NDLI: On the local correctness of $ℓ^{1}-minimization$ for dictionary learning

Content Provider	IEEE Xplore Digital Library
Author	Quan Geng Wright, J.
Copyright Year	2014
Description	Author affiliation: Dept. of Electr. Eng., Columbia Univ., New York, NY, USA (Wright, J.) \|\| Dept. of ECE, UIUC, Urbana, IL, USA (Quan Geng)
Abstract	The idea that many important classes of signals can be well-represented by linear combinations of a small set of atoms selected from a given dictionary has had dramatic impact on the theory and practice of signal processing. For practical problems in which an appropriate sparsifying dictionary is not known ahead of time, a very popular and successful heuristic is to search for a dictionary that minimizes an appropriate sparsity surrogate over a given set of sample data. While there is a body of empirical evidence suggesting this approach does learn very effective representations, there is little theoretical guarantee. In this paper, we show that under mild hypotheses, the dictionary learning problem is locally well-posed: the desired solution is indeed a local minimum of the $ℓ^{1}$ norm. Namely, if A ∈ $ℝ^{m×n}$ is an incoherent (and possibly overcomplete) dictionary, and the coefficients X ∈ $ℝ^{n×p}$ follow a random sparse model, then with high probability (A, X) is a local minimum of the $ℓ^{1}$ norm over the manifold of factorizations (A', X') satisfying A'X' = Y, provided the number of samples p = $Ω(n^{3}k).$ For overcomplete A, this is the first result showing that the dictionary learning problem is even locally solvable using $ℓ^{1}-minimization.$
Starting Page	3180
Ending Page	3184
File Size	503285
Page Count	5
File Format	PDF
ISBN	9781479951864
ISSN	21578117
DOI	10.1109/ISIT.2014.6875421
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-06-29
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Dictionaries Information theory Vectors Minimization Optimization Manifolds Mathematical model
Content Type	Text
Resource Type	Article

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