Sample Complexity Bounds for Low-Separation-Rank Dictionary Learning

2019 ◽  
Author(s):  
Mohsen Ghassemi ◽  
Zahra Shakeri ◽  
Waheed U. Bajwa ◽  
Anand D. Sarwate
Keyword(s):  
Dictionary Learning ◽  
Sample Complexity ◽  
Complexity Bounds

scholarly journals Convergence bounds for empirical nonlinear least-squares

2021 ◽  
Author(s):  
Philipp Trunschke ◽  
Martin Eigel ◽  
Reinhold Schneider
Keyword(s):  
Least Squares ◽  
Best Approximation ◽  
Approximation Error ◽  
Nonlinear Least Squares ◽  
Optimal Sampling ◽  
Sample Complexity ◽  
Complexity Bounds ◽  
Monte Carlo Estimate ◽  
General Bound ◽  
Carlo Estimate

We consider best approximation problems in a nonlinear subset  [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.

scholarly journals Sample Complexity of Dictionary Learning and Other Matrix Factorizations

2015 ◽  
Vol 61 (6) ◽  
pp. 3469-3486 ◽  
Author(s):  
Remi Gribonval ◽  
Rodolphe Jenatton ◽  
Francis Bach ◽  
Martin Kleinsteuber ◽  
Matthias Seibert
Keyword(s):  
Dictionary Learning ◽  
Matrix Factorizations ◽  
Sample Complexity

scholarly journals Size-independent sample complexity of neural networks

2020 ◽  
Vol 9 (2) ◽  
pp. 473-504 ◽  
Author(s):  
Noah Golowich ◽  
Alexander Rakhlin ◽  
Ohad Shamir
Keyword(s):  
Neural Networks ◽  
Network Size ◽  
Independent Interest ◽  
Sample Complexity ◽  
Complexity Bounds ◽  
Rademacher Complexity ◽  
Parameter Matrix ◽  
Additional Assumptions

Abstract We study the sample complexity of learning neural networks by providing new bounds on their Rademacher complexity, assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth and, under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest.

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