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Trading Group Theory for Randomness
, 1985
"... In a previous paper [BS] we proved, using the elements of the Clwory of nilyotenf yroupu, that some of the /undamcnla1 computational problems in mat & proup, belong to NP. These problems were also ahown to belong to CONP, assuming an unproven hypofhedi.9 concerning finilc simple Q ’ oup,. The a ..."
Abstract

Cited by 353 (9 self)
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prove th:rt. in spite of their analogy with the polynomial time hierarchy, the finite levrls of this hierarchy collapse t,o Afsf=Ah42). Using a combinatorial lemma on finite groups [IIE], we construct a game by whirh t.he nondeterministic player (Merlin) is able to coavlnre the random player (Arthur
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of . . .
 INSTITUTE FOR MATHEMATICAL SCIENCES, NATIONAL UNIVERSITY OF SINGAPORE, WORLD SCIENTIFIC
"... In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these area ..."
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Cited by 3 (1 self)
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in these areas. We begin with variations on König’s Lemma, and give an introduction to reverse mathematics and related parts of computability theory. We then focus on Ramsey’s Theorem as a case study in the computability theoretic and reverse mathematical analysis of combinatorial principles. We study Ramsey’s
Matrix factorization with Binary Components
"... Motivated by an application in computational biology, we consider lowrank matrix factorization with {0, 1}constraints on one of the factors and optionally convex constraints on the second one. In addition to the nonconvexity shared with other matrix factorization schemes, our problem is further ..."
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is further complicated by a combinatorial constraint set of size 2m·r, wherem is the dimension of the data points and r the rank of the factorization. Despite apparent intractability, we provide − in the line of recent work on nonnegative matrix factorization by Arora et al. (2012) − an algorithm