Connections Between Nuclear Norm and Frobenius Norm Based Representations

TL;DR

This paper establishes theoretical links between Frobenius-norm based and nuclear-norm based representations, explaining their similarities and differences in noisy data scenarios, especially in subspace clustering tasks.

Contribution

It provides the first theoretical analysis connecting FNR and NNR, showing their equivalence under certain conditions and their relation in noisy environments.

Findings

FNR equals NNR when the dictionary has sufficient representation capacity.

FNR and NNR are solutions on the column space of the dictionary in less ideal conditions.

Theoretical insights explain the empirical success of FNR in noisy data tasks.

Abstract

A lot of works have shown that frobenius-norm based representation (FNR) is competitive to sparse representation and nuclear-norm based representation (NNR) in numerous tasks such as subspace clustering. Despite the success of FNR in experimental studies, less theoretical analysis is provided to understand its working mechanism. In this paper, we fill this gap by building the theoretical connections between FNR and NNR. More specially, we prove that: 1) when the dictionary can provide enough representative capacity, FNR is exactly NNR even though the data set contains the Gaussian noise, Laplacian noise, or sample-specified corruption, 2) otherwise, FNR and NNR are two solutions on the column space of the dictionary.