A Likelihood Based Approach to Distribution Regression Using Conditional Deep Generative Models

In this work, we explore the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. More specifically, we study the large-sample properties of a likelihood-based approach for estimating these models. Our results lead to the convergence rate of a sieve maximum likelihood estimator (MLE) for estimating the conditional distribution (and its devolved counterpart) of the response given predictors in the Hellinger (Wasserstein) metric. Our rates depend solely on the intrinsic dimension and smoothness of the true conditional distribution. These findings provide an explanation of why conditional deep generative models can circumvent the curse of dimensionality from the perspective of statistical foundations and demonstrate that they can learn a broader class of nearly singular conditional distributions. Our analysis also emphasizes the importance of introducing a small noise perturbation to the data when they are supported sufficiently close to a manifold. Finally, in our numerical studies, we demonstrate the effective implementation of the proposed approach using both synthetic and real-world datasets, which also provide complementary validation to our theoretical findings.

💡 Research Summary

This paper investigates the statistical foundations of conditional deep generative models (CDGMs) within the framework of distribution regression, where the response variable Y resides in a high‑dimensional ambient space ℝ^D but is concentrated near a lower‑dimensional manifold of intrinsic dimension d ≤ D. The authors formalize the problem as Y|X = V|X + ε, with V|X representing the uncorrupted response supported on a manifold and ε ∼ N(0,σ²I_D) a full‑dimensional Gaussian noise. The conditional distribution of V|X is modeled by a conditional generator G*(Z, X), where Z is a latent variable with known distribution P_Z and G* is a deep neural network (DNN). Because G*(Z, X) lives on a d‑dimensional manifold, its distribution is singular with respect to Lebesgue measure in ℝ^D.

The central methodological contribution is a likelihood‑based estimator for the conditional distribution of Y given X. The authors define the conditional density p_{g,σ}(y|x) = ∫ φ_σ(y − g(z,x)) dP_Z(z), where φ_σ is the Gaussian kernel with variance σ². Given n i.i.d. pairs (X_i, Y_i), the log‑likelihood ℓ_n(g,σ) = (1/n)∑{i=1}^n log p{g,σ}(Y_i|X_i) is maximized over a sieve (restricted) class of generators g ∈ ℱ and noise levels σ ∈