Presentation 2014-11-21
Dynamics of compressed sensing at zero temperature
Jun-ichi INOUE,
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Abstract(in English) We discuss dynamics of signal restoring process in compressed sensing based on the L_r-minimizer. In particular, we will focus on the dynamics at 'zero temperature', which means that we utilize the MAP estimate for the solution by gradient descent learning of the cost function (logarithm of the posterior). When we regard the number of sampling as 'the number of examples', and the original signal to be reconstructed as 'synaptic weight vector', the dynamics of compressed sensing could be treated as a sort of 'learning from examples'. In this paper, we examine the case of r=2. Then, we find that the steady state of estimated signal vector by means of the gradient descent learning is written in terms of the pseudo inverse matrix, which has been well-known in the literature of 'AdaLine (Adaptive Linear Neuron) learning' for artificial neural networks. In order to evaluate the average-case performance of reconstruction dynamics, we derive coarse-grained dynamics with respect to several macroscopic quantities. We find that the dynamics contains an eigenvalue spectrum for correlation matrix whose elements are constructed by observation matrix. For the simplest choice of ensemble of observation matrix, we calculate the explicit form of the eigenvalue distribution and discuss the learning rate and hyper-parameter dependences of the mean-square error, and we also argue the residual error as a function of sparse and sampling rates.
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Keyword(in English) Compressed sensing / Signal reconstruction dynamics / Learning theory / Random matrix theory / Average-case performance
Paper # NC2014-29
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Committee NC
Conference Date 2014/11/14(1days)
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Registration To Neurocomputing (NC)
Language JPN
Title (in Japanese) (See Japanese page)
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Title (in English) Dynamics of compressed sensing at zero temperature
Sub Title (in English)
Keyword(1) Compressed sensing
Keyword(2) Signal reconstruction dynamics
Keyword(3) Learning theory
Keyword(4) Random matrix theory
Keyword(5) Average-case performance
1st Author's Name Jun-ichi INOUE
1st Author's Affiliation Graduate School of Information Science and Technology, Hokkaido University()
Date 2014-11-21
Paper # NC2014-29
Volume (vol) vol.114
Number (no) 326
Page pp.pp.-
#Pages 6
Date of Issue