Discriminative Learning of Similarity and Group Equivariant Representations

TL;DR

This paper advances metric learning by optimizing for k-NN accuracy, proposes a computationally efficient method for estimating metrics, and introduces SO(3)-equivariant neural networks for spherical data, addressing representation and symmetry issues.

Contribution

It introduces a new formulation for metric learning targeting k-NN accuracy, a simple gradient-based metric estimation method, and a Fourier space SO(3)-equivariant neural network architecture.

Findings

Improved k-NN accuracy through direct metric optimization.

Efficient gradient-based metric estimation method.

A novel SO(3)-equivariant neural network for spherical data.

Abstract

One of the most fundamental problems in machine learning is to compare examples: Given a pair of objects we want to return a value which indicates degree of (dis)similarity. Similarity is often task specific, and pre-defined distances can perform poorly, leading to work in metric learning. However, being able to learn a similarity-sensitive distance function also presupposes access to a rich, discriminative representation for the objects at hand. In this dissertation we present contributions towards both ends. In the first part of the thesis, assuming good representations for the data, we present a formulation for metric learning that makes a more direct attempt to optimize for the k-NN accuracy as compared to prior work. We also present extensions of this formulation to metric learning for kNN regression, asymmetric similarity learning and discriminative learning of Hamming distance. In the second part, we consider a situation where we are on a limited computational budget i.e. optimizing over a space of possible metrics would be infeasible, but access to a label aware distance metric is still desirable. We present a simple, and computationally inexpensive approach for estimating a well motivated metric that relies only on gradient estimates, discussing theoretical and experimental results. In the final part, we address representational issues, considering group equivariant convolutional neural networks (GCNNs). Equivariance to symmetry transformations is explicitly encoded in GCNNs; a classical CNN being the simplest example. In particular, we present a SO(3)-equivariant neural network architecture for spherical data, that operates entirely in Fourier space, while also providing a formalism for the design of fully Fourier neural networks that are equivariant to the action of any continuous compact group.