm and the impact of distance measure on
concentration and investigate on Minkowski norms and fractional norms, an
extension of the Minkowski norms with $p<1$, as a measure of distance among
data in the kernel space. Adaptive Minkowski metric allows to fight against
possible concentration in high-dimensional spaces and the weighting property
enables our algorithm to cover spherical and non-spherical (elliptical)
structures in feature space and getting the chance of finding even more
complex clusters in feature space.

The remainder of this article is organized as follows. Section 2, briefly
describes k-means and kernel k-means as the preliminary notions. Section 3
introduces Explicit Feature Maps and especially focused on Homogeneous
kernels. Section 4 presents our modified version of kernel K-means by
analyzing the alternative Minkowski distance for arbitrary
$p\in\mathbb{R}^{+}$ instead of Euclidean one in feature space. In Section 5
we describe the results of experiments four benchmark data sets. Section 6
concludes the paper.

##  2 K-means and Kernel K-means

The K-means method is designed to partition $N$ D-dimensional samples
$X=(x_{1},x_{2},\ldots,x_{N}).$ in to $K$ clusters
${C_{1}}{\rm{,}}{C_{2}}{\rm{,}}...{\rm{,}}{C_{K}}$ and return centroid vector
for each cluster $M=(m_{1},m_{2},\ldots,m_{K}).$ The batch mode K-means
algorithm would operate by the following iterative procedure:

  1. 1.

Initialize $K$ cluster center $m_{1},m_{2},\ldots,m_{K}.$

  2. 2.

Assign each sample $x_{i}$ to its closest center. Namely, compute the
indicator matrix ${\delta_{ik}},\>(1\leq k\leq K).$

${\delta_{ik}}{\rm{=}}\left\\{{\begin{array}[]{*{20}{c}}1&{d({{\rm{x}}_{\rm{i}}},{{\rm{m}}_{\rm{k}}})<d({{\rm{x}}_{\rm{i}}},{{\rm{m}}_{\rm{j}}})\>{\rm{for}}\>{\rm{all}}\>{\rm{j}}\neq
k}\\\ 0&{otherwise}\end{array}}\right.$ (1)

  3. 3.

Update the cluster centers.

${m_{k}}=\frac{1}{{|{c_{k}}|}}\sum\limits_{i=1}^{N}{{\delta_{ik}}}{x_{i}}$ (2)

where $|C_{k}|$ is the number of samples in $C_{k}$.

  4. 4.

Iterate between (2) and (3) until convergence.

  5. 5.

Return $m_{1},m_{2},\ldots,m_{K}.$

Note that, in (1), $d(x_{i},m_{k})$ is the Euclidean distance given by:

${d^{2}}({x_{i}},{m_{k}})=||{x_{i}}-{m_{k}}|{|^{2}}$ (3)

The preceding procedure actually is an iterative solution to optimization
problem that attempts to minimize the objective function as follows:

$\arg{\rm{}}\mathop{{\rm{min}}}\limits_{M,\delta}\sum\limits_{i=1}^{N}{\sum\limits_{k=1}^{K}{{\delta_{ik}}||{x_{i}}-{m_{k}}|{|^{2}}}}$
(4)

In most cases, the distance in use is the squared Euclidean distance. However,
the Euclidean distances tend to concentrate, when data are high dimensional.
This means that, all the pairwise distances may converge to the same value.
Accordingly, the relevance of Euclidean distance has been questioned in the
past and alternative norms, especially fractional norms ($L_{p}$ semi-norms
with $p<1$) were suggested to reduce concentration phenomenon [13, 1].
Obviously, by using different metrics, cluster centers don’t follow from the
equation in (2) anymore. Finding fractional and Minkowski’s centers, whose
components are minimizers of the summary corresponding distances, discussed in
Section 4.1.3.

The K-means algorithm with Euclidean distance generally works on ellipse-
shaped clusters. It is not applicable when elliptical regions does not hold.
By applying some kind of transformation to the data, mapping them to some new
space, the k-means algorithm may be achieve better performance than in
original space. Again, suppose we are given N samples of
$X=(x_{1},x_{2},\ldots,x_{N})\>x_{i}\in\mathbb{R}^{D}$, and mapping function
$\Phi$ that transforms $x_{i}$ from original space $\mathbb{R}^{D}$ to a high
dimensional feature space $\mathbb{H}$. Kernel functions are implicitly
defined as the dot product of two vectors in the new transformed feature
space.

$K(x_{i},x_{j})=\Phi(x_{i}).\Phi(x_{j})$ (5)

In the rest of paper, we use ${\Phi_{i}}$ instead of $\Phi({x_{i}})$ for ease
of description. Essentially, the transformation is defined implicitly, without
knowing the concrete form of $\Phi$ [22]. Computation of distances in the
transformed space is one of the most important issues when K-means is extended
to the kernel K-means. The squared Euclidean distance between $x_{i}$ and
$x_{j}$ in feature space would be as:

$\begin{array}[]{l}\begin{array}[]{l}\mathop{{d^{2}}}\limits_{Euc}({\Phi_{i}},{\Phi_{j}})=||{\Phi_{i}}-{\Phi_{j}}|{|^{2}}={\Phi_{i}}^{2}-2{\Phi_{i}}.{\Phi_{j}}+{\Phi_{j}}^{2}\\\
=K({x_{i}},{x_{i}})-2K({x_{i}},{x_{j}})+K({x_{j}},{x_{j}})\end{array}\end{array}$
(6)

The cluster center in transformed space can be calculated as given below:

${m^{\prime}_{k}}=\frac{1}{{|{C_{k}}|}}\sum\limits_{i=1}^{N}{{\delta_{ik}}}{\Phi_{i}}$
(7)

Therefore, the distance of each data point and the cluster center in new space
can be computed without knowing the transformation $\Phi$ explicitly.

$\begin{array}[]{*{20}{l}}{d{{}^{2}}({\Phi_{i}},{m^{\prime}_{k}})=||{\Phi_{i}}-\frac{1}{{|{C_{k}}|}}\sum\limits_{j=1}^{N}{{\delta^{\prime}_{jk}}.}{\Phi_{j}}|{|^{2}}}\\\
{=K({x_{i}},{x_{i}})-\frac{2}{{|{C_{k}}|}}\sum\nolimits_{{x_{j\in{C_{k}}}}}{K({x_{i}},{x_{j}})}+\frac{1}{{|{C_{k}}{|^{2}}}}\sum\nolimits_{{x_{j\in{C_{k}}}}}{\sum\nolimits_{{x_{l\in{C_{k}}}}}{K({x_{i}},{x_{l}})}}}\end{array}$
(8)

Where ${\delta^{\prime}}$ is the indicator matrix which
${{\delta^{\prime}_{jk}}}$ indicates whether ${{\Phi_{i}}}$ is assigned to
${{C_{k}}}$ or not. We will be moved from K-means to Kernel K-means by
applying(8) to the standard K-means.

The following notation is used in the rest of the paper. Multidimensional
additive kernels have been represented as $K(x_{i},x_{j})$, henceforth
$k(x_{i},x_{j})$ will be used for the scalar ones. Multidimensional kernel
function can be obtained from scalar ones as:
$K({x_{i}},{x_{j}})=\sum\limits_{l=1}^{D}k(x_{i}^{l},x_{j}^{l})$. The scalar
feature map is also denoted by $\varphi({x_{i}})$ and multidimensional ones
are given by ${\Phi_{i}}=\mathop{\oplus}\limits_{l=1}^{D}\varphi_{i}^{l}$, it
means ${\Phi_{i}}=[\varphi_{i}^{1},...\varphi_{i}^{D}]$.

##  3 Explicit Feature Maps

Namely, in kernel learning context, for each positive-definite (PD) kernel
$K(x_{i},x_{j})$ there exists a corresponding mapping function $\Phi$ to an
arbitrary dimensional space such that $K(x_{i},x_{j})=\Phi_{i}.\Phi_{j}.$ Even
though the explicit form of the mapping function is useful conceptually, it is
not often used in computations. Typically, these feature spaces are infinite-
dimensional, yet it is possible to find an efficient finite-dimensional
approximation of $\Phi$. In the following, we will introduce a class of
kernels which commonly used in computer vision, called homogeneous kernels.
Then describe deriving corresponding approximate explicit finite-dimensional
feature maps proposed by Vedaldi and Zisserman [23]. The main focus of this
paper, as in an influential paper [23], is a class of additive kernels such as
the Hellinger’s, ${\chi^{2}}$, intersection, and Jensen-Shannon ones, which
are frequently used in computer vision applications. All of the mentioned
kernels are mutual in two properties of additivity and homogeneity. Common
homogeneous kernels with their properties are listed in table 1.

Table 1: Well-known kernel functions and their corresponding distance , signature and closed form feature maps. This table is adopted from [23]. Kernel | $K(x,y)$ | $d(x,y)$ | Signature$\kappa(\lambda)$ | $\rho(\omega)$ | ${\varphi_{\omega}}({x^{i}})$
---|---|---|---|---|---
$\chi^{2}$ | $\sum{\frac{{{{({x^{i}}-{y^{i}})}^{2}}}}{{({x^{i}}-{y^{i}})}}}$ | $\sum{\frac{{2{x^{i}}{y^{j}}}}{{({x^{i}}+{y^{j}})}}}$ | ${\rm{sech(}}\lambda/2{\rm{)}}$ | ${\mathop{\rm sech}\nolimits}(\pi\omega)$ | ${e^{i\omega{\rm{log}}{{\rm{x}}^{i}}}}\sqrt{{x^{i}}{\rm{sech(}}\pi\omega{\rm{)}}}$
Hellinger | $2\sum{(\sqrt{{x^{i}}}}-\sqrt{{y^{i}}}{)^{2}}$ | $\sum{\sqrt{{x^{i}}{y^{i}}}}$ | $1$ | $\delta(\omega)$ | $\sqrt{{x^{i}}}$
JS | $\begin{array}[]{c}\sum{({x^{i}}{{\log}_{2}}(\frac{{2{x^{i}}}}{{{x^{i}}+{y^{j}}}})}+\\\ {y^{i}}{\log_{2}}(\frac{{2{y^{i}}}}{{{x^{i}}+{y^{j}}}}))\end{array}$ | $\begin{array}[]{c}\frac{1}{2}\sum{({x^{i}}{{\log}_{2}}(\frac{{{x^{i}}+{y^{j}}}}{{{x^{i}}}})}+\\\ {y^{j}}{\log_{2}}(\frac{{{x^{i}}+{y^{j}}}}{{{y^{j}}}}))\end{array}$ | $\begin{array}[]{c}\frac{{{e^{\frac{\lambda}{2}}}}}{2}{\log_{2}}(1+{e^{-\lambda}})+\\\ \frac{{{e^{\frac{{-\lambda}}{2}}}}}{2}{\log_{2}}(1+{e^{\lambda}})\end{array}$ | $\frac{2}{{\log 4}}\frac{{{\mathop{\rm sech}\nolimits}(\pi\omega)}}{{1+4{\omega^{2}}}}$ | ${e^{i\omega{\rm{log}}{{\rm{x}}^{i}}}}\sqrt{\frac{2}{{\log 4}}\frac{{{\mathop{\rm sech}\nolimits}(\pi\omega)}}{{1+4{\omega^{2}}}}}$
intersection | $\sum{|{x^{i}}-{y^{i}}|}$ | $\sum{\min({x^{i}},{y^{i}})}$ | ${e^{\frac{{-|\lambda|}}{2}}}$ | $\frac{2}{{\log 4}}\frac{{{\mathop{\rm sech}\nolimits}(\pi\omega)}}{{1+4{\omega^{2}}}}$ | ${e^{i\omega{\rm{log}}{{\rm{x}}^{i}}}}\sqrt{\frac{2}{\pi}\frac{1}{{1+4{\omega^{2}}}}}$

##### Homogeneous Kernels.

A kernel $k_{h}(a,b)$ called a homogeneous of degree $\gamma$ if

$\forall t\geq 0:{K_{h}}(ta,tb)={t^{\gamma}}{K_{h}}(a,b)$ (9)

Here $a,b\in\mathbb{R}$. If we set $t=\frac{1}{{\sqrt{ab}}}$, the homogeneous
kernel can be rewritten as :

$\begin{array}[]{l}{k_{h}}(a,b)={t^{-\gamma}}{k_{h}}(ta,tb)={(ab)^{\frac{\gamma}{2}}}{k_{h}}(\sqrt{\frac{b}{a}},\sqrt{\frac{a}{b}})\\\
={(ab)^{\frac{\gamma}{2}}}\kappa(\log b-\log a)\end{array}$ (10)

The scalar function $\kappa$ is called signature function which is used to
deriving associated feature map of homogeneous kernel functions. It is defined
as :

$\kappa(\lambda)={k_{h}}({e^{\frac{\lambda}{2}}},{e^{-\frac{\lambda}{2}}}),{\rm{}}\lambda\in{\rm{R,}}$
(11)

Bochner’s theorem by using Fourier transform of signature function,
$\kappa(\lambda)$ can address the problem of which mapping function made the
given homogeneous kernel. As a result, the feature map $\varphi$ for
$\gamma$-homogeneous kernels will be derived as:

${\varphi_{\omega}}(a)={e^{-i\omega\log a}}\sqrt{{a^{\gamma}}\rho(\omega)}$
(12)

where $\omega$ can be viewed as the index of vector dimension and
${\rho(\omega)}$ is the spectrum function and can be obtained from inverse
Fourier transform of the signature $\kappa(\lambda)$.

$\rho(\omega)=\frac{1}{{{{(2\pi)}^{D}}}}\int_{{R^{D}}}{{e^{-i(\omega,\lambda)}}}\kappa(\lambda)d\lambda.$
(13)

In (12) feature maps are continuous functions but still finite approximation
feature maps can be generated by sampling the continuous spectrum and
rescaling it. Closed-form of common kernel feature maps are described in table
1. Moreover, Hein and Bousquet [12] introduced a large class of
$\gamma$-homogeneous Hilbertian metrics and correspondent kernels which
encompasses all previously described kernels in table 1. This class of metrics
defined as follows with two parameters $\alpha$ and $\beta$ to be tuned.

$d_{\alpha|\beta}^{2}(a,b)=\frac{{{2^{\frac{1}{\beta}}}{{({a^{\alpha}}+{b^{\alpha}})}^{\frac{1}{\alpha}}}-{2^{\frac{1}{\alpha}}}{{({a^{\beta}}+{b^{\beta}})}^{\frac{1}{\beta}}}}}{{{2^{\frac{1}{\alpha}}}-{2^{\frac{1}{\beta}}}}}$
(14)

Function $d$ in Equation (14) is a $\gamma$-homogeneous metric on
${\mathbb{R}^{+}}$ with $\alpha\in[1,\infty]$ and
$\beta\in[\frac{1}{2},\alpha]$ or $\beta\in[-\infty,-1]$. The pointwise limit
of $d$ when $\alpha\to\beta$ is determined as:

$\begin{array}[]{l}\lim{\rm{d}}_{\alpha|\beta}^{2}(a,b)=\frac{{{\beta^{2}}{2^{\frac{1}{\beta}}}}}{{\log(2)}}\frac{\partial}{{\partial\beta}}{(\frac{{{a^{\beta}}+{b^{\beta}}}}{2})^{\frac{1}{\beta}}}\\\
={(\frac{{{a^{\beta}}+{b^{\beta}}}}{{\log(2)}})^{\frac{1}{\beta}}}\left[{\frac{{{a^{\beta}}}}{{{a^{\beta}}+{b^{\beta}}}}\log(\frac{{2{a^{\beta}}}}{{{a^{\beta}}+{b^{\beta}}}})+\frac{{{b^{\beta}}}}{{{b^{\beta}}+{b^{\beta}}}}\log(\frac{{2{b^{\beta}}}}{{{a^{\beta}}+{b^{\beta}}}})}\right]\end{array}$
(15)

Corresponding PD kernel for the above can be taken by:

$k(a,b)=\frac{1}{2}(-{d^{2}}(a,b)+{d^{2}}(a,0)+{d^{2}}(b,0))$ (16)

##  4 Adaptive Explicit Kernel K-means

In this section, we present the _Adaptive Explicit Kernel K-means_ method for
homogeneous kernels. Homogeneous kernels have been introduced in the previous
section and a data-independent method [23] for deriving approximate finite-
dimensional feature maps was described. This class of kernels is a frequently-
used measure for histogram image comparison due to its effectiveness rather
than linear kernel(Euclidean distance). Using explicit feature maps alleviates
issues around the storing huge similarity matrix or repeated calculation of
kernel values. Moreover, we use the explicit feature map to take advantage of
accessing through the data in feature space. We expect good matching between
K-means model and real data structure of data in transformed space by choosing
suitable kernel, but we still have this chance to get better fit by extending
from the squared Euclidean to arbitrary weighted Minkowski metric in new
space. In addition to adding more flexibility, adaptive Minkowski metric
allows to fight against possible concentration in high-dimensional spaces.
Furthermore, weighting property with reflect to within-cluster feature
variances,enables our algorithm to cover spherical and non-spherical
(elliptical) structures. In this way, features with smaller within-cluster
variances receive a larger weight and features which more evenly distributed
across the cluster get a smaller weight.

###  4.1 Minkowski Metric

The distance we consider here is the Minkowski distance. It can be seen as a
generalization of the Euclidean distance, which is defined as below:

${d_{Mink}}({x_{i}},{x_{j}})=(\sum\limits_{l=1}^{D}{|x_{i}^{l}-x_{j}^{l}{|^{p}}{)^{\frac{1}{p}}}}$
(17)

Note that, when $0<p<1$ does not hold metrics properties therefore it is not
actual metric and the corresponding norm are not are indeed norms. They are
not satisfying the triangle inequality. They are usually called prenorms or
fractional norms.

####  4.1.1 Concentration in high-dimensional spaces

Losing discriminative power of Euclidean distance to indexing points in high-
dimensional spaces had been shown in the past. This means, as dimension
increases, the distance to the nearest point appear to be as the same as the
farthest one. This phenomena is known as concentration of distances. This
inability of Euclidean distance to distinguish distances in high dimensions
caused to alternative distance measures were suggested. In [13] a theoretical
analysis of absolute difference between the farthest point distance $d_{max}$
and the closest point distance $d_{min}$ for Minkowski norms is presented. It
point out that for D-dimensional i.i.d random vectors $x_{i},(1\leq i\leq N)$

$C\leq\mathop{\lim}\limits_{d\to\infty}{\rm{E}}(\frac{{{{\max}_{i}}||{x_{i}}|{|_{p}}-{{\min}_{i}}||{x_{i}}|{|_{p}}}}{{{D^{\frac{1}{p}-\frac{1}{2}}}}})\leq(N-1).C$
(18)

Where C is some constant independent of the distribution of the $x_{i}$. This
means the contrast between closest and farthest neighbor on average grows as
$D^{\frac{1}{p}-\frac{1}{2}}$. The authors concluded that $l_{1}$ and $l_{2}$
norm may be more relevant than $l_{p}$ when $p\geq 3$. In fact for $l_{p}$
with $(p\geq 3)$, the difference between to the farthest and nearest neighbor
goes to $0$ as dimensionality increases. These results encouraged researchers
to examine fractional distances, $l_{p}$ distance with $p\in(0,1)$. In [1]
authors extended previous work and proposed using fractional distance metric.
They has been shown that fractional distance can provide higher relative
contrast and meaningful result under same conditions as in (18).
The obtained results in [13] and [1] cannot be used in general when the data
are not uniformly distributed. Indeed, it is quite rare that data spread
through such spaces.observation of real data shows that high dimensional
spaces are mostly empty. In these spaces, it is common that data spread on a
sub-manifold. In [10] data distribution instead of data set has been studied.
For that purpose the relative variance ratio is proposed which is defined as
follows:

$R{V_{F,p}}=\frac{{\sqrt{Var(||x_{i}|{|_{p}})}}}{{E(||x_{i}|{|_{p}})}}$ (19)

Similarly to the relative contrast (18), the relative variance can be seen as
a measure of concentration. Smaller values of $R{V_{F,p}}$ indicate less
concentration. We can see that the shape of distribution $F$ and the value of
$p$ might affect the value of the $R{V_{F,p}}$. Therefore, for adjusting the
value of $p$, the shape of distribution should be considered too. It is
completely possible that the higher relative variance acquired by higher-order
norms.

####  4.1.2 Choosing the Optimal Value of p

In supervised learning tasks like classification, the value of $p$ could be
chosen to maximize model accuracy. However, we do not have that access to the
class labels in unsupervised learning. In that case, a sensible way of
choosing $p$ could be investigating on use of $R{V_{F,p}}$ as an objective to
be maximized. Also in [9] the relation between the phenomenon of concentration
and hubness has been studied. The authors proposed an unsupervised approach
for choosing the value of $p$ by measuring hub and anti-hub occurrence as
defined in the paper. Although $R{V_{F,p}}$ and hubness can give a sense of
the concentration level, they are not excellent measures all the time. As
shown in Figure (1), a different value of $p$ reflects a distinctive measure
of distance in a Euclidean space. It can be seen, the rotation of the
coordinate system will also lead to changes in the distance measurement. The
only exception is the circular shape, $p=2$ (Figure 1(d)). By adopting $p\to
0$ being exact in one dimension has more value than have balance in two
(Figure 1(a)). conversely, by approaching $p\to\infty$, just the maximum
difference is matter (Figure 1(f)). So by going far from $p=2$, the distance
meaning completely changes therefore the value of $p$ should be chosen by
considering both being meaningful and reduction of the concentration. However,
ground truth annotation is often costly and inaccessible in real-world
applications, there are usually limited available class labels. In this case,
semi-supervised learning provides a better choice by leveraging unlabeled data
by using a small set of labeled data. We discover the optimal value of
exponent $p$, by uncovering only a few percents of data. We employ the entire
data set, either labeled or not to run a series of clustering experiments at
various values of $p$ as reported superior results rather than using only
labeled data in [5].

(a) p=0.25

(b) p=1

(c) p=1.5

(d) p=2

(e) p=4

(f) p=$\infty$

Figure 1: Isosimilarity contour lines with eight different values of $p$ in
the Minkowski distance formula

####  4.1.3 Finding Centers

After deriving the approximate feature map and an optimal value for $p$ then
the kernel clustering algorithm can be started in order to optimize the
objective function with considering Minkowski distance. It minimizes the sum
of Minkowski distances between instances and the related centers. The
Minkowski k-means objective function can be written as below by applying
Minkowski distance on (4).

$\arg\mathop{{\rm{min}}}\limits_{M^{\prime},\delta}\sum\limits_{i=1}^{N}{\sum\limits_{k=1}^{K}{{\delta_{ik}}||{\Phi_{i}}-{{m}^{\prime}_{k}}|{|^{p}}}}$
(20)

It should be considered that, it is not possible to use the definition of the
center which previously used as Eq (2) since it does not minimize (20) when
$\delta$ is given constant. In the other words, For finding Minkowski’s
centers, we need to find value of vector $m^{\prime}_{k}$ for each cluster,
minimizes the following function:

$\arg\mathop{{\rm{min}}}\limits_{{{m}^{\prime}_{k}}}{\rm{}}{\sum\limits_{{\Phi_{i}}\in{C_{k}}}{|{\Phi_{i}}-{{m}^{\prime}_{k}}|}^{p}}$
(21)

Being minimizer of (21) is required for vector $m^{\prime}_{k}$ in order to
prove the convergence of Minkowski K-means to a local optimum. In other words,
it lowers the cost in each iteration monotonically. By this way, it is proven
that the algorithm will be converged in a finite number of iterations since
the algorithm iterates a function whose domain is a finite set and the cost is
decreased monotonically.
The search space in order to find vector $m^{\prime}_{k}$ is too large for an
exhaustive search because the algorithm needs accurate solutions. Note that
finding vector $m^{\prime}_{k}$ is a single objective problem and elements on
different dimensions are independent. Various evolutionary algorithms can be
designed to find the best solutions. However, for $p>1$ the Eq (21) is a
convex function and more desirable algorithms like steepest descent can be
used to find the global minimizer [5].

###  4.2 Weighted Version of Explicit Kernel K-means

Using the feature weighted version of Minkowski distance enables the algorithm
to provide a better fit to the cluster structures than is possible with
Minkowski K-means alone. It allows our algorithm to find both spherical and
non-spherical (elliptical) structural clusters in feature space; accordingly,
gives a much better fit to arbitrary shaped clusters in original spaces.
Authors in [5] have extended weighted K-means variants work [3, 14, 15]
through transforming feature weights to be feature scale. This means Minkowski
exponent is assigned to feature weights too. The objective function for the
weighted version of Minkowski K-means in Equation (20) can be written as the
following equation:

$\arg\mathop{{\rm{min}}}\limits_{M^{\prime},\delta,W}\sum\limits_{i=1}^{N}{\sum\limits_{k=1}^{K}{\sum\limits_{l=1}^{D^{\prime}}{{\delta_{ik}}{{({w_{kl}})}^{p}}||\Phi_{i}^{l}-m_{k}^{{}^{\prime}l}|{|^{p}}}}}$
(22)

The weight $w_{kl}$ reflects the relevance of feature $l$ at the cluster $k$.
They are assigned base on inverse proportion of dispersion of a feature within
a certain cluster. Thus, a feature with small dispersion within a specific
cluster would have a higher weight, and vice versa. More precisely, $w_{kl}$
is updated on each iteration as Equation (23), where
${V_{kl}}=\sum\nolimits_{i\in{c_{k}}}{|\varphi({x_{iv}})-{m_{kv}}{|^{p}}}$

${w_{kv}}=\frac{1}{{\sum\nolimits_{u\in
V}{{{(\frac{{{V_{kl}}}}{{{V_{ku}}}})}^{\frac{1}{{p-1}}}}}}}$ (23)

To see the entire algorithm process easily we have also the following
flowchart (Fig. 2):

Figure 2: The process flowchart of the Explicit kernel MWK-means algorithm

###  4.3 Accelerate Minkowski K-means clustering

The Minkowski’s center computation would significantly decelerate the running
time. We know already, that at $p=2$ the center is the mean, and at $p=1$ it
is given by the component-wise median. Otherwise, it requires an iterative,
steepest descent process or an evolutionary computation that can be considered
in computation costs. A significant computational speed-up can be achieved by
appropriate initialization in order to lessen the number of iterations and
consequently decrease the time of searching for Minkowski centers as a process
of minimization. We use an output vector containing cluster centers of K-means
with $p$ equal 2 or 1 as an initialization. This initialization approach
results in an impressive reduction in the time required for clustering,
because all of these Minkowski’s distances share the same properties and data
would be half-clustered. In [11], the effect of the initialization of K-Means
has been studied. The authors found when the clusters overlap, considering
various strategies for initialization would not matter much on the results.

##  5 Experimental Results

In this section, We present the results of performance experiments of our
Explicit Kernel MWK-means. We have conducted our experiments on four benchmark
data sets: USPS dataset, MNIST dataset, caltech101, and MSRC-V1. Some relevant
statistics of them are shown in Table 2.

Table 2: Description of the data sets | # instances | # features | # classes
---|---|---|---
MNIST Digits | 70,000 | 784 | 10
USPS Digits | 11,000 | 256 | 10
Caltech7 | 441 |  60,000 | 7
MSRC-V1 | 210 | 68160 | 7

Two standard metrics were used to measure the performance of the image
clustering that is, Normalized Mutual Information (NMI) and Purity.

###  5.1 Data Descriptions

MNIST Dataset introduced by Yann Lecun and Corinna Cortes. The dataset
includes a total of 70,000 samples of digits 0-9. Each sample consists of 28
by 28 pixels which are within the range [ 0,255 ].

USPS dataset includes 11,000 0-9 digits instances. The dataset is known to be
very complicated with a recorded human error rate of 2.5% . The images are 16
by 16 gray-scale pixels.

Caltech101 dataset comprises 9144 images of items belonging to 101 classes and
one background class. The number of images in each class differs. The size of
each image is approximately 300$\times$200 pixels. We have selected the
commonly used 7 groups, i.e. Face, Motorbike, Dolla-Bill, Garfield, Snoopy,
Stop-sign and Windsor-chair for our experiments.

MSRC-V1 dataset is from Microsoft Research in Cambridge. This data set is
commonly used for scene recognition. We adopt Lee and Grauman’s approach [18]
to refine and getting 7 classes including tree, building, airplane, cow, face,
car and bicycle where each class has 30 images.

###  5.2 Setting of the Experiments

To figure out which value of the exponent $p$ has the best performance, a
semi-supervised manner was employed. The value of $p$ was derived from
uncovering the class of labels on a 15% data sample though clustering was
conducted over the whole dataset. After running a series of clustering
experiments at different values of $p$, the $p$ that produced the higher
clustering accuracy was picked. Each of the experiments repeated 50 times and
the average NMI is reported.

Images clustering by using raw pixels as features is unlikely to work
effectively. The standard practice is to use visual descriptors such as HOG
and SIFT. Both of HOG and SIFT use histograms of pixel intensity gradients in
their descriptors. The class of homogeneous kernels are popular due to its
effectiveness. Hog can describe the shape and edge information of object; and
SIFT features are invariant to image scale, rotation, noise and illumination
changes. For MNIST and USPS databases the HOG feature is used. We choose
$4\times 4$ grid cells, $2\times 2$ cells in one block and 1 cell spacing as
the parameter in HOG calculation. For the two other datasets, Caltech7 and
MSRC-V1, the key-points are extracted from each image then represented each
keypoint as a 128-dimensional SIFT descriptor. A randomly chosen subset of
SIFT features were clustered in order to form a visual vocabulary for either
of dataset. Each SIFT descriptor was then quantified into a visual word by
considering the nearest cluster center. A 500-dimensional vector
representation for each image is obtained.

###  5.3 Comparison Results

Table 3 clearly shows the match between the value of $p$ learned in semi-
supervised and fully supervised settings. The only exception is on MNIST with
Using ${\chi^{2}}$ kernel that there is no match between learned and optimal
value of exponent $p$. However, table 4 shows it has still better performance
than common kernel K-means with the exponent $p=2$.

Table 3: The effect of $p$ on clustering accuracy results with the semi-supervised learning by revealing the class of labels on a 15% of data | Exponent p | NMI
---|---|---
| learned | Optimal | learned | Optimal
MNIST (intersection) | 1.83 | 1.87 | 0.7862 | 0.7981
MNIST (JS) | 1.7 | 1.7 | 0.7692 | 0.7692
MNIST(${\chi^{2}}$) | 1.7 | 3.1 | 0.7772 | 0.7862
USPS (intersection) | 1.66 | 1.5 | 0.7212 | 0.7420
USPS (JS) | 1.4 | 1.4 | 0.7353 | 0.7353
USPS (${\chi^{2}}$) | 1.4 | 1.2 | 0.7291 | 0.7510
Caltech7 (intersection) | 1.1 | 1 | 0.6558 | 0.6661
Caltech7 (JS) | 1.1 | 0.9 | 0.6203 | 0.6367
Caltech7 (${\chi^{2}}$) | 0.9 | 1.2 | 0.6547 | 0.6706
MSRC-V1 (intersection) | 1.22 | 1 | 0.7228 | 0.7438
MSRC-V1 (JS) | 1.2 | 1.1 | 0.7358 | 0.7559
MSRC-V1 (${\chi^{2}}$)_ | 1.3 | 1.1 | 0.7565 | 0.7646

We compared the clustering performance of our method (Explicit kernel MWK-
means) with their respective equal weighted version and constant Euclidean
one. In addition, we also compared the results of our method with the exact
kernel K-means clustering. Table tables 4, 5, 6 and 7 demonstrate the
clustering results in terms of NMI and purity on all data sets. It can be seen
that, compared with common exact kernel K-means counterparts, our proposed
Explicit kernel MWK-means improve the clustering performance on all the data
sets.

Table 4: MNIST Digits: Adaptive Explicit kernel MWK-means after using HOG descriptor |  | intersection | ${\chi^{2}}$ | JS
---|---|---|---|---
| dm. | NMI | Purity | NMI | Purity | NMI | Purity
Exact Kernel | – | 0.7101$\pm$0.024 | 0.7388$\pm$0.040 | 0.7109$\pm$0.029 | 0.7047$\pm$0.044 | 0.7121$\pm$0.031 | 0.7007$\pm$0.041
| Explicit Kernel
---
K-means
3 | 0.7138$\pm$0.023 | 0.7388$\pm$0.040 | 0.7186$\pm$0.028 | 0.7107$\pm$0.045 | 0.7148$\pm$0.033 | 0.7027$\pm$0.042
| Explicit kernel
---
MK-means
3 | 0.7362$\pm$020 | 0.7380$\pm$0.038 | 0.7192$\pm$0.029 | 0.6868$\pm$0.043 | 0.7272$\pm$0.035 | 0.7127$\pm$0.045
| Explicit kernel
---
MWK-means
3 |  0.7862$\pm$0.026 |  0.8048$\pm$0.039 |  0.7772$\pm$0.031 |  0.7909$\pm$0.047 |  0.7692$\pm$0.038 |  0.7935$\pm$0.045
Table 5: USPS Digits: Adaptive Explicit kernel MWK-means after using HOG descriptor |  | intersection | ${\chi^{2}}$ | JS
---|---|---|---|---
| dm. | NMI | Purity | NMI | Purity | NMI | Purity
Exact Kernel | – | 0.6567$\pm$0.040 | 0.6981$\pm$0.052 | 0.6695$\pm$0.046 | 0.6932$\pm$0.057 | 0.6506$\pm$0.034 | 0.6976$\pm$0.045
| Explicit Kernel
---
K-means
3 | 0.6591$\pm$0.048 | 0.6986$\pm$0.056 | 0.6631$\pm$0.045 | 0.7006$\pm$0.057 | 0.6621$\pm$0.036 | 0.6923$\pm$0.044
| Explicit kernel
---
MK-means
3 | 0.68962$\pm$044 | 0.7087$\pm$0.055 | 0.6953$\pm$0.049 | 0.7107$\pm$0.059 | 0.6991$\pm$0.037 | 0.7029$\pm$0.045
| Explicit kernel
---
MWK-means
3 |  0.7212$\pm$0.045 |  0.7578$\pm$0.055 |  0.7291$\pm$0.043 |  0.7658$\pm$0.050 |  0.7353$\pm$0.039 |  0.7643$\pm$0.047
Table 6: Caltech7: Adaptive Explicit kernel MWK-means after using SIFT descriptor |  | intersection | ${\chi^{2}}$ | JS
---|---|---|---|---
| dm. | NMI | Purity | NMI | Purity | NMI | Purity
Exact Kernel | – | 0.6064$\pm$0.018 | 0.6581$\pm$0.028 | 0.5918$\pm$0.014 | 0.6849$\pm$0.023 | 0.5649$\pm$0.019 | 0.6710$\pm$0.025
| Explicit Kernel
---
K-means
3 | 0.5961$\pm$0.019 | 0.6941$\pm$0.027 | 0.5981$\pm$0.014 | 0.6858$\pm$0.021 | 0.5919$\pm$0.017 | 0.6735$\pm$0.023
| Explicit kernel
---
MK-means
3 | 0.6158$\pm$0.021 | 0.6987$\pm$0.025 | 0.6147$\pm$0.025 | 0.6926$\pm$0.028 | 0.5803$\pm$0.018 | 0.6635$\pm$0.022
| Explicit kernel
---
MWK-means
3 |  0.6558$\pm$0.024 |  0.7581$\pm$0.028 |  0.6547$\pm$0.026 |  0.7419$\pm$0.029 |  0.6203$\pm$0.021 |  0.7159$\pm$0.025
Table 7: MSRC-v1: Adaptive Explicit kernel MWK-means after using SIFT descriptor |  | intersection | ${\chi^{2}}$ | JS
---|---|---|---|---
| dm. | NMI | Purity | NMI | Purity | NMI | Purity
Exact Kernel | – | 0.6509$\pm$0.024 | 0.7201$\pm$0.029 | 0.6889$\pm$0.028 | 0.7301$\pm$0.031 | 0.6859$\pm$0.034 | 0.7407$\pm$0.039
| Explicit Kernel
---
K-means
3 | 0.6504$\pm$0.022 | 0.7221$\pm$0.023 | 0.6927$\pm$0.030 | 0.7343$\pm$0.034 | 0.6893$\pm$0.035 | 0.7436$\pm$0.040
| Explicit kernel
---
MK-means
3 | 0.6728$\pm$0.027 | 0.7256$\pm$0.032 | 0.7058$\pm$0.032 | 0.7390$\pm$0.034 | 0.7065$\pm$0.033 | 0.7541$\pm$0.037
| Explicit kernel
---
MWK-means
3 |  0.7228$\pm$0.026 |  0.8105$\pm$0.030 |  0.7556$\pm$0.033 |  0.8269$\pm$0.036 |  0.7358$\pm$0.030 |  0.8254$\pm$0.034

##  6 Conclusions

In this paper, we proposed a kernel K-means method which is based on explicit
feature maps with further matching in feature space. Using adaptive Minkowski
metric and feature weighting in feature space enable our algorithm to get high
clustering quality and show strong robustness to noise feature and
concentration phenomena. Applying approximate finite-dimensional feature maps
were only discussed in the Support Vector Machines (SVM) problems before. We
suggested using this method as alleviates storing huge kernel matrix in
memory, in addition to access the data explicitly in feature space. Our
experiments demonstrate that our proposed method consistently achieves
superiors clustering performances in terms of two standard metrics Normalized
Mutual Information (NMI), and Purity, evaluated on four standard bench-mark
data sets on object and scene recognition.

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