5,777 research outputs found
An MDL framework for sparse coding and dictionary learning
The power of sparse signal modeling with learned over-complete dictionaries
has been demonstrated in a variety of applications and fields, from signal
processing to statistical inference and machine learning. However, the
statistical properties of these models, such as under-fitting or over-fitting
given sets of data, are still not well characterized in the literature. As a
result, the success of sparse modeling depends on hand-tuning critical
parameters for each data and application. This work aims at addressing this by
providing a practical and objective characterization of sparse models by means
of the Minimum Description Length (MDL) principle -- a well established
information-theoretic approach to model selection in statistical inference. The
resulting framework derives a family of efficient sparse coding and dictionary
learning algorithms which, by virtue of the MDL principle, are completely
parameter free. Furthermore, such framework allows to incorporate additional
prior information to existing models, such as Markovian dependencies, or to
define completely new problem formulations, including in the matrix analysis
area, in a natural way. These virtues will be demonstrated with parameter-free
algorithms for the classic image denoising and classification problems, and for
low-rank matrix recovery in video applications
Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds
Recovery of the sparsity pattern (or support) of an unknown sparse vector
from a small number of noisy linear measurements is an important problem in
compressed sensing. In this paper, the high-dimensional setting is considered.
It is shown that if the measurement rate and per-sample signal-to-noise ratio
(SNR) are finite constants independent of the length of the vector, then the
optimal sparsity pattern estimate will have a constant fraction of errors.
Lower bounds on the measurement rate needed to attain a desired fraction of
errors are given in terms of the SNR and various key parameters of the unknown
vector. The tightness of the bounds in a scaling sense, as a function of the
SNR and the fraction of errors, is established by comparison with existing
achievable bounds. Near optimality is shown for a wide variety of practically
motivated signal models
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