From: Amara Graps (amara@amara.com)
Date: Sun Jun 03 2001 - 17:30:09 MDT
For your amusement: The following excerpt from the latest Wavelet Digest
describes a stochastic process called "The Spike Process."
Amara
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Date: Tue, 13 Mar 2001 21:21:42 -0800
From: Naoki Saito <saito@math.ucdavis.edu>
Subject: #6 Preprint: Sparsity vs. Statistical Independence
Dear Wavelet Digest subscribers,
The following preprint is available from my web page:
http://math.ucdavis.edu/~saito/publications/spikes.html
This contains some unexpected results and I am very interested in hearing
your comments!
Title: Sparsity vs. Statistical Independence in Adaptive Signal
Representations: A Case Study of the Spike Process
Authors: Bertrand Benichou and Naoki Saito
Institute: Department of Mathematics, University of California, Davis
Abstract: Finding a basis/coordinate system that can efficiently
represent an input data stream by viewing them as realizations of a
stochastic process is of tremendous importance in many fields
including data compression and computational neuroscience. Two popular
measures of such efficiency of a basis are sparsity (measured by the
expected $\ell^p$ norm) and statistical independence (measured by the
mutual information). Gaining deeper understanding of their intricate
relationship, however, remains elusive. Therefore, we chose to study
a simple synthetic stochastic process called the spike process, which
puts a unit impulse at a random location in an $n$-dimensional vector
for each realization. For this process, we obtained the following
results: 1) The standard basis is the best both in terms of sparsity
and statistical independence if $n \geq 5$ and the search of basis is
restricted within all possible orthonormal bases in ${\bf R}^n$; 2) If
we extend our basis search in all possible invertible linear
transformations in ${\bf R}^n$, then the best basis in statistical
independence differs from the one in sparsity; 3) In either of the
above, the best basis in statistical independence is not unique, and
there even exist those which make the inputs completely dense; 4)
There is no linear invertible transformation that achieves the true
statistical independence for $n > 2$.
Keywords: Sparse representation, statistical independence, data compression,
basis dictionary, best basis, spike process
Also, in my web page:
http://math.ucdavis.edu/~saito/publications/
there are several related papers you can retrieve.
Sincerely yours,
Naoki Saito
--
Naoki Saito, Ph.D., Associate Professor, Department of Mathematics
University of California, One Shields Avenue, Davis, CA 95616-8633 USA
Voice: 530-754-2121, Fax: 530-752-6635, Email: saito@math.ucdavis.edu
Home Page: http://math.ucdavis.edu/~saito/
********************************************************************
Amara Graps email: amara@amara.com
Computational Physics vita: finger agraps@shell5.ba.best.com
Multiplex Answers URL: http://www.amara.com/
********************************************************************
"There are strange things done in the midnight sun..."
-- Robert W. Service
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