Cosine Similarity Measure According to a Convex Cost Function

TL;DR

This paper introduces a novel vector similarity measure based on the angle between surface normals of a convex cost function, applicable even when the function is not differentiable everywhere.

Contribution

It proposes a new similarity measure using convex cost functions and subgradients, expanding the tools for vector comparison beyond traditional methods.

Findings

Applicable to non-differentiable convex functions

Uses surface normals at vectors to measure similarity

Can incorporate functions like negative entropy and total variation

Abstract

In this paper, we describe a new vector similarity measure associated with a convex cost function. Given two vectors, we determine the surface normals of the convex function at the vectors. The angle between the two surface normals is the similarity measure. Convex cost function can be the negative entropy function, total variation (TV) function and filtered variation function. The convex cost function need not be differentiable everywhere. In general, we need to compute the gradient of the cost function to compute the surface normals. If the gradient does not exist at a given vector, it is possible to use the subgradients and the normal producing the smallest angle between the two vectors is used to compute the similarity measure.