18,599 research outputs found
The Generalized Spike Process, Sparsity, and Statistical Independence
A basis under which a given set of realizations of a stochastic process can
be represented most sparsely (the so-called best sparsifying basis (BSB)) and
the one under which such a set becomes as less statistically dependent as
possible (the so-called least statistically-dependent basis (LSDB)) are
important for data compression and have generated interests among computational
neuroscientists as well as applied mathematicians. Here we consider these bases
for a particularly simple stochastic process called ``generalized spike
process'', which puts a single spike--whose amplitude is sampled from the
standard normal distribution--at a random location in the zero vector of length
\ndim for each realization.
Unlike the ``simple spike process'' which we dealt with in our previous paper
and whose amplitude is constant, we need to consider the kurtosis-maximizing
basis (KMB) instead of the LSDB due to the difficulty of evaluating
differential entropy and mutual information of the generalized spike process.
By computing the marginal densities and moments, we prove that: 1) the BSB and
the KMB selects the standard basis if we restrict our basis search within all
possible orthonormal bases in ; 2) if we extend our basis search
to all possible volume-preserving invertible linear transformations, then the
BSB exists and is again the standard basis whereas the KMB does not exist.
Thus, the KMB is rather sensitive to the orthonormality of the transformations
under consideration whereas the BSB is insensitive to that. Our results once
again support the preference of the BSB over the LSDB/KMB for data compression
applications as our previous work did.Comment: 26 pages, 2 figure
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
Sparsity vs. Statistical Independence in Adaptive Signal Representations: A Case Study of the Spike Process
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 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
-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 and the search of basis is
restricted within all possible orthonormal bases in ; 2) If we extend our
basis search in all possible invertible linear transformations in , 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 .Comment: 39 pages, 7 figures, submitted to Annals of the Institute of
Statistical Mathematic
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
Structured Sparsity Models for Multiparty Speech Recovery from Reverberant Recordings
We tackle the multi-party speech recovery problem through modeling the
acoustic of the reverberant chambers. Our approach exploits structured sparsity
models to perform room modeling and speech recovery. We propose a scheme for
characterizing the room acoustic from the unknown competing speech sources
relying on localization of the early images of the speakers by sparse
approximation of the spatial spectra of the virtual sources in a free-space
model. The images are then clustered exploiting the low-rank structure of the
spectro-temporal components belonging to each source. This enables us to
identify the early support of the room impulse response function and its unique
map to the room geometry. To further tackle the ambiguity of the reflection
ratios, we propose a novel formulation of the reverberation model and estimate
the absorption coefficients through a convex optimization exploiting joint
sparsity model formulated upon spatio-spectral sparsity of concurrent speech
representation. The acoustic parameters are then incorporated for separating
individual speech signals through either structured sparse recovery or inverse
filtering the acoustic channels. The experiments conducted on real data
recordings demonstrate the effectiveness of the proposed approach for
multi-party speech recovery and recognition.Comment: 31 page
Spectro-Perfectionism: An Algorithmic Framework for Photon Noise-Limited Extraction of Optical Fiber Spectroscopy
We describe a new algorithm for the "perfect" extraction of one-dimensional
spectra from two-dimensional (2D) digital images of optical fiber
spectrographs, based on accurate 2D forward modeling of the raw pixel data. The
algorithm is correct for arbitrarily complicated 2D point-spread functions
(PSFs), as compared to the traditional optimal extraction algorithm, which is
only correct for a limited class of separable PSFs. The algorithm results in
statistically independent extracted samples in the 1D spectrum, and preserves
the full native resolution of the 2D spectrograph without degradation. Both the
statistical errors and the 1D resolution of the extracted spectrum are
accurately determined, allowing a correct chi-squared comparison of any model
spectrum with the data. Using a model PSF similar to that found in the red
channel of the Sloan Digital Sky Survey spectrograph, we compare the
performance of our algorithm to that of cross-section based optimal extraction,
and also demonstrate that our method allows coaddition and foreground
estimation to be carried out as an integral part of the extraction step. This
work demonstrates the feasibility of current- and next-generation multi-fiber
spectrographs for faint galaxy surveys even in the presence of strong night-sky
foregrounds. We describe the handling of subtleties arising from fiber-to-fiber
crosstalk, discuss some of the likely challenges in deploying our method to the
analysis of a full-scale survey, and note that our algorithm could be
generalized into an optimal method for the rectification and combination of
astronomical imaging data.Comment: 9 pages, 4 figures, emulateapj; minor corrections and clarifications;
to be published in the PAS
Machine Learning for Fluid Mechanics
The field of fluid mechanics is rapidly advancing, driven by unprecedented
volumes of data from field measurements, experiments and large-scale
simulations at multiple spatiotemporal scales. Machine learning offers a wealth
of techniques to extract information from data that could be translated into
knowledge about the underlying fluid mechanics. Moreover, machine learning
algorithms can augment domain knowledge and automate tasks related to flow
control and optimization. This article presents an overview of past history,
current developments, and emerging opportunities of machine learning for fluid
mechanics. It outlines fundamental machine learning methodologies and discusses
their uses for understanding, modeling, optimizing, and controlling fluid
flows. The strengths and limitations of these methods are addressed from the
perspective of scientific inquiry that considers data as an inherent part of
modeling, experimentation, and simulation. Machine learning provides a powerful
information processing framework that can enrich, and possibly even transform,
current lines of fluid mechanics research and industrial applications.Comment: To appear in the Annual Reviews of Fluid Mechanics, 202
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