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Start Over Topic approximation theory Topic constraint satisfaction Topic mathematical optimization Publisher academic press inc.
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1. Intractability and approximation of optimization theories of cognition

Author

van Rooij, Iris and Wareham, Todd

Subjects
*COGNITION, *MATHEMATICAL optimization, *MATHEMATICAL models, *HOPFIELD networks, *MATHEMATICAL functions, *APPROXIMATION theory
Abstract

Abstract: Many computational- or rational-level theories of human cognition suffer from computational intractability: the postulated optimization functions are impossible to compute in a reasonable time by a finite mind/brain, or any other computational mechanism. It has been proposed that such intractable theories can nevertheless have explanatory force if we assume that human cognitive processes somehow approximate the optimal function. This raises the question of when a cognitive process can be said to approximate an optimal function. In this paper, we distinguish between two notions of approximation, called value-approximation and structure-approximation respectively, and show that they are not equivalent. Although a mathematical framework for assessing degrees of tractable value-approximability has long been available, no such framework previously existed for structure-approximability. In this paper, we present a framework consisting of definitions and proof techniques for assessing degrees of structure-approximability. We illustrate the use of our framework for a particular intractable cognitive theory, i.e., Coherence model, known to be equivalent to harmony maximization in Hopfield networks. We discuss implications of our findings for this class of theories, as well as explain how similar results may be derived for other intractable optimization theories of cognition. [Copyright &y& Elsevier]

Published
2012
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2. Sparse recovery with coherent tight frames via analysis Dantzig selector and analysis LASSO.

Author

Lin, Junhong and Li, Song

Subjects
*APPROXIMATION theory, *MATHEMATICAL optimization, *CONSTRAINT satisfaction, *ISOMETRICS (Mathematics), *ERROR analysis in mathematics, *RANDOM noise theory, *ALGORITHMS
Abstract

Abstract: This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly constrained -analysis optimization problem, called analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a tight frame provided that the measurement matrix satisfies a restricted isometry property adapted to the tight frame. As a special case, we consider the Gaussian noise. Further, under a sparsity scenario, with high probability, the recovery error from noisy data is within a log-like factor of the minimax risk over the class of vectors which are at most s sparse in terms of the tight frame. Similar results for the analysis LASSO are shown. The above two algorithms provide guarantees only for noise that is bounded or bounded with high probability (for example, Gaussian noise). However, when the underlying measurements are corrupted by sparse noise, these algorithms perform suboptimally. We propose new algorithms for reconstructing signals that are nearly sparse in terms of a tight frame in the presence of bounded noise combined with sparse noise, and present corresponding recovery guarantees. The analysis in this paper is based on the restricted isometry property adapted to a tight frame, which is a natural extension to the standard restricted isometry property. [Copyright &y& Elsevier]

Published
2014
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