4,484 research outputs found
Learning parametric dictionaries for graph signals
In sparse signal representation, the choice of a dictionary often involves a
tradeoff between two desirable properties -- the ability to adapt to specific
signal data and a fast implementation of the dictionary. To sparsely represent
signals residing on weighted graphs, an additional design challenge is to
incorporate the intrinsic geometric structure of the irregular data domain into
the atoms of the dictionary. In this work, we propose a parametric dictionary
learning algorithm to design data-adapted, structured dictionaries that
sparsely represent graph signals. In particular, we model graph signals as
combinations of overlapping local patterns. We impose the constraint that each
dictionary is a concatenation of subdictionaries, with each subdictionary being
a polynomial of the graph Laplacian matrix, representing a single pattern
translated to different areas of the graph. The learning algorithm adapts the
patterns to a training set of graph signals. Experimental results on both
synthetic and real datasets demonstrate that the dictionaries learned by the
proposed algorithm are competitive with and often better than unstructured
dictionaries learned by state-of-the-art numerical learning algorithms in terms
of sparse approximation of graph signals. In contrast to the unstructured
dictionaries, however, the dictionaries learned by the proposed algorithm
feature localized atoms and can be implemented in a computationally efficient
manner in signal processing tasks such as compression, denoising, and
classification
Graph learning under sparsity priors
Graph signals offer a very generic and natural representation for data that
lives on networks or irregular structures. The actual data structure is however
often unknown a priori but can sometimes be estimated from the knowledge of the
application domain. If this is not possible, the data structure has to be
inferred from the mere signal observations. This is exactly the problem that we
address in this paper, under the assumption that the graph signals can be
represented as a sparse linear combination of a few atoms of a structured graph
dictionary. The dictionary is constructed on polynomials of the graph
Laplacian, which can sparsely represent a general class of graph signals
composed of localized patterns on the graph. We formulate a graph learning
problem, whose solution provides an ideal fit between the signal observations
and the sparse graph signal model. As the problem is non-convex, we propose to
solve it by alternating between a signal sparse coding and a graph update step.
We provide experimental results that outline the good graph recovery
performance of our method, which generally compares favourably to other recent
network inference algorithms
Matrix of Polynomials Model based Polynomial Dictionary Learning Method for Acoustic Impulse Response Modeling
We study the problem of dictionary learning for signals that can be
represented as polynomials or polynomial matrices, such as convolutive signals
with time delays or acoustic impulse responses. Recently, we developed a method
for polynomial dictionary learning based on the fact that a polynomial matrix
can be expressed as a polynomial with matrix coefficients, where the
coefficient of the polynomial at each time lag is a scalar matrix. However, a
polynomial matrix can be also equally represented as a matrix with polynomial
elements. In this paper, we develop an alternative method for learning a
polynomial dictionary and a sparse representation method for polynomial signal
reconstruction based on this model. The proposed methods can be used directly
to operate on the polynomial matrix without having to access its coefficients
matrices. We demonstrate the performance of the proposed method for acoustic
impulse response modeling.Comment: 5 pages, 2 figure
Simple, Efficient, and Neural Algorithms for Sparse Coding
Sparse coding is a basic task in many fields including signal processing,
neuroscience and machine learning where the goal is to learn a basis that
enables a sparse representation of a given set of data, if one exists. Its
standard formulation is as a non-convex optimization problem which is solved in
practice by heuristics based on alternating minimization. Re- cent work has
resulted in several algorithms for sparse coding with provable guarantees, but
somewhat surprisingly these are outperformed by the simple alternating
minimization heuristics. Here we give a general framework for understanding
alternating minimization which we leverage to analyze existing heuristics and
to design new ones also with provable guarantees. Some of these algorithms seem
implementable on simple neural architectures, which was the original motivation
of Olshausen and Field (1997a) in introducing sparse coding. We also give the
first efficient algorithm for sparse coding that works almost up to the
information theoretic limit for sparse recovery on incoherent dictionaries. All
previous algorithms that approached or surpassed this limit run in time
exponential in some natural parameter. Finally, our algorithms improve upon the
sample complexity of existing approaches. We believe that our analysis
framework will have applications in other settings where simple iterative
algorithms are used.Comment: 37 pages, 1 figur
Entropy of Overcomplete Kernel Dictionaries
In signal analysis and synthesis, linear approximation theory considers a
linear decomposition of any given signal in a set of atoms, collected into a
so-called dictionary. Relevant sparse representations are obtained by relaxing
the orthogonality condition of the atoms, yielding overcomplete dictionaries
with an extended number of atoms. More generally than the linear decomposition,
overcomplete kernel dictionaries provide an elegant nonlinear extension by
defining the atoms through a mapping kernel function (e.g., the gaussian
kernel). Models based on such kernel dictionaries are used in neural networks,
gaussian processes and online learning with kernels.
The quality of an overcomplete dictionary is evaluated with a diversity
measure the distance, the approximation, the coherence and the Babel measures.
In this paper, we develop a framework to examine overcomplete kernel
dictionaries with the entropy from information theory. Indeed, a higher value
of the entropy is associated to a further uniform spread of the atoms over the
space. For each of the aforementioned diversity measures, we derive lower
bounds on the entropy. Several definitions of the entropy are examined, with an
extensive analysis in both the input space and the mapped feature space.Comment: 10 page
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