On Universal Features for High-Dimensional Learning and Inference

TL;DR

This paper introduces a unified information geometric framework for identifying universal low-dimensional features in high-dimensional data, enhancing understanding and optimization of various learning systems.

Contribution

It develops a novel theoretical framework connecting multiple classical and modern analysis tools for high-dimensional feature extraction and inference.

Findings

Unified geometric framework for feature extraction

Connections among SVD, CCA, information bottleneck, and more

Applications to neural networks, matrix factorization, and semi-supervised learning

Abstract

We consider the problem of identifying universal low-dimensional features from high-dimensional data for inference tasks in settings involving learning. For such problems, we introduce natural notions of universality and we show a local equivalence among them. Our analysis is naturally expressed via information geometry, and represents a conceptually and computationally useful analysis. The development reveals the complementary roles of the singular value decomposition, Hirschfeld-Gebelein-R\'enyi maximal correlation, the canonical correlation and principle component analyses of Hotelling and Pearson, Tishby's information bottleneck, Wyner's common information, Ky Fan $k$-norms, and Brieman and Friedman's alternating conditional expectations algorithm. We further illustrate how this framework facilitates understanding and optimizing aspects of learning systems, including multinomial logistic (softmax) regression and the associated neural network architecture, matrix factorization methods for collaborative filtering and other applications, rank-constrained multivariate linear regression, and forms of semi-supervised learning.