Inverse Scale Space Decomposition

TL;DR

This paper explores the inverse scale space flow as a method for decomposing data into generalized singular vectors, establishing conditions for successful decomposition and providing numerical validation.

Contribution

It introduces new conditions under which inverse scale space flow can decompose data into singular vectors and proves when it can recover singular vectors from arbitrary data.

Findings

Decomposition into singular vectors is possible under orthogonality and subgradient inclusion conditions.

The inverse scale space flow can recover singular vectors from arbitrary data if a dual condition is met.

Numerical results confirm the theoretical conditions and highlight their importance.

Abstract

We investigate the inverse scale space flow as a decomposition method for decomposing data into generalised singular vectors. We show that the inverse scale space flow, based on convex and absolutely one-homogeneous regularisation functionals, can decompose data represented by the application of a forward operator to a linear combination of generalised singular vectors into its individual singular vectors. We verify that for this decomposition to hold true, two additional conditions on the singular vectors are sufficient: orthogonality in the data space and inclusion of partial sums of the subgradients of the singular vectors in the subdifferential of the regularisation functional at zero. We also address the converse question of when the inverse scale space flow returns a generalised singular vector given that the initial data is arbitrary (and therefore not necessarily in the range of the forward operator). We prove that the inverse scale space flow is guaranteed to return a singular vector if the data satisfies a novel dual singular vector condition. We conclude the paper with numerical results that validate the theoretical results and that demonstrate the importance of the additional conditions required to guarantee the decomposition result.