Geodesics of learned representations

TL;DR

This paper introduces a method to visualize and analyze the invariance properties of learned image representations by constructing geodesics in the representation space, revealing limitations and enabling improvements in linearizing geometric transformations.

Contribution

The paper proposes a novel technique to assess and enhance the linearization of transformations in learned representations through geodesic synthesis and refinement.

Findings

State-of-the-art network does not linearize transformations as expected.

Refined representations better linearize geometric transformations.

Method reveals invariance failures and guides improvements.

Abstract

We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional subspace. Given two reference images (typically, differing by some transformation), we synthesize a sequence of images lying on a path between them that is of minimal length in the space of the representation (a "representational geodesic"). If the transformation relating the two reference images is linearized by the representation, this sequence should follow the gradual evolution of this transformation. We use this method to assess the invariance properties of a state-of-the-art image classification network and find that geodesics generated for image pairs differing by translation, rotation, and dilation do not evolve according to their associated transformations. Our method also suggests a remedy for these failures, and following this prescription, we show that the modified representation is able to linearize a variety of geometric image transformations.