New algorithms for learning incoherent and overcomplete dictionaries
Name
Moitra_New algorithms.pdf
Size
272.4 KB
Format
Adobe PDF
Checksum (MD5)
141b6061ea1079a9e3afa0f3465b9db9
Author(s)
Arora, Sanjeev
Ge, Rong
Moitra, Ankur
Date Issued
May 2014
Journal
Journal of Machine Learning Research: Workshop and Conference Proceedings
Publisher
Journal of Machine Learning Research
Citation
Arora, Sanjeev, Rong Ge, and Ankur Moitra. "New Algorithms for Learning Incoherent and Overcomplete Dictionaries." Journal of Machine Learning Research: Workshop and Conference Proceedings, 35 (2014), 779-806.
Version
Original manuscript
Abstract
In sparse recovery we are given a matrix A∈R[superscript n×m] (“the dictionary”) and a vector of the form AX where X is sparse, and the goal is to recover X. This is a central notion in signal processing, statistics and machine learning. But in applications such as sparse coding, edge detection, compression and super resolution, the dictionary A is unknown and has to be learned from random examples of the form Y=AX where X is drawn from an appropriate distribution — this is the dictionary learning problem. In most settings, A is overcomplete: it has more columns than rows. This paper presents a polynomial-time algorithm for learning overcomplete dictionaries; the only previously known algorithm with provable guarantees is the recent work of Spielman et al. (2012) who gave an algorithm for the undercomplete case, which is rarely the case in applications. Our algorithm applies to incoherent dictionaries which have been a central object of study since they were introduced in seminal work of Donoho and Huo (1999). In particular, a dictionary is μ-incoherent if each pair of columns has inner product at most μ/√n. The algorithm makes natural stochastic assumptions about the unknown sparse vector X, which can contain k≤cmin(√/n/μlogn,m[superscript 1/2−η]) non-zero entries (for any η>0). This is close to the best k allowable by the best sparse recovery algorithms even if one knows the dictionary A exactly. Moreover, both the running time and sample complexity depend on log1/ϵ, where ϵ is the target accuracy, and so our algorithms converge very quickly to the true dictionary. Our algorithm can also tolerate substantial amounts of noise provided it is incoherent with respect to the dictionary (e.g., Gaussian). In the noisy setting, our running time and sample complexity depend polynomially on 1/ϵ, and this is necessary.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
http://www.jmlr.org/proceedings/papers/v35/
