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Connections Between Nuclear-Norm and Frobenius-Norm-Based Representations.
IEEE Transactions on Neural Networks and Learning Systems
|October 11, 2016
Summary
Frobenius-norm-based representation (FNR) is theoretically shown to be equivalent to nuclear-norm-based representation (NNR) under sufficient dictionary capacity, even with noise. Otherwise, both methods yield solutions within the dictionary
Area of Science:
- Machine Learning
- Computer Vision
- Data Science
Background:
- Frobenius-norm-based representation (FNR) shows promise in tasks like subspace clustering, comparable to sparse and nuclear-norm-based representation (NNR).
- Existing research primarily focuses on FNR's experimental success, with limited theoretical understanding of its mechanisms.
Purpose of the Study:
- To establish theoretical connections between Frobenius-norm-based representation (FNR) and nuclear-norm-based representation (NNR).
- To elucidate the working principles of FNR through mathematical analysis.
Main Methods:
- Theoretical analysis to connect FNR and NNR.
- Mathematical proofs to demonstrate the relationship under varying conditions.
Main Results:
- FNR is proven to be equivalent to NNR when the dictionary has sufficient representative capacity, robust to various noise types (Gaussian, Laplacian, sample-specific corruption).
- When dictionary capacity is insufficient, FNR and NNR provide distinct solutions within the dictionary's column space.
Conclusions:
- The theoretical equivalence between FNR and NNR is established under specific conditions, clarifying FNR's behavior.
- This study bridges the gap between experimental observations and theoretical understanding of FNR in data representation and clustering.
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