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Yedidia J.S.,Disney Research Boston
Journal of Statistical Physics | Year: 2011

Message-passing algorithms can solve a wide variety of optimization, inference, and constraint satisfaction problems. The algorithms operate on factor graphs that visually represent and specify the structure of the problems. After describing some of their applications, I survey the family of belief propagation (BP) algorithms, beginning with a detailed description of the min-sum algorithm and its exactness on tree factor graphs, and then turning to a variety of more sophisticated BP algorithms, including free-energy based BP algorithms, "splitting" BP algorithms that generalize "tree-reweighted" BP, and the various BP algorithms that have been proposed to deal with problems with continuous variables. The Divide and Concur (DC) algorithm is a projection-based constraint satisfaction algorithm that deals naturally with continuous variables, and converges to exact answers for problems where the solution sets of the constraints are convex. I show how it exploits the "difference-map" dynamics to avoid traps that cause more naive alternating projection algorithms to fail for non-convex problems, and explain that it is a message-passing algorithm that can also be applied to optimization problems. The BP and DC algorithms are compared, both in terms of their fundamental justifications and their strengths and weaknesses. © 2011 Springer Science+Business Media, LLC. Source

Wang Y.,MItsubishi Electric | Wang Y.,South China University of Technology | Draper S.C.,MItsubishi Electric | Draper S.C.,University of Toronto | And 2 more authors.
IEEE Transactions on Information Theory | Year: 2013

We present an approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation, and compatible with some desired input code structure defined using a protograph. The approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi-cyclic (HQC) LDPC codes. Whereas the parity check matrices of standard quasi-cyclic (QC) LDPC codes are composed of circulant submatrices, those of HQC LDPC codes are composed of a hierarchy of circulant submatrices that are, in turn, constructed from circulant submatrices, and so on, through some number of levels. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code, subject to bounds imposed by the fact that HQC codes are still quasi-cyclic. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a "squashing" procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with three design examples of QC LDPC codes - two girth-10 codes of rates 1/3 and 0.45 and one girth-8 code of rate 0.7 - all of which are obtained from protographs of one-sided spatially coupled codes. © 2013 IEEE. Source

Alexa M.,TU Berlin | Alexa M.,University of Toronto | Alexa M.,Disney Research Boston | Matusik W.,Massachusetts Institute of Technology | Matusik W.,Disney Research Boston
Computers and Graphics (Pergamon) | Year: 2012

We propose a complete system for designing, simulating, and fabricating surfaces with shading due to self-occlusion that induce desired input images. Our work is based on a simple observation. Consider a cylindrical hole (a pit) in a planar surface. As the depth of the hole increases, the radiance emitted from the surface patch that contains the hole decreases. This is because more light is trapped and absorbed in the hole. First, we propose a measurement-based approach that derives a mapping between average albedo of the surface patch containing the hole and the hole depth. Given this mapping and an input image, we show how to produce a distribution of holes with varied depth that approximates the image well. We demonstrate that by aligning holes with image features, while still maintaining an irregular distribution, we can obtain reproductions that look better than those resulting from regular hole patterns - despite using less holes. We validate this method on a variety of images and corresponding surfaces fabricated with a computer-controlled milling machine and a 3D printer. © 2012 Elsevier Ltd. Source

Derbinsky N.,Disney Research Boston | Bento J.,Disney Research Boston | Yedidia J.S.,Disney Research Boston
Biologically Inspired Cognitive Architectures | Year: 2014

In this paper we consider optimization as an approach for quickly and flexibly developing hybrid cognitive capabilities that are efficient, scalable, and can exploit task knowledge to improve solution speed and quality. Given this context, we focus on the Three-Weight Algorithm, which is interruptible, scalable, and aims to solve general optimization problems. We propose novel methods by which to integrate diverse forms of task knowledge with this algorithm in order to improve expressiveness, efficiency, and scaling across a variety of problems. To demonstrate these techniques, we focus on two large-scale constraint-satisfaction domains, Sudoku and circle packing. In Sudoku, we efficiently and dynamically integrate knowledge of logically deduced sub-problem solutions; this integration leads to improved system reactivity and greatly reduced solution time for large problem instances. In circle packing, we efficiently integrate knowledge of task dynamics, as well as real-time human guidance via mouse gestures; these integrations lead to greatly improved system reactivity, as well as world-record-breaking solutions on very large packing problems. These results exemplify how cognitive architecture can integrate high-level knowledge with powerful optimization techniques in order to effectively and efficiently contend with a variety of cognitive tasks. © 2014 Elsevier B.V. All rights reserved. Source

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