Connections Between Nuclear Norm and Frobenius Norm Based Representations
This provides theoretical insights for researchers in subspace clustering and representation learning, though it is incremental as it builds on existing empirical studies.
The paper tackles the lack of theoretical understanding of Frobenius-norm based representation (FNR) by proving connections to nuclear-norm based representation (NNR), showing that FNR equals NNR under sufficient dictionary capacity even with noise, and otherwise they are solutions on the same column space.
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.