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DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing
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Proposes DUAL, which learns a diverse pool of kernels for aggregated U-statistic MMD/HSIC tests by maximizing a covariance-normalized (Mahalanobis-style) aggregated statistic on training data, then selects kernels via train/test signum alignment with wild-bootstrap thresholds.
Problem
Kernel aggregation for two-sample and independence testing can yield redundant or weak kernels that dilute test power, so naively aggregating many kernels does not reliably increase power.
Approach
Aggregate kernel U-statistics with a covariance-normalized Mahalanobis-style statistic that rewards kernel diversity, learn the kernels on training split by gradient ascent, and then keep only kernels whose train and test signs agree (selection inference), with wild-bootstrap calibration to control type-I error.
Key results
- MMD-DUAL and HSIC-DUAL report the highest test power against all baselines across the six benchmarks at alpha=0.05 (averaged over 1,000 repetitions).
- Ablations show removing either kernel diversity or selection inference lowers power, and both variants still beat plain aggregation.
- Under the null, each kernel is selected with roughly 50% probability, while under the alternative the selection concentrates on informative kernels (approaching ~1.0 on Higgs).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MMD-DUAL instead of the compared baselines | MMD two-sample testing · BLOB, MNIST, ImageNet | test power | Helped |
| Use HSIC-DUAL instead of the compared baselines | HSIC independence testing · Higgs, MNIST, CIFAR-10 | test power | Helped |
| Add selection inference to diverse-kernel aggregation (ablation) | MMD-DUAL · BLOB | test power | Helped |
| Add diversity to aggregation with selection inference (ablation) | MMD-DUAL · BLOB | test power | Helped |
| Add diversity and selection inference to plain aggregation (ablation) | MMD-DUAL · BLOB | test power | Helped |
| Add selection inference to diverse-kernel aggregation (ablation) | HSIC-DUAL · Higgs | test power | Helped |
| Add diversity to aggregation with selection inference (ablation) | HSIC-DUAL · Higgs | test power | Helped |
| Add diversity and selection inference to plain aggregation (ablation) | HSIC-DUAL · Higgs | test power | Helped |
Surprising findings
- Aggregating a subset of kernels can outperform aggregating the full 20-kernel set, showing redundant kernels can dilute power.
- Aggregating kernels with high individual power does not guarantee higher power if the kernels are highly correlated.
Limitations
- Computational cost is quadratic in sample size and polynomial in number of kernels, so large kernel pools are costly.
- Data splitting reduces effective sample size for the test, though the authors argue learned kernels compensate.
- Theory requires positive-definite covariance (regularized in practice) and an assumption on sign alignment between training and population statistics.
Experimental setup
Datasets: BLOB MNIST ImageNet ImageNetV2 DCGAN-generated MNIST Higgs CIFAR-10
Benchmarks: BLOB MNIST vs DCGAN ImageNet vs ImageNetV2 Higgs MNIST independence CIFAR-10 independence
Metrics: test power type-I error
Baselines: AutoTST MMD-Agg MMD-D MMD-FUSE FSIC HSIC-Agg HSIC-O
Builds on: MMD HSIC MMD-Agg MMD-FUSE AutoML two-sample test
Closest papers by meaning
- Meta Two-Sample Testing: Learning Kernels for Testing with Limited Data (NeurIPS 2021 · similarity 0.92)
- Learning Deep Kernels for Non-Parametric Two-Sample Tests (ICML 2020 · similarity 0.90)
- Kernel Selection is Model Selection: A Unified Complexity-Penalised Approach for MMD Two-Sample Tests (NeurIPS 2026 · similarity 0.89)
- On the Hardness of Conditional Independence Testing In Practice (NeurIPS 2025 · similarity 0.86)
- Kernel conditional tests from learning-theoretic bounds (NeurIPS 2025 · similarity 0.85)
- Anchor-based Maximum Discrepancy for Relative Similarity Testing (NeurIPS 2025 · similarity 0.85)
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