Maximum Class Separation as Inductive Bias in One Matrix

Abstract

Maximizing the separation between classes constitutes a well-known inductive bias in machine learning and a pillar of many traditional algorithms. By default, deep networks are not equipped with this inductive bias and therefore many alternative solutions have been proposed through differential optimization. Current approaches tend to optimize classification and separation jointly: aligning inputs with class vectors and separating class vectors angularly. This paper proposes a simple alternative: encoding maximum separation as an inductive bias in the network by adding one fixed matrix multiplication before computing the softmax activations. The main observation behind our approach is that separation does not require optimization but can be solved in closed-form prior to training and plugged into a network. We outline a recursive approach to obtain the matrix consisting of maximally separable vectors for any number of classes, which can be added with negligible engineering effort and computational overhead. Despite its simple nature, this one matrix multiplication provides real impact. We show that our proposal directly boosts classification, long-tailed recognition, out-of-distribution detection, and open-set recognition, from CIFAR to ImageNet. We find empirically that maximum separation works best as a fixed bias; making the matrix learnable adds nothing to the performance. The closed-form implementation and code to reproduce the experiments are on github.Preprint. Under review.

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“…Since directly using the unhinged loss will lead to volatile effects, which is mainly reflected in the rapid increase of feature norms and the imbalance between class prototypes when training DNNs with the stochastic gradient method, as shown in Figure 19. Inspired by recent works (Zhou et al, 2022c;Kasarla et al, 2022;Yang et al, 2022) that use the Neural Collapse structure as an inductive bias (also called prototyping-anchored learning, PAL), we fix prototypes W as a simplex ETF during training, i.e.,…”