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

2024-10-02

arXiv.org

AI Summary

This paper addresses conditional distribution regression when the response variable lies on a low-dimensional manifold embedded in a high-dimensional input space. Methodologically, it proposes a likelihood-based framework for estimating conditional deep generative models, employing a sieve maximum likelihood estimator augmented with manifold perturbation regularization to ensure training stability. Theoretical analysis establishes convergence rates under both the Hellinger and Wasserstein metrics. The key contribution is the first derivation of minimax-optimal convergence rates depending solely on the intrinsic dimension of the underlying manifold and the smoothness of the target conditional density—thereby rigorously characterizing how the model circumvents the “curse of dimensionality.” This statistical mechanism is especially advantageous for nearly singular conditional distributions. Empirical validation on synthetic and real-world datasets confirms both the method’s effectiveness and its alignment with theoretical predictions.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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.

Problem

• Estimating conditional distributions using deep generative models in regression
• Analyzing convergence rates of sieve MLE for high-dimensional manifold data
• Overcoming dimensionality curse by learning nearly singular distributions

Innovation

• Uses conditional deep generative models
• Employs sieve maximum likelihood estimation
• Introduces small noise perturbation technique