[Paper Review] Deep Dimension Reduction for Supervised Representation Learning

Jian Huang, Yuling Jiao+3 more|arXiv (Cornell University)|Jun 10, 2020

Machine Learning and Data Classification71 references4 citations

TL;DR

This paper proposes Deep Dimension Reduction (DDR), a nonparametric deep learning framework that learns low-dimensional, disentangled, and sufficient representations for supervised learning by minimizing an objective based on conditional independence and f-divergence. The method ensures consistency in excess risk and outperforms existing dimension reduction and deep learning methods in classification and regression tasks.

ABSTRACT

The goal of supervised representation learning is to construct effective data representations for prediction. Among all the characteristics of an ideal nonparametric representation of high-dimensional complex data, sufficiency, low dimensionality and disentanglement are some of the most essential ones. We propose a deep dimension reduction approach to learning representations with these characteristics. The proposed approach is a nonparametric generalization of the sufficient dimension reduction method. We formulate the ideal representation learning task as that of finding a nonparametric representation that minimizes an objective function characterizing conditional independence and promoting disentanglement at the population level. We then estimate the target representation at the sample level nonparametrically using deep neural networks. We show that the estimated deep nonparametric representation is consistent in the sense that its excess risk converges to zero. Our extensive numerical experiments using simulated and real benchmark data demonstrate that the proposed methods have better performance than several existing dimension reduction methods and the standard deep learning models in the context of classification and regression.

Motivation & Objective

• To develop a principled deep learning approach for supervised representation learning that ensures sufficiency, low dimensionality, and disentanglement in learned representations.
• To generalize sufficient dimension reduction (SDR) to the nonlinear, nonparametric setting using deep neural networks.
• To formulate representation learning as minimizing an objective function that enforces conditional independence and promotes disentanglement at the population level.
• To establish theoretical consistency of the learned representation by proving convergence of excess risk to zero.
• To demonstrate superior empirical performance compared to standard deep learning and existing dimension reduction methods on benchmark datasets.

Proposed method

• Formulates the ideal representation learning task as minimizing an objective function that characterizes conditional independence and promotes disentanglement via f-divergence and distance covariance.
• Uses deep neural networks to nonparametrically estimate the target representation at the sample level, enabling flexible modeling of complex, nonlinear relationships.
• Employs a two-stage learning framework: a representation network $ D_{m{ heta}} $ and a prediction network $ R_{m{ heta}} $, jointly optimized to minimize excess risk.
• Applies symmetrization and Rademacher complexity techniques to bound the empirical process and establish generalization error bounds.
• Uses covering numbers and VC-dimension-based complexity control to manage the capacity of the deep network classes and ensure consistency.
• Derives theoretical consistency by showing that the excess risk of the estimated representation converges to zero as sample size increases.

Experimental results

Research questions

• RQ1 Can a nonparametric deep learning framework be designed to learn representations that are sufficient, low-dimensional, and disentangled for supervised learning?
• RQ2 How can conditional independence between input and response given the representation be enforced in a nonlinear, nonparametric setting?
• RQ3 What theoretical guarantees can be established for the consistency of deep representation learning under such a framework?
• RQ4 How does the proposed DDR method compare in performance to standard deep learning and existing dimension reduction techniques in classification and regression tasks?
• RQ5 Can the method effectively handle high-dimensional, complex data such as images and natural language without strong parametric assumptions?

Key findings

• The proposed DDR method achieves consistent estimation, with the excess risk of the learned representation converging to zero as the sample size increases.
• Extensive experiments on simulated and real benchmark datasets show DDR outperforms several existing dimension reduction methods and standard deep learning models in both classification and regression tasks.
• The method successfully learns low-dimensional, disentangled representations that preserve all relevant information about the response, satisfying the conditional independence principle.
• Theoretical analysis confirms that the generalization error is controlled via Rademacher complexity and covering number bounds, ensuring robustness and consistency.
• DDR demonstrates strong performance on high-dimensional data, including image-like datasets, where linear methods fail due to nonlinear structure.
• The framework generalizes classical SDR by extending it to nonlinear, nonparametric settings using deep neural networks without requiring linearity or constant covariance assumptions.