PATE-GAN ($\epsilon=1$) | $0.139\scriptscriptstyle\pm\scriptstyle.002$ | $0.171\scriptscriptstyle\pm\scriptstyle.002$ | $0.173\scriptscriptstyle\pm\scriptstyle.002$ | $0.155\scriptscriptstyle\pm\scriptstyle.005$ | $0.215\scriptscriptstyle\pm\scriptstyle.004$ | $0.523\scriptscriptstyle\pm\scriptstyle.004$ | $0.089\scriptscriptstyle\pm\scriptstyle.004$ | $0.478\scriptscriptstyle\pm\scriptstyle.007$ | $0.353\scriptscriptstyle\pm\scriptstyle.009$ | $0.061\scriptscriptstyle\pm\scriptstyle.002$ | $0.386\scriptscriptstyle\pm\scriptstyle.011$ | $0.250\scriptscriptstyle\pm\scriptstyle.003$
TableDiffusion ($\epsilon=1$) | $0.179\scriptscriptstyle\pm\scriptstyle.002$ | $0.210\scriptscriptstyle\pm\scriptstyle.002$ | $0.121\scriptscriptstyle\pm\scriptstyle.002$ | $0.132\scriptscriptstyle\pm\scriptstyle.005$ | $0.390\scriptscriptstyle\pm\scriptstyle.004$ | $0.149\scriptscriptstyle\pm\scriptstyle.004$ | $0.338\scriptscriptstyle\pm\scriptstyle.005$ | $0.203\scriptscriptstyle\pm\scriptstyle.002$ | $0.226\scriptscriptstyle\pm\scriptstyle.007$ | $0.128\scriptscriptstyle\pm\scriptstyle.001$ | $0.366\scriptscriptstyle\pm\scriptstyle.008$ | $0.098\scriptscriptstyle\pm\scriptstyle.001$
HALF | $0.022\scriptscriptstyle\pm\scriptstyle.002$ | $0.023\scriptscriptstyle\pm\scriptstyle.002$ | $0.016\scriptscriptstyle\pm\scriptstyle.003$ | $0.011\scriptscriptstyle\pm\scriptstyle.003$ | $0.042\scriptscriptstyle\pm\scriptstyle.005$ | $0.015\scriptscriptstyle\pm\scriptstyle.003$ | $0.041\scriptscriptstyle\pm\scriptstyle.006$ | $0.023\scriptscriptstyle\pm\scriptstyle.005$ | $0.028\scriptscriptstyle\pm\scriptstyle.007$ | $0.010\scriptscriptstyle\pm\scriptstyle.002$ | $0.060\scriptscriptstyle\pm\scriptstyle.007$ | $0.014\scriptscriptstyle\pm\scriptstyle.001$
HISTOGRAM | $0.199\scriptscriptstyle\pm\scriptstyle.017$ | $0.101\scriptscriptstyle\pm\scriptstyle.001$ | $0.030\scriptscriptstyle\pm\scriptstyle.001$ | $0.048\scriptscriptstyle\pm\scriptstyle.002$ | $0.113\scriptscriptstyle\pm\scriptstyle.006$ | $0.080\scriptscriptstyle\pm\scriptstyle.003$ | $0.066\scriptscriptstyle\pm\scriptstyle.002$ | $0.065\scriptscriptstyle\pm\scriptstyle.002$ | $0.094\scriptscriptstyle\pm\scriptstyle.009$ | $0.059\scriptscriptstyle\pm\scriptstyle.006$ | $0.081\scriptscriptstyle\pm\scriptstyle.004$ | $0.076\scriptscriptstyle\pm\scriptstyle.001$

Table E18. Fidelity evaluation of one-way categorical marginals on train data
$\mathcal{D}_{\text{train}}$. The best result of each category is in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.015\scriptscriptstyle\pm\scriptstyle.002$ | $0.012\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.005\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.004\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.011\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.008\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.005\scriptscriptstyle\pm\scriptstyle.000}$ | $0.018\scriptscriptstyle\pm\scriptstyle.007$ | -
PrivSyn | $\mathbf{0.014\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.011\scriptscriptstyle\pm\scriptstyle.001}$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.007\scriptscriptstyle\pm\scriptstyle.005$ | $0.028\scriptscriptstyle\pm\scriptstyle.006$ | $0.016\scriptscriptstyle\pm\scriptstyle.005$ | $0.014\scriptscriptstyle\pm\scriptstyle.002$ | $0.048\scriptscriptstyle\pm\scriptstyle.026$ | $0.018\scriptscriptstyle\pm\scriptstyle.007$ | $0.005\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.016\scriptscriptstyle\pm\scriptstyle.002}$ | -
TVAE | $0.089\scriptscriptstyle\pm\scriptstyle.004$ | $0.193\scriptscriptstyle\pm\scriptstyle.001$ | $0.018\scriptscriptstyle\pm\scriptstyle.001$ | $0.012\scriptscriptstyle\pm\scriptstyle.006$ | $0.050\scriptscriptstyle\pm\scriptstyle.010$ | $0.055\scriptscriptstyle\pm\scriptstyle.010$ | $0.040\scriptscriptstyle\pm\scriptstyle.005$ | $0.058\scriptscriptstyle\pm\scriptstyle.000$ | $0.055\scriptscriptstyle\pm\scriptstyle.006$ | $0.084\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.009$ | -
CTGAN | $0.067\scriptscriptstyle\pm\scriptstyle.002$ | $0.067\scriptscriptstyle\pm\scriptstyle.002$ | $0.048\scriptscriptstyle\pm\scriptstyle.002$ | $0.265\scriptscriptstyle\pm\scriptstyle.011$ | $0.121\scriptscriptstyle\pm\scriptstyle.006$ | $0.207\scriptscriptstyle\pm\scriptstyle.008$ | $0.061\scriptscriptstyle\pm\scriptstyle.003$ | $0.136\scriptscriptstyle\pm\scriptstyle.006$ | $0.015\scriptscriptstyle\pm\scriptstyle.006$ | $0.036\scriptscriptstyle\pm\scriptstyle.002$ | $0.042\scriptscriptstyle\pm\scriptstyle.003$ | -
TabDDPM | $0.017\scriptscriptstyle\pm\scriptstyle.001$ | $0.014\scriptscriptstyle\pm\scriptstyle.002$ | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.005$ | $0.016\scriptscriptstyle\pm\scriptstyle.003$ | $0.017\scriptscriptstyle\pm\scriptstyle.003$ | $0.010\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.003$ | $0.013\scriptscriptstyle\pm\scriptstyle.006$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.018\scriptscriptstyle\pm\scriptstyle.006$ | -
GReaT | $0.056\scriptscriptstyle\pm\scriptstyle.003$ | $0.044\scriptscriptstyle\pm\scriptstyle.002$ | $0.058\scriptscriptstyle\pm\scriptstyle.001$ | $0.005\scriptscriptstyle\pm\scriptstyle.004$ | $0.034\scriptscriptstyle\pm\scriptstyle.008$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $0.038\scriptscriptstyle\pm\scriptstyle.004$ | $0.054\scriptscriptstyle\pm\scriptstyle.004$ | $0.018\scriptscriptstyle\pm\scriptstyle.007$ | - | $0.108\scriptscriptstyle\pm\scriptstyle.006$ | -
PGM ($\epsilon=1$) | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.019\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.009\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.039\scriptscriptstyle\pm\scriptstyle.006}$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.007}$ | $\mathbf{0.026\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.030\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.016\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.032\scriptscriptstyle\pm\scriptstyle.004}$ | -
PrivSyn ($\epsilon=1$) | $0.028\scriptscriptstyle\pm\scriptstyle.002$ | $0.041\scriptscriptstyle\pm\scriptstyle.006$ | $0.020\scriptscriptstyle\pm\scriptstyle.002$ | $0.041\scriptscriptstyle\pm\scriptstyle.005$ | $0.095\scriptscriptstyle\pm\scriptstyle.009$ | $0.105\scriptscriptstyle\pm\scriptstyle.012$ | $0.113\scriptscriptstyle\pm\scriptstyle.010$ | $0.069\scriptscriptstyle\pm\scriptstyle.035$ | $0.047\scriptscriptstyle\pm\scriptstyle.014$ | $0.029\scriptscriptstyle\pm\scriptstyle.002$ | $0.059\scriptscriptstyle\pm\scriptstyle.005$ | -
PATE-GAN ($\epsilon=1$) | $0.152\scriptscriptstyle\pm\scriptstyle.001$ | $0.189\scriptscriptstyle\pm\scriptstyle.002$ | $0.136\scriptscriptstyle\pm\scriptstyle.002$ | $0.176\scriptscriptstyle\pm\scriptstyle.009$ | $0.080\scriptscriptstyle\pm\scriptstyle.003$ | $0.630\scriptscriptstyle\pm\scriptstyle.007$ | $0.070\scriptscriptstyle\pm\scriptstyle.003$ | $0.494\scriptscriptstyle\pm\scriptstyle.003$ | $0.189\scriptscriptstyle\pm\scriptstyle.010$ | $0.044\scriptscriptstyle\pm\scriptstyle.002$ | $0.177\scriptscriptstyle\pm\scriptstyle.003$ | -
TableDiffusion ($\epsilon=1$) | $0.195\scriptscriptstyle\pm\scriptstyle.002$ | $0.243\scriptscriptstyle\pm\scriptstyle.002$ | $0.091\scriptscriptstyle\pm\scriptstyle.002$ | $0.131\scriptscriptstyle\pm\scriptstyle.006$ | $0.360\scriptscriptstyle\pm\scriptstyle.004$ | $0.147\scriptscriptstyle\pm\scriptstyle.008$ | $0.206\scriptscriptstyle\pm\scriptstyle.003$ | $0.140\scriptscriptstyle\pm\scriptstyle.002$ | $0.283\scriptscriptstyle\pm\scriptstyle.008$ | $0.103\scriptscriptstyle\pm\scriptstyle.001$ | $0.310\scriptscriptstyle\pm\scriptstyle.003$ | -
HALF | $0.019\scriptscriptstyle\pm\scriptstyle.002$ | $0.018\scriptscriptstyle\pm\scriptstyle.001$ | $0.008\scriptscriptstyle\pm\scriptstyle.002$ | $0.010\scriptscriptstyle\pm\scriptstyle.007$ | $0.014\scriptscriptstyle\pm\scriptstyle.002$ | $0.022\scriptscriptstyle\pm\scriptstyle.006$ | $0.012\scriptscriptstyle\pm\scriptstyle.003$ | $0.008\scriptscriptstyle\pm\scriptstyle.002$ | $0.015\scriptscriptstyle\pm\scriptstyle.006$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.021\scriptscriptstyle\pm\scriptstyle.004$ | -
HISTOGRAM | $0.017\scriptscriptstyle\pm\scriptstyle.002$ | $0.015\scriptscriptstyle\pm\scriptstyle.001$ | $0.008\scriptscriptstyle\pm\scriptstyle.001$ | $0.010\scriptscriptstyle\pm\scriptstyle.003$ | $0.008\scriptscriptstyle\pm\scriptstyle.002$ | $0.032\scriptscriptstyle\pm\scriptstyle.004$ | $0.008\scriptscriptstyle\pm\scriptstyle.001$ | $0.006\scriptscriptstyle\pm\scriptstyle.004$ | $0.008\scriptscriptstyle\pm\scriptstyle.002$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | -

Table E19. Fidelity evaluation of one-way numerical marginals on train data
$\mathcal{D}_{\text{train}}$. The best result of each category is in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.212\scriptscriptstyle\pm\scriptstyle.012$ | $0.097\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.037\scriptscriptstyle\pm\scriptstyle.000$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.039\scriptscriptstyle\pm\scriptstyle.001$ | $0.040\scriptscriptstyle\pm\scriptstyle.000$ | $0.034\scriptscriptstyle\pm\scriptstyle.000$ | $0.065\scriptscriptstyle\pm\scriptstyle.001$ | $0.029\scriptscriptstyle\pm\scriptstyle.000$ | $0.042\scriptscriptstyle\pm\scriptstyle.000$
PrivSyn | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | $0.023\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.010\scriptscriptstyle\pm\scriptstyle.000$ | $0.019\scriptscriptstyle\pm\scriptstyle.002$ | $0.026\scriptscriptstyle\pm\scriptstyle.000$ | $0.019\scriptscriptstyle\pm\scriptstyle.002$ | $0.029\scriptscriptstyle\pm\scriptstyle.001$ | $0.010\scriptscriptstyle\pm\scriptstyle.000$ | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.000}$ | $0.023\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.000$
TVAE | $0.024\scriptscriptstyle\pm\scriptstyle.000$ | $0.014\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.014\scriptscriptstyle\pm\scriptstyle.000$ | $0.020\scriptscriptstyle\pm\scriptstyle.001$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $0.026\scriptscriptstyle\pm\scriptstyle.002$ | $0.021\scriptscriptstyle\pm\scriptstyle.001$ | $0.017\scriptscriptstyle\pm\scriptstyle.001$ | $0.018\scriptscriptstyle\pm\scriptstyle.000$ | $0.024\scriptscriptstyle\pm\scriptstyle.004$ | $0.025\scriptscriptstyle\pm\scriptstyle.000$
CTGAN | $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.000}$ | $\mathbf{0.008\scriptscriptstyle\pm\scriptstyle.000}$ | - | $0.023\scriptscriptstyle\pm\scriptstyle.001$ | $0.039\scriptscriptstyle\pm\scriptstyle.001$ | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $0.021\scriptscriptstyle\pm\scriptstyle.000$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.036\scriptscriptstyle\pm\scriptstyle.001$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $0.062\scriptscriptstyle\pm\scriptstyle.003$ | $0.021\scriptscriptstyle\pm\scriptstyle.001$
TabDDPM | $0.007\scriptscriptstyle\pm\scriptstyle.001$ | $0.012\scriptscriptstyle\pm\scriptstyle.000$ | - | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.014\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.005\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.009\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.008\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.001}$ | $0.037\scriptscriptstyle\pm\scriptstyle.000$ | $0.013\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.000}$
GReaT | $0.012\scriptscriptstyle\pm\scriptstyle.001$ | $0.021\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $0.026\scriptscriptstyle\pm\scriptstyle.002$ | $0.010\scriptscriptstyle\pm\scriptstyle.001$ | $0.032\scriptscriptstyle\pm\scriptstyle.002$ | $0.026\scriptscriptstyle\pm\scriptstyle.001$ | $0.011\scriptscriptstyle\pm\scriptstyle.002$ | - | $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.002}$ | $0.013\scriptscriptstyle\pm\scriptstyle.001$
PGM ($\epsilon=1$) | $0.216\scriptscriptstyle\pm\scriptstyle.015$ | $0.098\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.035\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.040\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.002}$ | $0.043\scriptscriptstyle\pm\scriptstyle.000$ | $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.000}$ | $\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.001}$ | $0.067\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.031\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.000}$
PrivSyn ($\epsilon=1$) | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.001}$ | $0.044\scriptscriptstyle\pm\scriptstyle.002$ | - | $\mathbf{0.028\scriptscriptstyle\pm\scriptstyle.001}$ | $0.168\scriptscriptstyle\pm\scriptstyle.004$ | $0.072\scriptscriptstyle\pm\scriptstyle.003$ | $0.098\scriptscriptstyle\pm\scriptstyle.003$ | $0.129\scriptscriptstyle\pm\scriptstyle.004$ | $0.093\scriptscriptstyle\pm\scriptstyle.004$ | $0.056\scriptscriptstyle\pm\scriptstyle.001$ | $0.078\scriptscriptstyle\pm\scriptstyle.002$ | $0.103\scriptscriptstyle\pm\scriptstyle.003$
PATE-GAN ($\epsilon=1$) | $0.028\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.037\scriptscriptstyle\pm\scriptstyle.001}$ | - | $0.063\scriptscriptstyle\pm\scriptstyle.001$ | $0.150\scriptscriptstyle\pm\scriptstyle.000$ | $0.204\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.035\scriptscriptstyle\pm\scriptstyle.001}$ | $0.230\scriptscriptstyle\pm\scriptstyle.000$ | $0.236\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.040\scriptscriptstyle\pm\scriptstyle.000}$ | $0.324\scriptscriptstyle\pm\scriptstyle.001$ | $0.168\scriptscriptstyle\pm\scriptstyle.000$
TableDiffusion ($\epsilon=1$) | $0.046\scriptscriptstyle\pm\scriptstyle.001$ | $0.050\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.067\scriptscriptstyle\pm\scriptstyle.000$ | $0.126\scriptscriptstyle\pm\scriptstyle.000$ | $0.072\scriptscriptstyle\pm\scriptstyle.002$ | $0.244\scriptscriptstyle\pm\scriptstyle.002$ | $0.132\scriptscriptstyle\pm\scriptstyle.000$ | $0.090\scriptscriptstyle\pm\scriptstyle.001$ | $0.067\scriptscriptstyle\pm\scriptstyle.001$ | $0.156\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.001$
HALF | $0.004\scriptscriptstyle\pm\scriptstyle.000$ | $0.003\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.004\scriptscriptstyle\pm\scriptstyle.001$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.004\scriptscriptstyle\pm\scriptstyle.001$ | $0.008\scriptscriptstyle\pm\scriptstyle.001$ | $0.005\scriptscriptstyle\pm\scriptstyle.000$ | $0.004\scriptscriptstyle\pm\scriptstyle.001$ | $0.004\scriptscriptstyle\pm\scriptstyle.000$ | $0.009\scriptscriptstyle\pm\scriptstyle.002$ | $0.003\scriptscriptstyle\pm\scriptstyle.000$
HISTOGRAM | $0.226\scriptscriptstyle\pm\scriptstyle.016$ | $0.099\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.035\scriptscriptstyle\pm\scriptstyle.002$ | $0.037\scriptscriptstyle\pm\scriptstyle.000$ | $0.035\scriptscriptstyle\pm\scriptstyle.001$ | $0.039\scriptscriptstyle\pm\scriptstyle.000$ | $0.041\scriptscriptstyle\pm\scriptstyle.000$ | $0.034\scriptscriptstyle\pm\scriptstyle.000$ | $0.068\scriptscriptstyle\pm\scriptstyle.002$ | $0.029\scriptscriptstyle\pm\scriptstyle.001$ | $0.042\scriptscriptstyle\pm\scriptstyle.000$

Table E20. Fidelity evaluation of two-way categorical-categorical marginals on
train data $\mathcal{D}_{\text{train}}$. The best result of each category is
in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.049\scriptscriptstyle\pm\scriptstyle.001$ | $0.044\scriptscriptstyle\pm\scriptstyle.002$ | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.087\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.020\scriptscriptstyle\pm\scriptstyle.001$ | - | - | $0.025\scriptscriptstyle\pm\scriptstyle.001$ | $0.037\scriptscriptstyle\pm\scriptstyle.010$ | -
PrivSyn | $0.038\scriptscriptstyle\pm\scriptstyle.003$ | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$ | - | $0.151\scriptscriptstyle\pm\scriptstyle.016$ | - | $0.038\scriptscriptstyle\pm\scriptstyle.003$ | - | - | $0.022\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.035\scriptscriptstyle\pm\scriptstyle.005}$ | -
TVAE | $0.150\scriptscriptstyle\pm\scriptstyle.004$ | $0.336\scriptscriptstyle\pm\scriptstyle.001$ | $0.030\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.087\scriptscriptstyle\pm\scriptstyle.014$ | - | $0.076\scriptscriptstyle\pm\scriptstyle.006$ | - | - | $0.161\scriptscriptstyle\pm\scriptstyle.002$ | $0.135\scriptscriptstyle\pm\scriptstyle.010$ | -
CTGAN | $0.127\scriptscriptstyle\pm\scriptstyle.001$ | $0.138\scriptscriptstyle\pm\scriptstyle.001$ | $0.077\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.259\scriptscriptstyle\pm\scriptstyle.006$ | - | $0.126\scriptscriptstyle\pm\scriptstyle.003$ | - | - | $0.062\scriptscriptstyle\pm\scriptstyle.002$ | $0.101\scriptscriptstyle\pm\scriptstyle.005$ | -
TabDDPM | $\mathbf{0.036\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.030\scriptscriptstyle\pm\scriptstyle.002}$ | $0.019\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.032\scriptscriptstyle\pm\scriptstyle.004}$ | - | $\mathbf{0.019\scriptscriptstyle\pm\scriptstyle.001}$ | - | - | $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.001}$ | $0.037\scriptscriptstyle\pm\scriptstyle.007$ | -
GReaT | $0.092\scriptscriptstyle\pm\scriptstyle.003$ | $0.074\scriptscriptstyle\pm\scriptstyle.002$ | $0.094\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.074\scriptscriptstyle\pm\scriptstyle.007$ | - | $0.064\scriptscriptstyle\pm\scriptstyle.006$ | - | - | - | $0.205\scriptscriptstyle\pm\scriptstyle.006$ | -
PGM ($\epsilon=1$) | $0.081\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.074\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.038\scriptscriptstyle\pm\scriptstyle.001}$ | - | $\mathbf{0.154\scriptscriptstyle\pm\scriptstyle.003}$ | - | $\mathbf{0.066\scriptscriptstyle\pm\scriptstyle.001}$ | - | - | $\mathbf{0.027\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.112\scriptscriptstyle\pm\scriptstyle.002}$ | -
PrivSyn ($\epsilon=1$) | $\mathbf{0.077\scriptscriptstyle\pm\scriptstyle.003}$ | $0.118\scriptscriptstyle\pm\scriptstyle.007$ | $0.046\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.267\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.205\scriptscriptstyle\pm\scriptstyle.016$ | - | - | $0.066\scriptscriptstyle\pm\scriptstyle.003$ | $0.143\scriptscriptstyle\pm\scriptstyle.011$ | -
PATE-GAN ($\epsilon=1$) | $0.278\scriptscriptstyle\pm\scriptstyle.002$ | $0.347\scriptscriptstyle\pm\scriptstyle.001$ | $0.210\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.253\scriptscriptstyle\pm\scriptstyle.006$ | - | $0.129\scriptscriptstyle\pm\scriptstyle.005$ | - | - | $0.073\scriptscriptstyle\pm\scriptstyle.003$ | $0.273\scriptscriptstyle\pm\scriptstyle.005$ | -
TableDiffusion ($\epsilon=1$) | $0.323\scriptscriptstyle\pm\scriptstyle.002$ | $0.361\scriptscriptstyle\pm\scriptstyle.003$ | $0.155\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.620\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.348\scriptscriptstyle\pm\scriptstyle.004$ | - | - | $0.196\scriptscriptstyle\pm\scriptstyle.001$ | $0.466\scriptscriptstyle\pm\scriptstyle.003$ | -
HALF | $0.045\scriptscriptstyle\pm\scriptstyle.004$ | $0.040\scriptscriptstyle\pm\scriptstyle.002$ | $0.014\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.027\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.023\scriptscriptstyle\pm\scriptstyle.004$ | - | - | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.041\scriptscriptstyle\pm\scriptstyle.006$ | -
HISTOGRAM | $0.116\scriptscriptstyle\pm\scriptstyle.002$ | $0.072\scriptscriptstyle\pm\scriptstyle.002$ | $0.046\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.188\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.053\scriptscriptstyle\pm\scriptstyle.002$ | - | - | $0.027\scriptscriptstyle\pm\scriptstyle.002$ | $0.044\scriptscriptstyle\pm\scriptstyle.003$ | -

Table E21. Fidelity evaluation of two-way categorical-numerical marginals on
train data $\mathcal{D}_{\text{train}}$. The best result of each category is
in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.230\scriptscriptstyle\pm\scriptstyle.012$ | $0.110\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.038\scriptscriptstyle\pm\scriptstyle.002$ | $0.058\scriptscriptstyle\pm\scriptstyle.002$ | $0.045\scriptscriptstyle\pm\scriptstyle.003$ | $0.051\scriptscriptstyle\pm\scriptstyle.001$ | $0.049\scriptscriptstyle\pm\scriptstyle.004$ | $0.040\scriptscriptstyle\pm\scriptstyle.004$ | $0.072\scriptscriptstyle\pm\scriptstyle.001$ | $0.048\scriptscriptstyle\pm\scriptstyle.007$ | -
PrivSyn | $0.029\scriptscriptstyle\pm\scriptstyle.001$ | $0.036\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.018\scriptscriptstyle\pm\scriptstyle.005$ | $0.065\scriptscriptstyle\pm\scriptstyle.004$ | $0.045\scriptscriptstyle\pm\scriptstyle.005$ | $0.043\scriptscriptstyle\pm\scriptstyle.002$ | $0.100\scriptscriptstyle\pm\scriptstyle.022$ | $0.032\scriptscriptstyle\pm\scriptstyle.007$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.001}$ | $0.041\scriptscriptstyle\pm\scriptstyle.002$ | -
TVAE | $0.113\scriptscriptstyle\pm\scriptstyle.003$ | $0.207\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.027\scriptscriptstyle\pm\scriptstyle.005$ | $0.074\scriptscriptstyle\pm\scriptstyle.010$ | $0.077\scriptscriptstyle\pm\scriptstyle.010$ | $0.070\scriptscriptstyle\pm\scriptstyle.005$ | $0.082\scriptscriptstyle\pm\scriptstyle.001$ | $0.085\scriptscriptstyle\pm\scriptstyle.005$ | $0.103\scriptscriptstyle\pm\scriptstyle.002$ | $0.103\scriptscriptstyle\pm\scriptstyle.008$ | -
CTGAN | $0.077\scriptscriptstyle\pm\scriptstyle.002$ | $0.077\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.288\scriptscriptstyle\pm\scriptstyle.011$ | $0.165\scriptscriptstyle\pm\scriptstyle.007$ | $0.246\scriptscriptstyle\pm\scriptstyle.009$ | $0.092\scriptscriptstyle\pm\scriptstyle.003$ | $0.175\scriptscriptstyle\pm\scriptstyle.006$ | $0.059\scriptscriptstyle\pm\scriptstyle.006$ | $0.053\scriptscriptstyle\pm\scriptstyle.001$ | $0.111\scriptscriptstyle\pm\scriptstyle.004$ | -
TabDDPM | $\mathbf{0.025\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.027\scriptscriptstyle\pm\scriptstyle.002}$ | - | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.006}$ | $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.023\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.022\scriptscriptstyle\pm\scriptstyle.007}$ | $0.043\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.035\scriptscriptstyle\pm\scriptstyle.006}$ | -
GReaT | $0.068\scriptscriptstyle\pm\scriptstyle.002$ | $0.065\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.041\scriptscriptstyle\pm\scriptstyle.003$ | $0.064\scriptscriptstyle\pm\scriptstyle.008$ | $0.030\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.006$ | $0.084\scriptscriptstyle\pm\scriptstyle.005$ | $0.033\scriptscriptstyle\pm\scriptstyle.006$ | - | $0.120\scriptscriptstyle\pm\scriptstyle.006$ | -
PGM ($\epsilon=1$) | $0.243\scriptscriptstyle\pm\scriptstyle.017$ | $0.119\scriptscriptstyle\pm\scriptstyle.002$ | - | $\mathbf{0.042\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.099\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.060\scriptscriptstyle\pm\scriptstyle.006}$ | $\mathbf{0.081\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.080\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.059\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.075\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.088\scriptscriptstyle\pm\scriptstyle.001}$ | -
PrivSyn ($\epsilon=1$) | $\mathbf{0.055\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.089\scriptscriptstyle\pm\scriptstyle.005}$ | - | $0.070\scriptscriptstyle\pm\scriptstyle.005$ | $0.269\scriptscriptstyle\pm\scriptstyle.010$ | $0.220\scriptscriptstyle\pm\scriptstyle.008$ | $0.216\scriptscriptstyle\pm\scriptstyle.012$ | $0.212\scriptscriptstyle\pm\scriptstyle.037$ | $0.146\scriptscriptstyle\pm\scriptstyle.014$ | $0.086\scriptscriptstyle\pm\scriptstyle.003$ | $0.161\scriptscriptstyle\pm\scriptstyle.005$ | -
PATE-GAN ($\epsilon=1$) | $0.182\scriptscriptstyle\pm\scriptstyle.001$ | $0.228\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.240\scriptscriptstyle\pm\scriptstyle.009$ | $0.237\scriptscriptstyle\pm\scriptstyle.003$ | $0.837\scriptscriptstyle\pm\scriptstyle.009$ | $0.113\scriptscriptstyle\pm\scriptstyle.003$ | $0.724\scriptscriptstyle\pm\scriptstyle.004$ | $0.427\scriptscriptstyle\pm\scriptstyle.010$ | $0.085\scriptscriptstyle\pm\scriptstyle.002$ | $0.501\scriptscriptstyle\pm\scriptstyle.004$ | -
TableDiffusion ($\epsilon=1$) | $0.242\scriptscriptstyle\pm\scriptstyle.003$ | $0.293\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.197\scriptscriptstyle\pm\scriptstyle.006$ | $0.487\scriptscriptstyle\pm\scriptstyle.004$ | $0.226\scriptscriptstyle\pm\scriptstyle.008$ | $0.450\scriptscriptstyle\pm\scriptstyle.003$ | $0.274\scriptscriptstyle\pm\scriptstyle.002$ | $0.373\scriptscriptstyle\pm\scriptstyle.009$ | $0.171\scriptscriptstyle\pm\scriptstyle.000$ | $0.467\scriptscriptstyle\pm\scriptstyle.003$ | -
HALF | $0.025\scriptscriptstyle\pm\scriptstyle.002$ | $0.022\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.015\scriptscriptstyle\pm\scriptstyle.007$ | $0.021\scriptscriptstyle\pm\scriptstyle.002$ | $0.028\scriptscriptstyle\pm\scriptstyle.006$ | $0.021\scriptscriptstyle\pm\scriptstyle.004$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $0.020\scriptscriptstyle\pm\scriptstyle.005$ | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.033\scriptscriptstyle\pm\scriptstyle.005$ | -
HISTOGRAM | $0.253\scriptscriptstyle\pm\scriptstyle.015$ | $0.117\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.060\scriptscriptstyle\pm\scriptstyle.003$ | $0.078\scriptscriptstyle\pm\scriptstyle.001$ | $0.163\scriptscriptstyle\pm\scriptstyle.002$ | $0.064\scriptscriptstyle\pm\scriptstyle.001$ | $0.085\scriptscriptstyle\pm\scriptstyle.003$ | $0.094\scriptscriptstyle\pm\scriptstyle.001$ | $0.076\scriptscriptstyle\pm\scriptstyle.002$ | $0.061\scriptscriptstyle\pm\scriptstyle.001$ | -

Table E22. Fidelity evaluation of two-way numerical-numerical marginals on
train data $\mathcal{D}_{\text{train}}$. The best result of each category is
in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.425\scriptscriptstyle\pm\scriptstyle.023$ | $0.195\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.070\scriptscriptstyle\pm\scriptstyle.001$ | $0.083\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.001$ | $0.085\scriptscriptstyle\pm\scriptstyle.001$ | $0.103\scriptscriptstyle\pm\scriptstyle.000$ | $0.069\scriptscriptstyle\pm\scriptstyle.001$ | $0.135\scriptscriptstyle\pm\scriptstyle.002$ | $0.060\scriptscriptstyle\pm\scriptstyle.001$ | $0.090\scriptscriptstyle\pm\scriptstyle.001$
PrivSyn | $0.027\scriptscriptstyle\pm\scriptstyle.001$ | $0.048\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.024\scriptscriptstyle\pm\scriptstyle.000$ | $0.057\scriptscriptstyle\pm\scriptstyle.002$ | $0.054\scriptscriptstyle\pm\scriptstyle.000$ | $0.057\scriptscriptstyle\pm\scriptstyle.002$ | $0.084\scriptscriptstyle\pm\scriptstyle.001$ | $0.036\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.031\scriptscriptstyle\pm\scriptstyle.000}$ | $0.052\scriptscriptstyle\pm\scriptstyle.001$ | $0.026\scriptscriptstyle\pm\scriptstyle.000$
TVAE | $0.047\scriptscriptstyle\pm\scriptstyle.001$ | $0.028\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.032\scriptscriptstyle\pm\scriptstyle.001$ | $0.045\scriptscriptstyle\pm\scriptstyle.002$ | $0.039\scriptscriptstyle\pm\scriptstyle.002$ | $0.061\scriptscriptstyle\pm\scriptstyle.002$ | $0.050\scriptscriptstyle\pm\scriptstyle.001$ | $0.037\scriptscriptstyle\pm\scriptstyle.001$ | $0.037\scriptscriptstyle\pm\scriptstyle.001$ | $0.055\scriptscriptstyle\pm\scriptstyle.007$ | $0.052\scriptscriptstyle\pm\scriptstyle.000$
CTGAN | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.018\scriptscriptstyle\pm\scriptstyle.000}$ | - | $0.051\scriptscriptstyle\pm\scriptstyle.002$ | $0.084\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.001$ | $0.058\scriptscriptstyle\pm\scriptstyle.001$ | $0.080\scriptscriptstyle\pm\scriptstyle.001$ | $0.088\scriptscriptstyle\pm\scriptstyle.002$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.134\scriptscriptstyle\pm\scriptstyle.005$ | $0.044\scriptscriptstyle\pm\scriptstyle.001$
TabDDPM | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$ | $0.026\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.026\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.023\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.016\scriptscriptstyle\pm\scriptstyle.002}$ | $0.076\scriptscriptstyle\pm\scriptstyle.001$ | $0.035\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$
GReaT | $0.024\scriptscriptstyle\pm\scriptstyle.002$ | $0.042\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.067\scriptscriptstyle\pm\scriptstyle.003$ | $0.055\scriptscriptstyle\pm\scriptstyle.004$ | $0.023\scriptscriptstyle\pm\scriptstyle.001$ | $0.068\scriptscriptstyle\pm\scriptstyle.004$ | $0.055\scriptscriptstyle\pm\scriptstyle.002$ | $0.025\scriptscriptstyle\pm\scriptstyle.004$ | - | $\mathbf{0.028\scriptscriptstyle\pm\scriptstyle.002}$ | $0.026\scriptscriptstyle\pm\scriptstyle.002$
PGM ($\epsilon=1$) | $0.432\scriptscriptstyle\pm\scriptstyle.030$ | $0.203\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.084\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.098\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.079\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.099\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.109\scriptscriptstyle\pm\scriptstyle.000}$ | $\mathbf{0.099\scriptscriptstyle\pm\scriptstyle.001}$ | $0.137\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.090\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.092\scriptscriptstyle\pm\scriptstyle.001}$
PrivSyn ($\epsilon=1$) | $\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.002}$ | $0.091\scriptscriptstyle\pm\scriptstyle.003$ | - | $\mathbf{0.068\scriptscriptstyle\pm\scriptstyle.001}$ | $0.340\scriptscriptstyle\pm\scriptstyle.009$ | $0.171\scriptscriptstyle\pm\scriptstyle.005$ | $0.205\scriptscriptstyle\pm\scriptstyle.005$ | $0.268\scriptscriptstyle\pm\scriptstyle.007$ | $0.221\scriptscriptstyle\pm\scriptstyle.006$ | $0.115\scriptscriptstyle\pm\scriptstyle.002$ | $0.176\scriptscriptstyle\pm\scriptstyle.005$ | $0.209\scriptscriptstyle\pm\scriptstyle.005$
PATE-GAN ($\epsilon=1$) | $0.056\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.078\scriptscriptstyle\pm\scriptstyle.002}$ | - | $0.131\scriptscriptstyle\pm\scriptstyle.001$ | $0.302\scriptscriptstyle\pm\scriptstyle.001$ | $0.409\scriptscriptstyle\pm\scriptstyle.005$ | $0.082\scriptscriptstyle\pm\scriptstyle.002$ | $0.461\scriptscriptstyle\pm\scriptstyle.001$ | $0.474\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.082\scriptscriptstyle\pm\scriptstyle.001}$ | $0.649\scriptscriptstyle\pm\scriptstyle.002$ | $0.335\scriptscriptstyle\pm\scriptstyle.000$
TableDiffusion ($\epsilon=1$) | $0.092\scriptscriptstyle\pm\scriptstyle.001$ | $0.099\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.133\scriptscriptstyle\pm\scriptstyle.001$ | $0.252\scriptscriptstyle\pm\scriptstyle.001$ | $0.147\scriptscriptstyle\pm\scriptstyle.003$ | $0.488\scriptscriptstyle\pm\scriptstyle.004$ | $0.265\scriptscriptstyle\pm\scriptstyle.001$ | $0.181\scriptscriptstyle\pm\scriptstyle.002$ | $0.136\scriptscriptstyle\pm\scriptstyle.001$ | $0.314\scriptscriptstyle\pm\scriptstyle.001$ | $0.145\scriptscriptstyle\pm\scriptstyle.001$
HALF | $0.009\scriptscriptstyle\pm\scriptstyle.001$ | $0.007\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | $0.014\scriptscriptstyle\pm\scriptstyle.001$ | $0.012\scriptscriptstyle\pm\scriptstyle.001$ | $0.021\scriptscriptstyle\pm\scriptstyle.002$ | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.002$ | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.025\scriptscriptstyle\pm\scriptstyle.003$ | $0.008\scriptscriptstyle\pm\scriptstyle.000$
HISTOGRAM | $0.453\scriptscriptstyle\pm\scriptstyle.031$ | $0.205\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.098\scriptscriptstyle\pm\scriptstyle.004$ | $0.096\scriptscriptstyle\pm\scriptstyle.001$ | $0.120\scriptscriptstyle\pm\scriptstyle.002$ | $0.093\scriptscriptstyle\pm\scriptstyle.001$ | $0.113\scriptscriptstyle\pm\scriptstyle.000$ | $0.142\scriptscriptstyle\pm\scriptstyle.000$ | $0.140\scriptscriptstyle\pm\scriptstyle.003$ | $0.080\scriptscriptstyle\pm\scriptstyle.002$ | $0.094\scriptscriptstyle\pm\scriptstyle.000$

Table E23. Fidelity evaluation of one-way categorical marginals on test data
$\mathcal{D}_{\text{test}}$. The best result of each category is in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.023\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.012\scriptscriptstyle\pm\scriptstyle.001}$ | $0.009\scriptscriptstyle\pm\scriptstyle.004$ | $0.044\scriptscriptstyle\pm\scriptstyle.007$ | $\mathbf{0.017\scriptscriptstyle\pm\scriptstyle.004}$ | $0.031\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.012\scriptscriptstyle\pm\scriptstyle.007}$ | $0.030\scriptscriptstyle\pm\scriptstyle.011$ | $\mathbf{0.005\scriptscriptstyle\pm\scriptstyle.001}$ | $0.062\scriptscriptstyle\pm\scriptstyle.012$ | -
PrivSyn | $0.015\scriptscriptstyle\pm\scriptstyle.001$ | $0.023\scriptscriptstyle\pm\scriptstyle.004$ | $0.012\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.007\scriptscriptstyle\pm\scriptstyle.004}$ | $0.040\scriptscriptstyle\pm\scriptstyle.004$ | $0.018\scriptscriptstyle\pm\scriptstyle.006$ | $0.031\scriptscriptstyle\pm\scriptstyle.007$ | $0.016\scriptscriptstyle\pm\scriptstyle.005$ | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.005}$ | $0.008\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.057\scriptscriptstyle\pm\scriptstyle.009}$ | -
TVAE | $0.090\scriptscriptstyle\pm\scriptstyle.003$ | $0.191\scriptscriptstyle\pm\scriptstyle.003$ | $0.021\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.005$ | $0.083\scriptscriptstyle\pm\scriptstyle.024$ | $0.060\scriptscriptstyle\pm\scriptstyle.007$ | $0.036\scriptscriptstyle\pm\scriptstyle.003$ | $0.062\scriptscriptstyle\pm\scriptstyle.012$ | $0.049\scriptscriptstyle\pm\scriptstyle.009$ | $0.084\scriptscriptstyle\pm\scriptstyle.001$ | $0.115\scriptscriptstyle\pm\scriptstyle.009$ | -
CTGAN | $0.069\scriptscriptstyle\pm\scriptstyle.004$ | $0.065\scriptscriptstyle\pm\scriptstyle.002$ | $0.054\scriptscriptstyle\pm\scriptstyle.001$ | $0.251\scriptscriptstyle\pm\scriptstyle.008$ | $0.102\scriptscriptstyle\pm\scriptstyle.011$ | $0.205\scriptscriptstyle\pm\scriptstyle.008$ | $0.073\scriptscriptstyle\pm\scriptstyle.005$ | $0.135\scriptscriptstyle\pm\scriptstyle.014$ | $0.028\scriptscriptstyle\pm\scriptstyle.009$ | $0.036\scriptscriptstyle\pm\scriptstyle.001$ | $0.068\scriptscriptstyle\pm\scriptstyle.010$ | -
TabDDPM | $0.017\scriptscriptstyle\pm\scriptstyle.001$ | $0.025\scriptscriptstyle\pm\scriptstyle.001$ | $0.014\scriptscriptstyle\pm\scriptstyle.001$ | $0.009\scriptscriptstyle\pm\scriptstyle.003$ | $0.055\scriptscriptstyle\pm\scriptstyle.011$ | $\mathbf{0.017\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.029\scriptscriptstyle\pm\scriptstyle.004}$ | $0.022\scriptscriptstyle\pm\scriptstyle.006$ | $0.033\scriptscriptstyle\pm\scriptstyle.022$ | $0.006\scriptscriptstyle\pm\scriptstyle.001$ | $0.059\scriptscriptstyle\pm\scriptstyle.010$ | -
GReaT | $0.057\scriptscriptstyle\pm\scriptstyle.002$ | $0.053\scriptscriptstyle\pm\scriptstyle.004$ | $0.055\scriptscriptstyle\pm\scriptstyle.001$ | $0.008\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.031\scriptscriptstyle\pm\scriptstyle.007}$ | $0.018\scriptscriptstyle\pm\scriptstyle.007$ | $0.038\scriptscriptstyle\pm\scriptstyle.008$ | $0.058\scriptscriptstyle\pm\scriptstyle.010$ | $0.032\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.112\scriptscriptstyle\pm\scriptstyle.010$ | -
PGM ($\epsilon=1$) | $\mathbf{0.016\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.025\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.000}$ | $0.011\scriptscriptstyle\pm\scriptstyle.007$ | $\mathbf{0.066\scriptscriptstyle\pm\scriptstyle.007}$ | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.015}$ | $0.051\scriptscriptstyle\pm\scriptstyle.016$ | $0.007\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.053\scriptscriptstyle\pm\scriptstyle.006}$ | -
PrivSyn ($\epsilon=1$) | $0.032\scriptscriptstyle\pm\scriptstyle.003$ | $0.051\scriptscriptstyle\pm\scriptstyle.002$ | $0.018\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.003\scriptscriptstyle\pm\scriptstyle.002}$ | $0.138\scriptscriptstyle\pm\scriptstyle.025$ | $0.077\scriptscriptstyle\pm\scriptstyle.016$ | $0.085\scriptscriptstyle\pm\scriptstyle.012$ | $0.086\scriptscriptstyle\pm\scriptstyle.016$ | $\mathbf{0.029\scriptscriptstyle\pm\scriptstyle.015}$ | $\mathbf{0.012\scriptscriptstyle\pm\scriptstyle.001}$ | $0.104\scriptscriptstyle\pm\scriptstyle.009$ | -
PATE-GAN ($\epsilon=1$) | $0.152\scriptscriptstyle\pm\scriptstyle.003$ | $0.184\scriptscriptstyle\pm\scriptstyle.002$ | $0.136\scriptscriptstyle\pm\scriptstyle.002$ | $0.183\scriptscriptstyle\pm\scriptstyle.009$ | $0.079\scriptscriptstyle\pm\scriptstyle.003$ | $0.636\scriptscriptstyle\pm\scriptstyle.005$ | $0.073\scriptscriptstyle\pm\scriptstyle.004$ | $0.496\scriptscriptstyle\pm\scriptstyle.012$ | $0.204\scriptscriptstyle\pm\scriptstyle.018$ | $0.042\scriptscriptstyle\pm\scriptstyle.001$ | $0.152\scriptscriptstyle\pm\scriptstyle.007$ | -
TableDiffusion ($\epsilon=1$) | $0.194\scriptscriptstyle\pm\scriptstyle.002$ | $0.247\scriptscriptstyle\pm\scriptstyle.002$ | $0.089\scriptscriptstyle\pm\scriptstyle.002$ | $0.133\scriptscriptstyle\pm\scriptstyle.008$ | $0.371\scriptscriptstyle\pm\scriptstyle.004$ | $0.150\scriptscriptstyle\pm\scriptstyle.007$ | $0.194\scriptscriptstyle\pm\scriptstyle.004$ | $0.140\scriptscriptstyle\pm\scriptstyle.003$ | $0.254\scriptscriptstyle\pm\scriptstyle.010$ | $0.102\scriptscriptstyle\pm\scriptstyle.000$ | $0.320\scriptscriptstyle\pm\scriptstyle.007$ | -
HALF | $0.020\scriptscriptstyle\pm\scriptstyle.003$ | $0.023\scriptscriptstyle\pm\scriptstyle.003$ | $0.012\scriptscriptstyle\pm\scriptstyle.003$ | $0.011\scriptscriptstyle\pm\scriptstyle.005$ | $0.037\scriptscriptstyle\pm\scriptstyle.007$ | $0.019\scriptscriptstyle\pm\scriptstyle.005$ | $0.030\scriptscriptstyle\pm\scriptstyle.005$ | $0.018\scriptscriptstyle\pm\scriptstyle.009$ | $0.028\scriptscriptstyle\pm\scriptstyle.008$ | $0.008\scriptscriptstyle\pm\scriptstyle.002$ | $0.048\scriptscriptstyle\pm\scriptstyle.011$ | -
HISTOGRAM | $0.017\scriptscriptstyle\pm\scriptstyle.002$ | $0.020\scriptscriptstyle\pm\scriptstyle.001$ | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.011\scriptscriptstyle\pm\scriptstyle.003$ | $0.046\scriptscriptstyle\pm\scriptstyle.010$ | $0.017\scriptscriptstyle\pm\scriptstyle.005$ | $0.025\scriptscriptstyle\pm\scriptstyle.001$ | $0.012\scriptscriptstyle\pm\scriptstyle.004$ | $0.038\scriptscriptstyle\pm\scriptstyle.020$ | $0.009\scriptscriptstyle\pm\scriptstyle.001$ | $0.047\scriptscriptstyle\pm\scriptstyle.003$ | -

Table E24. Fidelity evaluation of one-way numerical marginals on test data
$\mathcal{D}_{\text{test}}$. The best result of each category is in bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.195\scriptscriptstyle\pm\scriptstyle.004$ | $0.095\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.033\scriptscriptstyle\pm\scriptstyle.000$ | $0.054\scriptscriptstyle\pm\scriptstyle.003$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.043\scriptscriptstyle\pm\scriptstyle.001$ | $0.042\scriptscriptstyle\pm\scriptstyle.000$ | $0.046\scriptscriptstyle\pm\scriptstyle.003$ | $0.052\scriptscriptstyle\pm\scriptstyle.004$ | $0.036\scriptscriptstyle\pm\scriptstyle.002$ | $0.048\scriptscriptstyle\pm\scriptstyle.001$
PrivSyn | $0.013\scriptscriptstyle\pm\scriptstyle.000$ | $0.024\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.011\scriptscriptstyle\pm\scriptstyle.000$ | $0.032\scriptscriptstyle\pm\scriptstyle.005$ | $0.027\scriptscriptstyle\pm\scriptstyle.000$ | $0.027\scriptscriptstyle\pm\scriptstyle.002$ | $0.031\scriptscriptstyle\pm\scriptstyle.003$ | $0.019\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.011\scriptscriptstyle\pm\scriptstyle.000}$ | $0.031\scriptscriptstyle\pm\scriptstyle.003$ | $0.012\scriptscriptstyle\pm\scriptstyle.001$
TVAE | $0.025\scriptscriptstyle\pm\scriptstyle.000$ | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.014\scriptscriptstyle\pm\scriptstyle.000$ | $0.028\scriptscriptstyle\pm\scriptstyle.003$ | $0.017\scriptscriptstyle\pm\scriptstyle.001$ | $0.033\scriptscriptstyle\pm\scriptstyle.002$ | $0.024\scriptscriptstyle\pm\scriptstyle.001$ | $0.016\scriptscriptstyle\pm\scriptstyle.001$ | $0.017\scriptscriptstyle\pm\scriptstyle.001$ | $0.033\scriptscriptstyle\pm\scriptstyle.005$ | $0.029\scriptscriptstyle\pm\scriptstyle.001$
CTGAN | $0.009\scriptscriptstyle\pm\scriptstyle.000$ | $\mathbf{0.009\scriptscriptstyle\pm\scriptstyle.001}$ | - | $0.024\scriptscriptstyle\pm\scriptstyle.000$ | $0.047\scriptscriptstyle\pm\scriptstyle.002$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.027\scriptscriptstyle\pm\scriptstyle.001$ | $0.037\scriptscriptstyle\pm\scriptstyle.003$ | $0.046\scriptscriptstyle\pm\scriptstyle.001$ | $0.016\scriptscriptstyle\pm\scriptstyle.000$ | $0.072\scriptscriptstyle\pm\scriptstyle.003$ | $0.022\scriptscriptstyle\pm\scriptstyle.001$
TabDDPM | $\mathbf{0.008\scriptscriptstyle\pm\scriptstyle.000}$ | $0.012\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.006\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.021\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.013\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.017\scriptscriptstyle\pm\scriptstyle.004}$ | $0.034\scriptscriptstyle\pm\scriptstyle.000$ | $0.024\scriptscriptstyle\pm\scriptstyle.006$ | $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.001}$
GReaT | $0.013\scriptscriptstyle\pm\scriptstyle.000$ | $0.021\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.036\scriptscriptstyle\pm\scriptstyle.002$ | $0.009\scriptscriptstyle\pm\scriptstyle.000$ | $0.039\scriptscriptstyle\pm\scriptstyle.004$ | $0.027\scriptscriptstyle\pm\scriptstyle.002$ | $0.019\scriptscriptstyle\pm\scriptstyle.003$ | - | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.006}$ | $0.015\scriptscriptstyle\pm\scriptstyle.001$
PGM ($\epsilon=1$) | $0.196\scriptscriptstyle\pm\scriptstyle.004$ | $0.094\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.055\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.034\scriptscriptstyle\pm\scriptstyle.001}$ | $0.045\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.043\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.046\scriptscriptstyle\pm\scriptstyle.002}$ | $0.053\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.037\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.047\scriptscriptstyle\pm\scriptstyle.001}$
PrivSyn ($\epsilon=1$) | $\mathbf{0.024\scriptscriptstyle\pm\scriptstyle.000}$ | $0.044\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.026\scriptscriptstyle\pm\scriptstyle.002}$ | $0.146\scriptscriptstyle\pm\scriptstyle.006$ | $0.069\scriptscriptstyle\pm\scriptstyle.004$ | $0.106\scriptscriptstyle\pm\scriptstyle.005$ | $0.128\scriptscriptstyle\pm\scriptstyle.004$ | $0.086\scriptscriptstyle\pm\scriptstyle.002$ | $0.040\scriptscriptstyle\pm\scriptstyle.001$ | $0.079\scriptscriptstyle\pm\scriptstyle.007$ | $0.085\scriptscriptstyle\pm\scriptstyle.003$
PATE-GAN ($\epsilon=1$) | $0.027\scriptscriptstyle\pm\scriptstyle.000$ | $\mathbf{0.036\scriptscriptstyle\pm\scriptstyle.001}$ | - | $0.063\scriptscriptstyle\pm\scriptstyle.001$ | $0.166\scriptscriptstyle\pm\scriptstyle.003$ | $0.204\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.003}$ | $0.231\scriptscriptstyle\pm\scriptstyle.001$ | $0.250\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.037\scriptscriptstyle\pm\scriptstyle.002}$ | $0.341\scriptscriptstyle\pm\scriptstyle.009$ | $0.167\scriptscriptstyle\pm\scriptstyle.002$
TableDiffusion ($\epsilon=1$) | $0.046\scriptscriptstyle\pm\scriptstyle.000$ | $0.048\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.066\scriptscriptstyle\pm\scriptstyle.001$ | $0.146\scriptscriptstyle\pm\scriptstyle.001$ | $0.072\scriptscriptstyle\pm\scriptstyle.001$ | $0.243\scriptscriptstyle\pm\scriptstyle.002$ | $0.132\scriptscriptstyle\pm\scriptstyle.000$ | $0.099\scriptscriptstyle\pm\scriptstyle.002$ | $0.060\scriptscriptstyle\pm\scriptstyle.001$ | $0.174\scriptscriptstyle\pm\scriptstyle.005$ | $0.065\scriptscriptstyle\pm\scriptstyle.001$
HALF | $0.005\scriptscriptstyle\pm\scriptstyle.001$ | $0.003\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.004\scriptscriptstyle\pm\scriptstyle.001$ | $0.015\scriptscriptstyle\pm\scriptstyle.002$ | $0.004\scriptscriptstyle\pm\scriptstyle.001$ | $0.018\scriptscriptstyle\pm\scriptstyle.003$ | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.013\scriptscriptstyle\pm\scriptstyle.003$ | $0.005\scriptscriptstyle\pm\scriptstyle.001$ | $0.018\scriptscriptstyle\pm\scriptstyle.003$ | $0.007\scriptscriptstyle\pm\scriptstyle.001$
HISTOGRAM | $0.208\scriptscriptstyle\pm\scriptstyle.020$ | $0.094\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.033\scriptscriptstyle\pm\scriptstyle.000$ | $0.051\scriptscriptstyle\pm\scriptstyle.001$ | $0.034\scriptscriptstyle\pm\scriptstyle.000$ | $0.043\scriptscriptstyle\pm\scriptstyle.001$ | $0.042\scriptscriptstyle\pm\scriptstyle.000$ | $0.047\scriptscriptstyle\pm\scriptstyle.002$ | $0.061\scriptscriptstyle\pm\scriptstyle.007$ | $0.039\scriptscriptstyle\pm\scriptstyle.004$ | $0.048\scriptscriptstyle\pm\scriptstyle.001$

Table E25. Fidelity evaluation of two-way categorical-categorical marginals on
test data $\mathcal{D}_{\text{test}}$. The best result of each category is in
bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.048\scriptscriptstyle\pm\scriptstyle.003$ | $0.063\scriptscriptstyle\pm\scriptstyle.002$ | $0.041\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.122\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.059\scriptscriptstyle\pm\scriptstyle.004$ | - | - | $0.025\scriptscriptstyle\pm\scriptstyle.001$ | $0.122\scriptscriptstyle\pm\scriptstyle.021$ | -
PrivSyn | $\mathbf{0.038\scriptscriptstyle\pm\scriptstyle.001}$ | $0.058\scriptscriptstyle\pm\scriptstyle.004$ | $\mathbf{0.023\scriptscriptstyle\pm\scriptstyle.002}$ | - | $0.150\scriptscriptstyle\pm\scriptstyle.010$ | - | $0.064\scriptscriptstyle\pm\scriptstyle.009$ | - | - | $0.024\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.107\scriptscriptstyle\pm\scriptstyle.008}$ | -
TVAE | $0.153\scriptscriptstyle\pm\scriptstyle.003$ | $0.333\scriptscriptstyle\pm\scriptstyle.003$ | $0.035\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.118\scriptscriptstyle\pm\scriptstyle.025$ | - | $0.072\scriptscriptstyle\pm\scriptstyle.002$ | - | - | $0.161\scriptscriptstyle\pm\scriptstyle.001$ | $0.203\scriptscriptstyle\pm\scriptstyle.012$ | -
CTGAN | $0.128\scriptscriptstyle\pm\scriptstyle.005$ | $0.137\scriptscriptstyle\pm\scriptstyle.002$ | $0.083\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.256\scriptscriptstyle\pm\scriptstyle.005$ | - | $0.147\scriptscriptstyle\pm\scriptstyle.008$ | - | - | $0.063\scriptscriptstyle\pm\scriptstyle.002$ | $0.132\scriptscriptstyle\pm\scriptstyle.009$ | -
TabDDPM | $0.039\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.054\scriptscriptstyle\pm\scriptstyle.001}$ | $0.025\scriptscriptstyle\pm\scriptstyle.002$ | - | $\mathbf{0.083\scriptscriptstyle\pm\scriptstyle.010}$ | - | $\mathbf{0.055\scriptscriptstyle\pm\scriptstyle.005}$ | - | - | $\mathbf{0.010\scriptscriptstyle\pm\scriptstyle.002}$ | $0.113\scriptscriptstyle\pm\scriptstyle.010$ | -
GReaT | $0.094\scriptscriptstyle\pm\scriptstyle.002$ | $0.091\scriptscriptstyle\pm\scriptstyle.005$ | $0.089\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.092\scriptscriptstyle\pm\scriptstyle.017$ | - | $0.067\scriptscriptstyle\pm\scriptstyle.011$ | - | - | - | $0.203\scriptscriptstyle\pm\scriptstyle.012$ | -
PGM ($\epsilon=1$) | $\mathbf{0.074\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.083\scriptscriptstyle\pm\scriptstyle.002}$ | $0.043\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.172\scriptscriptstyle\pm\scriptstyle.007}$ | - | $\mathbf{0.085\scriptscriptstyle\pm\scriptstyle.006}$ | - | - | $\mathbf{0.027\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.144\scriptscriptstyle\pm\scriptstyle.010}$ | -
PrivSyn ($\epsilon=1$) | $0.082\scriptscriptstyle\pm\scriptstyle.004$ | $0.135\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.003}$ | - | $0.310\scriptscriptstyle\pm\scriptstyle.019$ | - | $0.166\scriptscriptstyle\pm\scriptstyle.022$ | - | - | $0.031\scriptscriptstyle\pm\scriptstyle.002$ | $0.190\scriptscriptstyle\pm\scriptstyle.018$ | -
PATE-GAN ($\epsilon=1$) | $0.278\scriptscriptstyle\pm\scriptstyle.003$ | $0.342\scriptscriptstyle\pm\scriptstyle.004$ | $0.211\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.247\scriptscriptstyle\pm\scriptstyle.010$ | - | $0.140\scriptscriptstyle\pm\scriptstyle.004$ | - | - | $0.070\scriptscriptstyle\pm\scriptstyle.001$ | $0.258\scriptscriptstyle\pm\scriptstyle.012$ | -
TableDiffusion($\epsilon=1$) | $0.321\scriptscriptstyle\pm\scriptstyle.002$ | $0.365\scriptscriptstyle\pm\scriptstyle.003$ | $0.153\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.620\scriptscriptstyle\pm\scriptstyle.007$ | - | $0.332\scriptscriptstyle\pm\scriptstyle.006$ | - | - | $0.195\scriptscriptstyle\pm\scriptstyle.000$ | $0.494\scriptscriptstyle\pm\scriptstyle.008$ | -
HALF | $0.049\scriptscriptstyle\pm\scriptstyle.003$ | $0.052\scriptscriptstyle\pm\scriptstyle.004$ | $0.021\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.060\scriptscriptstyle\pm\scriptstyle.008$ | - | $0.056\scriptscriptstyle\pm\scriptstyle.009$ | - | - | $0.013\scriptscriptstyle\pm\scriptstyle.003$ | $0.099\scriptscriptstyle\pm\scriptstyle.008$ | -
HISTOGRAM | $0.116\scriptscriptstyle\pm\scriptstyle.002$ | $0.081\scriptscriptstyle\pm\scriptstyle.000$ | $0.049\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.217\scriptscriptstyle\pm\scriptstyle.007$ | - | $0.072\scriptscriptstyle\pm\scriptstyle.002$ | - | - | $0.030\scriptscriptstyle\pm\scriptstyle.001$ | $0.096\scriptscriptstyle\pm\scriptstyle.003$ | -

Table E26. Fidelity evaluation of two-way categorical-numerical marginals on
test data $\mathcal{D}_{\text{test}}$. The best result of each category is in
bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.212\scriptscriptstyle\pm\scriptstyle.005$ | $0.119\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.043\scriptscriptstyle\pm\scriptstyle.004$ | $0.106\scriptscriptstyle\pm\scriptstyle.007$ | $0.053\scriptscriptstyle\pm\scriptstyle.005$ | $0.080\scriptscriptstyle\pm\scriptstyle.004$ | $0.058\scriptscriptstyle\pm\scriptstyle.006$ | $0.077\scriptscriptstyle\pm\scriptstyle.012$ | $0.060\scriptscriptstyle\pm\scriptstyle.005$ | $0.106\scriptscriptstyle\pm\scriptstyle.010$ | -
PrivSyn | $0.029\scriptscriptstyle\pm\scriptstyle.001$ | $0.049\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.019\scriptscriptstyle\pm\scriptstyle.004$ | $0.090\scriptscriptstyle\pm\scriptstyle.005$ | $0.048\scriptscriptstyle\pm\scriptstyle.006$ | $0.069\scriptscriptstyle\pm\scriptstyle.006$ | $0.079\scriptscriptstyle\pm\scriptstyle.006$ | $0.042\scriptscriptstyle\pm\scriptstyle.006$ | $\mathbf{0.020\scriptscriptstyle\pm\scriptstyle.001}$ | $0.097\scriptscriptstyle\pm\scriptstyle.008$ | -
TVAE | $0.115\scriptscriptstyle\pm\scriptstyle.003$ | $0.205\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.025\scriptscriptstyle\pm\scriptstyle.005$ | $0.115\scriptscriptstyle\pm\scriptstyle.023$ | $0.082\scriptscriptstyle\pm\scriptstyle.007$ | $0.075\scriptscriptstyle\pm\scriptstyle.004$ | $0.092\scriptscriptstyle\pm\scriptstyle.011$ | $0.081\scriptscriptstyle\pm\scriptstyle.007$ | $0.101\scriptscriptstyle\pm\scriptstyle.001$ | $0.158\scriptscriptstyle\pm\scriptstyle.011$ | -
CTGAN | $0.080\scriptscriptstyle\pm\scriptstyle.003$ | $0.076\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.275\scriptscriptstyle\pm\scriptstyle.008$ | $0.157\scriptscriptstyle\pm\scriptstyle.012$ | $0.244\scriptscriptstyle\pm\scriptstyle.009$ | $0.109\scriptscriptstyle\pm\scriptstyle.005$ | $0.178\scriptscriptstyle\pm\scriptstyle.015$ | $0.081\scriptscriptstyle\pm\scriptstyle.008$ | $0.053\scriptscriptstyle\pm\scriptstyle.001$ | $0.150\scriptscriptstyle\pm\scriptstyle.009$ | -
TabDDPM | $\mathbf{0.027\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.039\scriptscriptstyle\pm\scriptstyle.001}$ | - | $\mathbf{0.016\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.080\scriptscriptstyle\pm\scriptstyle.012}$ | $\mathbf{0.025\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.055\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.041\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.051\scriptscriptstyle\pm\scriptstyle.024}$ | $0.040\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.090\scriptscriptstyle\pm\scriptstyle.009}$ | -
GReaT | $0.070\scriptscriptstyle\pm\scriptstyle.002$ | $0.074\scriptscriptstyle\pm\scriptstyle.004$ | - | $0.045\scriptscriptstyle\pm\scriptstyle.004$ | $0.078\scriptscriptstyle\pm\scriptstyle.006$ | $0.032\scriptscriptstyle\pm\scriptstyle.006$ | $0.081\scriptscriptstyle\pm\scriptstyle.008$ | $0.089\scriptscriptstyle\pm\scriptstyle.010$ | $0.055\scriptscriptstyle\pm\scriptstyle.004$ | - | $0.138\scriptscriptstyle\pm\scriptstyle.012$ | -
PGM ($\epsilon=1$) | $0.219\scriptscriptstyle\pm\scriptstyle.005$ | $0.121\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.044\scriptscriptstyle\pm\scriptstyle.007$ | $\mathbf{0.139\scriptscriptstyle\pm\scriptstyle.006}$ | $\mathbf{0.059\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.098\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.093\scriptscriptstyle\pm\scriptstyle.013}$ | $\mathbf{0.102\scriptscriptstyle\pm\scriptstyle.016}$ | $0.061\scriptscriptstyle\pm\scriptstyle.002$ | $\mathbf{0.117\scriptscriptstyle\pm\scriptstyle.004}$ | -
PrivSyn ($\epsilon=1$) | $\mathbf{0.060\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.098\scriptscriptstyle\pm\scriptstyle.003}$ | - | $\mathbf{0.033\scriptscriptstyle\pm\scriptstyle.002}$ | $0.290\scriptscriptstyle\pm\scriptstyle.022$ | $0.201\scriptscriptstyle\pm\scriptstyle.015$ | $0.197\scriptscriptstyle\pm\scriptstyle.017$ | $0.226\scriptscriptstyle\pm\scriptstyle.017$ | $0.124\scriptscriptstyle\pm\scriptstyle.017$ | $\mathbf{0.053\scriptscriptstyle\pm\scriptstyle.001}$ | $0.204\scriptscriptstyle\pm\scriptstyle.010$ | -
PATE-GAN ($\epsilon=1$) | $0.182\scriptscriptstyle\pm\scriptstyle.002$ | $0.221\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.246\scriptscriptstyle\pm\scriptstyle.009$ | $0.251\scriptscriptstyle\pm\scriptstyle.002$ | $0.842\scriptscriptstyle\pm\scriptstyle.006$ | $0.115\scriptscriptstyle\pm\scriptstyle.004$ | $0.726\scriptscriptstyle\pm\scriptstyle.012$ | $0.455\scriptscriptstyle\pm\scriptstyle.018$ | $0.080\scriptscriptstyle\pm\scriptstyle.003$ | $0.494\scriptscriptstyle\pm\scriptstyle.011$ | -
TableDiffusion ($\epsilon=1$) | $0.242\scriptscriptstyle\pm\scriptstyle.003$ | $0.295\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.199\scriptscriptstyle\pm\scriptstyle.008$ | $0.519\scriptscriptstyle\pm\scriptstyle.005$ | $0.229\scriptscriptstyle\pm\scriptstyle.007$ | $0.437\scriptscriptstyle\pm\scriptstyle.006$ | $0.275\scriptscriptstyle\pm\scriptstyle.004$ | $0.353\scriptscriptstyle\pm\scriptstyle.012$ | $0.163\scriptscriptstyle\pm\scriptstyle.000$ | $0.495\scriptscriptstyle\pm\scriptstyle.008$ | -
HALF | $0.027\scriptscriptstyle\pm\scriptstyle.003$ | $0.029\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.016\scriptscriptstyle\pm\scriptstyle.005$ | $0.056\scriptscriptstyle\pm\scriptstyle.006$ | $0.025\scriptscriptstyle\pm\scriptstyle.006$ | $0.052\scriptscriptstyle\pm\scriptstyle.008$ | $0.035\scriptscriptstyle\pm\scriptstyle.008$ | $0.043\scriptscriptstyle\pm\scriptstyle.010$ | $0.013\scriptscriptstyle\pm\scriptstyle.002$ | $0.077\scriptscriptstyle\pm\scriptstyle.011$ | -
HISTOGRAM | $0.236\scriptscriptstyle\pm\scriptstyle.019$ | $0.117\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.053\scriptscriptstyle\pm\scriptstyle.003$ | $0.123\scriptscriptstyle\pm\scriptstyle.009$ | $0.151\scriptscriptstyle\pm\scriptstyle.005$ | $0.084\scriptscriptstyle\pm\scriptstyle.001$ | $0.092\scriptscriptstyle\pm\scriptstyle.003$ | $0.122\scriptscriptstyle\pm\scriptstyle.012$ | $0.071\scriptscriptstyle\pm\scriptstyle.007$ | $0.106\scriptscriptstyle\pm\scriptstyle.005$ | -

Table E27. Fidelity evaluation of two-way numerical-numerical marginals on
test data $\mathcal{D}_{\text{test}}$. The best result of each category is in
bold.

| Adult | Shoppers | Phishing | Magic | Faults | Bean | Obesity | Robot | Abalone | News | Insurance | Wine
---|---|---|---|---|---|---|---|---|---|---|---|---
PGM | $0.392\scriptscriptstyle\pm\scriptstyle.008$ | $0.191\scriptscriptstyle\pm\scriptstyle.002$ | - | $0.069\scriptscriptstyle\pm\scriptstyle.000$ | $0.120\scriptscriptstyle\pm\scriptstyle.006$ | $0.073\scriptscriptstyle\pm\scriptstyle.001$ | $0.097\scriptscriptstyle\pm\scriptstyle.002$ | $0.107\scriptscriptstyle\pm\scriptstyle.001$ | $0.094\scriptscriptstyle\pm\scriptstyle.006$ | $0.108\scriptscriptstyle\pm\scriptstyle.009$ | $0.088\scriptscriptstyle\pm\scriptstyle.003$ | $0.101\scriptscriptstyle\pm\scriptstyle.002$
PrivSyn | $0.027\scriptscriptstyle\pm\scriptstyle.000$ | $0.050\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.026\scriptscriptstyle\pm\scriptstyle.001$ | $0.083\scriptscriptstyle\pm\scriptstyle.009$ | $0.056\scriptscriptstyle\pm\scriptstyle.001$ | $0.073\scriptscriptstyle\pm\scriptstyle.002$ | $0.089\scriptscriptstyle\pm\scriptstyle.003$ | $0.049\scriptscriptstyle\pm\scriptstyle.006$ | $\mathbf{0.026\scriptscriptstyle\pm\scriptstyle.001}$ | $0.079\scriptscriptstyle\pm\scriptstyle.004$ | $0.032\scriptscriptstyle\pm\scriptstyle.001$
TVAE | $0.049\scriptscriptstyle\pm\scriptstyle.001$ | $0.026\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $0.064\scriptscriptstyle\pm\scriptstyle.005$ | $0.041\scriptscriptstyle\pm\scriptstyle.001$ | $0.078\scriptscriptstyle\pm\scriptstyle.003$ | $0.058\scriptscriptstyle\pm\scriptstyle.002$ | $0.037\scriptscriptstyle\pm\scriptstyle.002$ | $0.034\scriptscriptstyle\pm\scriptstyle.001$ | $0.082\scriptscriptstyle\pm\scriptstyle.007$ | $0.061\scriptscriptstyle\pm\scriptstyle.001$
CTGAN | $0.018\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.019\scriptscriptstyle\pm\scriptstyle.002}$ | - | $0.052\scriptscriptstyle\pm\scriptstyle.001$ | $0.103\scriptscriptstyle\pm\scriptstyle.004$ | $0.074\scriptscriptstyle\pm\scriptstyle.002$ | $0.069\scriptscriptstyle\pm\scriptstyle.001$ | $0.087\scriptscriptstyle\pm\scriptstyle.004$ | $0.109\scriptscriptstyle\pm\scriptstyle.002$ | $0.033\scriptscriptstyle\pm\scriptstyle.001$ | $0.157\scriptscriptstyle\pm\scriptstyle.008$ | $0.047\scriptscriptstyle\pm\scriptstyle.002$
TabDDPM | $\mathbf{0.017\scriptscriptstyle\pm\scriptstyle.000}$ | $0.025\scriptscriptstyle\pm\scriptstyle.001$ | - | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.049\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.015\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.054\scriptscriptstyle\pm\scriptstyle.002}$ | $\mathbf{0.035\scriptscriptstyle\pm\scriptstyle.003}$ | $\mathbf{0.035\scriptscriptstyle\pm\scriptstyle.006}$ | $0.070\scriptscriptstyle\pm\scriptstyle.001$ | $0.064\scriptscriptstyle\pm\scriptstyle.008$ | $\mathbf{0.023\scriptscriptstyle\pm\scriptstyle.001}$
GReaT | $0.026\scriptscriptstyle\pm\scriptstyle.001$ | $0.042\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.068\scriptscriptstyle\pm\scriptstyle.002$ | $0.079\scriptscriptstyle\pm\scriptstyle.003$ | $0.023\scriptscriptstyle\pm\scriptstyle.001$ | $0.086\scriptscriptstyle\pm\scriptstyle.008$ | $0.059\scriptscriptstyle\pm\scriptstyle.004$ | $0.040\scriptscriptstyle\pm\scriptstyle.005$ | - | $\mathbf{0.060\scriptscriptstyle\pm\scriptstyle.008}$ | $0.033\scriptscriptstyle\pm\scriptstyle.002$
PGM ($\epsilon=1$) | $0.393\scriptscriptstyle\pm\scriptstyle.009$ | $0.194\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.080\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.130\scriptscriptstyle\pm\scriptstyle.004}$ | $\mathbf{0.079\scriptscriptstyle\pm\scriptstyle.001}$ | $0.108\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.113\scriptscriptstyle\pm\scriptstyle.001}$ | $\mathbf{0.121\scriptscriptstyle\pm\scriptstyle.004}$ | $0.109\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.114\scriptscriptstyle\pm\scriptstyle.005}$ | $\mathbf{0.102\scriptscriptstyle\pm\scriptstyle.001}$
PrivSyn ($\epsilon=1$) | $\mathbf{0.049\scriptscriptstyle\pm\scriptstyle.001}$ | $0.091\scriptscriptstyle\pm\scriptstyle.002$ | - | $\mathbf{0.062\scriptscriptstyle\pm\scriptstyle.003}$ | $0.297\scriptscriptstyle\pm\scriptstyle.012$ | $0.167\scriptscriptstyle\pm\scriptstyle.007$ | $0.219\scriptscriptstyle\pm\scriptstyle.010$ | $0.268\scriptscriptstyle\pm\scriptstyle.008$ | $0.205\scriptscriptstyle\pm\scriptstyle.007$ | $0.083\scriptscriptstyle\pm\scriptstyle.002$ | $0.181\scriptscriptstyle\pm\scriptstyle.010$ | $0.174\scriptscriptstyle\pm\scriptstyle.006$
PATE-GAN ($\epsilon=1$) | $0.055\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.074\scriptscriptstyle\pm\scriptstyle.001}$ | - | $0.130\scriptscriptstyle\pm\scriptstyle.001$ | $0.334\scriptscriptstyle\pm\scriptstyle.005$ | $0.408\scriptscriptstyle\pm\scriptstyle.003$ | $\mathbf{0.083\scriptscriptstyle\pm\scriptstyle.005}$ | $0.461\scriptscriptstyle\pm\scriptstyle.002$ | $0.501\scriptscriptstyle\pm\scriptstyle.001$ | $\mathbf{0.077\scriptscriptstyle\pm\scriptstyle.004}$ | $0.682\scriptscriptstyle\pm\scriptstyle.017$ | $0.334\scriptscriptstyle\pm\scriptstyle.003$
TableDiffusion ($\epsilon=1$) | $0.093\scriptscriptstyle\pm\scriptstyle.001$ | $0.096\scriptscriptstyle\pm\scriptstyle.001$ | - | $0.132\scriptscriptstyle\pm\scriptstyle.002$ | $0.294\scriptscriptstyle\pm\scriptstyle.002$ | $0.147\scriptscriptstyle\pm\scriptstyle.002$ | $0.486\scriptscriptstyle\pm\scriptstyle.005$ | $0.265\scriptscriptstyle\pm\scriptstyle.001$ | $0.200\scriptscriptstyle\pm\scriptstyle.005$ | $0.121\scriptscriptstyle\pm\scriptstyle.001$ | $0.348\scriptscriptstyle\pm\scriptstyle.011$ | $0.132\scriptscriptstyle\pm\scriptstyle.001$
HALF | $0.011\scriptscriptstyle\pm\scriptstyle.001$ | $0.008\scriptscriptstyle\pm\scriptstyle.000$ | - | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | $0.039\scriptscriptstyle\pm\scriptstyle.003$ | $0.013\scriptscriptstyle\pm\scriptstyle.001$ | $0.050\scriptscriptstyle\pm\scriptstyle.004$ | $0.028\scriptscriptstyle\pm\scriptstyle.001$ | $0.027\scriptscriptstyle\pm\scriptstyle.005$ | $0.012\scriptscriptstyle\pm\scriptstyle.001$ | $0.059\scriptscriptstyle\pm\scriptstyle.004$ | $0.020\scriptscriptstyle\pm\scriptstyle.001$
HISTOGRAM | $0.417\scriptscriptstyle\pm\scriptstyle.041$ | $0.194\scriptscriptstyle\pm\scriptstyle.003$ | - | $0.093\scriptscriptstyle\pm\scriptstyle.001$ | $0.127\scriptscriptstyle\pm\scriptstyle.003$ | $0.118\scriptscriptstyle\pm\scriptstyle.001$ | $0.104\scriptscriptstyle\pm\scriptstyle.002$ | $0.116\scriptscriptstyle\pm\scriptstyle.000$ | $0.168\scriptscriptstyle\pm\scriptstyle.003$ | $0.125\scriptscriptstyle\pm\scriptstyle.015$ | $0.115\scriptscriptstyle\pm\scriptstyle.006$ | $0.104\scriptscriptstyle\pm\scriptstyle.002$

##  Appendix F Discussion of proposed privacy metric

The definition of the proposed membership inference score (MDS) aligns closely
with concepts of memorization in neural networks (Feldman, 2020; Zhang et al.,
2023a) and the leave-one-out notion of stability in machine learning (Bousquet
and Elisseeff, 2002). However, it diverges in three crucial ways: (i) We
measure the worst-case discourse risk as the privacy metric, whereas other
studies focus on the difference of individuals or average cases. (ii) Our work
specifically addresses privacy concerns in data synthesis, as opposed to other
studies that explore discriminative models like classification. (iii) Our
approach emphasizes the discrepancy caused by the presence or absence of
target data in training, in contrast to other works that highlight performance
gains from adding samples to the training set.

##  Appendix G Misleading conclusion

We find that some statements in data synthesis may be misleading or even
incorrect due to limitations of evaluation metrics or methodologies. We
highlight some of them as below:

  * •

In (Kotelnikov et al., 2023), authors show that the machine learning
performance on TabDDPM is even better than that on real data, which implies
that synthetic data can be a perfect (even better) substitute for real data.
However, this statement may be incorrect due to inadequate model tuning and
improper data shuffling practices. Our evaluations show that even simple
models, such as linear regression, can achieve better performance on real data
than on high-quality synthetic data (_i.e.,_ generated by TabDDPM).

  * •

Some studies (Kotelnikov et al., 2023; Lee et al., 2023; Kim et al., 2022;
Jordon et al., 2018) prioritize machine learning efficacy as the primary (if
not only) fidelity evaluation metric. This approach is problematic because
data synthesis can be biased to label attributes, and a high machine learning
efficacy score does not necessarily equate to high fidelity in synthetic data.

  * •

Extensive studies (Zhao et al., 2021; Lee et al., 2023; Kim et al., 2022;
Kotelnikov et al., 2023; Zhang et al., 2023b) use Distance to Closest Records
(DCR) to evaluate the privacy of synthetic data and assert their models are
safe. However, in this paper we show that DCR fails to serve as an adequate
measure of privacy. We also provide evidence that many recently introduced HP
methods exhibit significant risks of membership disclosure, which are often
ignored by HP synthesis algorithms.

##  Appendix H Computing Wasserstein Distance

When the cost matrix is large and dense, computing this metric can be
prohibitively time-consuming. Several options are provided to address this
problem. (i) Sinkhorn distance (Cuturi, 2013) provides a fast approximation to
the Wasserstein distance by penalizing the objective with an entropy term.
(ii) Sliced-Wasserstein distance (Bonneel et al., 2015), which uses Radon
transform to linearly project data into one dimension, can be efficiently
computed. (iii) Reducing the size of the cost matrix by randomly sampling a
small set of points from the probability densities. These computational
strategies ensure the Wasserstein distance remains a feasible solution for
fidelity evaluation, even for complex and high-dimensional marginals. In
practice, when the domain of target probability distribution is large (_i.e.,_
$n\geq 5,000$), we find that randomly sampling half of the elements is both
effective and efficient.