Reducibility and computational lower bounds for problems with planted sparse structure
Name
1051460504-MIT.pdf
Description
Full printable version
Size
33.41 MB
Format
Adobe PDF
Checksum (MD5)
f22f45daf9b7c70169979a316390db1b
Author(s)
Brennan, Matthew (Matthew Stewart)
Advisor(s)
Guy Bresler.
Date Issued
2018
Publisher
Massachusetts Institute of Technology
Abstract
Recently, research in unsupervised learning has gravitated towards exploring statistical-computational gaps induced by sparsity. A line of work initiated by Berthet and Rigollet has aimed to explain these gaps through reductions to conjecturally hard problems from complexity theory. However, the delicate nature of average-case reductions has limited the development of techniques and often led to weaker hardness results that only apply to algorithms that are robust to different noise distributions or that do not need to know the parameters of the problem. We introduce several new techniques to give a web of average-case reductions showing strong computational lower bounds based on the planted clique conjecture for planted independent set, planted dense subgraph, biclustering, sparse rank-1 submatrix, sparse PCA and the subgraph stochastic block model. Our results demonstrate that, despite the delicate nature of average-case reductions, using natural problems as intermediates can often be beneficial, as is the case in worst-case complexity. Our main technical contribution is to introduce a set of techniques for average-case reductions that: (1) maintain the level of signal in an instance of a problem; (2) alter its planted structure; and (3) map two initial high-dimensional distributions simultaneously to two target distributions approximately under total variation. We also give algorithms matching our lower bounds and identify the information-theoretic limits of the models we consider.
Description
Thesis: S.M. in Computer Science and Engineering, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 145-155).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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