The Expected Jacobian Outerproduct: Theory and Empirics

TL;DR

This paper introduces the expected Jacobian outerproduct (EJOP) for multi-class regression, providing a simple estimator that is statistically consistent and useful for improving classification tasks through metric learning.

Contribution

It extends the EGOP concept to multi-class settings, proposes a consistent estimator for EJOP, and demonstrates its practical utility in classification and metric learning.

Findings

The proposed EJOP estimator is statistically consistent.

Eigenvalues and eigenspaces of EJOP are reliably estimated.

Using EJOP as a metric improves classification performance.

Abstract

The expected gradient outerproduct (EGOP) of an unknown regression function is an operator that arises in the theory of multi-index regression, and is known to recover those directions that are most relevant to predicting the output. However, work on the EGOP, including that on its cheap estimators, is restricted to the regression setting. In this work, we adapt this operator to the multi-class setting, which we dub the expected Jacobian outerproduct (EJOP). Moreover, we propose a simple rough estimator of the EJOP and show that somewhat surprisingly, it remains statistically consistent under mild assumptions. Furthermore, we show that the eigenvalues and eigenspaces also remain consistent. Finally, we show that the estimated EJOP can be used as a metric to yield improvements in real-world non-parametric classification tasks: both by its use as a metric, and also as cheap initialization in metric learning tasks.