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Table 2 Pareto MTL with MINE versus Pareto MTL with MMD

From: Trade-off between conservation of biological variation and batch effect removal in deep generative modeling for single-cell transcriptomics

Dataset

Percentage

Hypervolume

NDC

Train

Test

Train

Test

Train

Test

MINE

MMD

MINE

MMD

MINE

MMD

MINE

MMD

MINE

MMD

MINE

MMD

\({\overline{U}}_n(\varvec{\phi }, \varvec{\theta })\) versus \(\varvec{NN}\)

TM-MARROW

\(\varvec{0.85} \pm 0.07\)

\(0.48 \pm 0.07\)

\(\varvec{0.58} \pm 0.18\)

\(0.35 \pm 0.12\)

\(\varvec{4.35} \pm 0.04\)

\(3.91 \pm 0.16\)

\(\varvec{4.29} \pm 0.06\)

\(3.85 \pm 0.21\)

\(\varvec{7.7} \pm 0.82\)

\(5.4 \pm 0.52\)

\(\varvec{5.6} \pm 1.35\)

\(4.1 \pm 1.20\)

MACAQUE-RETINA

\(\varvec{0.72} \pm 0.10\)

\(0.27 \pm 0.08\)

\(\varvec{0.47} \pm 0.13\)

\(0.14 \pm 0.04\)

\(\varvec{293.82} \pm 1.77\)

\(273.48 \pm 4.23\)

\(\varvec{295.15} \pm 4.17\)

\(257.92 \pm 8.90\)

\(\varvec{4.1} \pm 0.74\)

\(2.0 \pm 0.00\)

\(\varvec{3.8} \pm 1.55\)

\(1.2 \pm 0.42\)

-ASW versus -BE

TM-MARROW

\(\varvec{0.45} \pm 0.06\)

\(0.43 \pm 0.09\)

\(\varvec{0.48} \pm 0.10\)

\(0.41 \pm 0.05\)

\(\varvec{ 0.29} \pm 0.02\)

\(0.16 \pm 0.02\)

\(\varvec{0.38} \pm 0.01\)

\(0.30 \pm 0.02\)

\(\varvec{4.6} \pm 0.84\)

\(4.2 \pm 0.79\)

\(\varvec{4.9} \pm 1.37\)

\(3.7 \pm 0.48\)

MACAQUE-RETINA

\(\varvec{0.61} \pm 0.07\)

\(0.33 \pm 0.10\)

\(\varvec{0.64} \pm 0.10\)

\(0.34 \pm 0.11\)

\(\varvec{0.32} \pm 0.01\)

\(0.15 \pm 0.02\)

\(\varvec{0.30} \pm 0.01\)

\(0.19 \pm 0.03\)

\(\varvec{4.4} \pm 0.84\)

\(3.1 \pm 0.57\)

\(\varvec{5.5} \pm 0.85\)

\(2.6 \pm 0.70\)

-NMI versus -BE

TM-MARROW

\(\varvec{0.56} \pm 0.08\)

\(0.49 \pm 0.08\)

\(\varvec{0.58} \pm 0.12\)

\(0.40 \pm 0.09\)

\(\varvec{0.28} \pm 0.01\)

\(0.16 \pm 0.02\)

\(\varvec{0.37} \pm 0.01\)

\(0.31 \pm 0.02\)

\(\varvec{6.0} \pm 0.67\)

\(4.5 \pm 0.85\)

\(\varvec{5.0} \pm 1.33\)

\(4.3 \pm 0.95\)

MACAQUE-RETINA

\(\varvec{0.38} \pm 0.11\)

\(0.23 \pm 0.11\)

\(\varvec{0.49} \pm 0.18\)

\(0.33 \pm 0.12\)

\(\varvec{0.26} \pm 0.02\)

\(0.11 \pm 0.02\)

\(\varvec{0.25} \pm 0.01\)

\(0.14 \pm 0.02\)

\(\varvec{3.5} \pm 0.85\)

\(2.5 \pm 0.85\)

\(\varvec{4.5} \pm 1.27\)

\(3.5 \pm 1.08\)

  1. The Pareto fronts—generative loss \(\overline{U}_n\) versus mutual information estimator NN and negative ASW/NMI versus negative BE from Pareto MTL with MINE and Pareto MTL with MMD are compared
  2. Better Pareto front has higher percentage/hypervolume/NDC
  3. The values are mean ± standard deviation over 10 Monte Carlos of training-testing split of the dataset. Bold illustrates higher mean in comparison