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A Likelihood Based Approach to Distribution Regression Using Conditional Deep Generative Models
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Theoretical analysis of a likelihood-based sieve MLE for conditional deep generative models in distribution regression, giving Hellinger and Wasserstein convergence rates that depend only on intrinsic dimension and smoothness, plus small numerical validation on synthetic and MNIST data.
Problem
Estimating conditional distributions of a high-dimensional response that concentrates near a lower-dimensional manifold (with full-dimensional noise), and understanding why conditional deep generative models avoid the curse of dimensionality statistically.
Approach
Models the response as a conditional pushforward of a latent seed through a deep ReLU generator plus Gaussian noise, and analyzes a sieve MLE over sparse or fully connected networks; rates depend on intrinsic dimension and smoothness, and small noise injection (data perturbation) is proposed to handle near-singular supports.</1-2 sentences>
Key results
- On FD1-FD3 the sieve MLE has lowest MSE for mean in all three settings (e.g., FD1 mean 0.0379 vs CKDE 1.0053 and FlexCode 1.1660) and lowest SD MSE in FD1 and FD2, but not FD3 SD (1.6394 vs CKDE 0.6334).
- Theory gives Hellinger rate n^{-(beta_*-t_*alpha)/(2beta_*+t_*)} log^2 n, and Wasserstein rate governed by intrinsic dimension and smoothness; noise perturbation improves the rate when noise is small.
- On MNIST, generated-image W1 distances rise from about 1.9 at zero noise to about 4.1 at noise 0.5, with sparse and fully connected architectures performing similarly.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Sieve MLE instead of CKDE or FlexCode | conditional mean estimation · FD1, FD2, and FD3 | mean MSE: FD1: 0.0379 ± 0.0170 vs 1.0053 ± 0.1004 and 1.1660 ± 0.1076; FD2: 0.1943 ± 0.0427 vs 0.2640 ± 0.0515 and 0.3954 ± 0.0571 | Helped |
| Use Sieve MLE instead of CKDE or FlexCode | conditional standard deviation estimation · FD1 and FD2 | SD MSE: FD1: 0.0280 ± 0.0045 vs 0.9887 ± 0.0347 and 1.2000 ± 0.0126; FD2: 0.2843 ± 0.0093 vs 0.2853 ± 0.0213 and 5.8278 ± 0.1607 | Helped |
| Use Sieve MLE instead of CKDE for standard deviation estimation | conditional distribution estimation · FD3 | SD MSE: 1.6394 ± 0.0861 vs 0.6334 ± 0.0460 | Hurt |
| Increase training sample size | Sieve MLE standard deviation estimation · FD3 | SD MSE | Helped |
| Add Gaussian noise to MNIST training data (ablation) | sparse and fully connected architectures · MNIST test images | W1: Sparse: 1.9555 ± 0.7182 at 0 to 4.0880 ± 0.7518 at 0.5; fully connected: 1.8859 ± 0.7355 at 0 to 4.0995 ± 0.7633 at 0.5 | Hurt |
| Use sparse rather than fully connected architecture (ablation) | conditional MNIST generation · MNIST test images | W1: Noise 0: 1.9555 ± 0.7182 vs 1.8859 ± 0.7355; noise 0.005: 1.9478 ± 0.7329 vs 1.8663 ± 0.6251; noise 0.01: 1.9503 ± 0.729 | Mixed |
| Increase sample size (ablation) | Sieve MLE on manifold data · Two Moon and Ellipse (σ* = 0.01) | W1 median: 4000: 0.251 and 0.295; 10000: 0.196 and 0.251 | Helped |
Surprising findings
- On FD3 the sieve MLE has worse standard-deviation MSE (1.6394) than CKDE (0.6334), the exception to its otherwise superior performance.
- The Wasserstein rate has a regime where the noise-level term sigma_* sqrt(log n) dominates, so small noise can hurt estimation without perturbation.
Limitations
- Experiments are limited to small synthetic settings and a single MNIST study with no reported sample-size details.
- Rates depend on constants and assumptions (composite Holder structure, identifiability with reach bound) that may be restrictive.
- Optimization error and the practical VAE-based algorithm are not analyzed theoretically.
Experimental setup
Datasets: MNIST synthetic FD1-FD3 synthetic manifold M1 (two moons) synthetic manifold M2 (ellipses)
Benchmarks: MNIST synthetic FD1-FD3 synthetic manifolds M1 and M2
Metrics: MSE of conditional mean MSE of conditional standard deviation W1 distance
Baselines: CKDE FlexCode
Builds on: sieve MLE noise outsourcing lemma FlexCode conditional kernel density estimation
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