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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 /undamcn-la1 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
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Cited by 353 (9 self)
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prove th:rt. in spite of their analogy with the polynomial time hierarchy, the finite lev-rls of this hierarchy collapse t,o Afsf=Ah42). Using a com-binatorial 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 ..."
Abstract
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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 com-binatorial principles. We study Ramsey’s
Matrix factorization with Binary Components
"... Motivated by an application in computational biology, we consider low-rank ma-trix factorization with {0, 1}-constraints on one of the factors and optionally con-vex constraints on the second one. In addition to the non-convexity shared with other matrix factorization schemes, our problem is further ..."
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is further complicated by a com-binatorial 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 non-negative matrix factorization by Arora et al. (2012) − an algorithm
