Connections Between Nuclear Norm and Frobenius Norm Based Representation

Several recent works have shown that Frobenius-Norm based Representation (FNR) is comparable with Sparse Representation (SR) and Nuclear-Norm based Representation (NNR) in face recognition and 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 bridging FNR and NNR. More specially, we prove that: 1) when the dictionary can provide enough representative capacity, FNR is exactly the NNR; 2) Otherwise, FNR and NNR are two solutions on the column space of the dictionary. The first result provides a novel theoretical explanation towards some existing FNR based methods by crediting their success to low rank property. The second result provides a new insight to understand FNR and NNR under a unified framework.