1,609 research outputs found
Sparse and Smooth Prior for Bayesian Linear Regression with Application to ETEX Data
Sparsity of the solution of a linear regression model is a common
requirement, and many prior distributions have been designed for this purpose.
A combination of the sparsity requirement with smoothness of the solution is
also common in application, however, with considerably fewer existing prior
models. In this paper, we compare two prior structures, the Bayesian fused
lasso (BFL) and least-squares with adaptive prior covariance matrix (LS-APC).
Since only variational solution was published for the latter, we derive a Gibbs
sampling algorithm for its inference and Bayesian model selection. The method
is designed for high dimensional problems, therefore, we discuss numerical
issues associated with evaluation of the posterior. In simulation, we show that
the LS-APC prior achieves results comparable to that of the Bayesian Fused
Lasso for piecewise constant parameter and outperforms the BFL for parameters
of more general shapes. Another advantage of the LS-APC priors is revealed in
real application to estimation of the release profile of the European Tracer
Experiment (ETEX). Specifically, the LS-APC model provides more conservative
uncertainty bounds when the regressor matrix is not informative
Online Low-Rank Subspace Learning from Incomplete Data: A Bayesian View
Extracting the underlying low-dimensional space where high-dimensional
signals often reside has long been at the center of numerous algorithms in the
signal processing and machine learning literature during the past few decades.
At the same time, working with incomplete (partly observed) large scale
datasets has recently been commonplace for diverse reasons. This so called {\it
big data era} we are currently living calls for devising online subspace
learning algorithms that can suitably handle incomplete data. Their envisaged
objective is to {\it recursively} estimate the unknown subspace by processing
streaming data sequentially, thus reducing computational complexity, while
obviating the need for storing the whole dataset in memory. In this paper, an
online variational Bayes subspace learning algorithm from partial observations
is presented. To account for the unawareness of the true rank of the subspace,
commonly met in practice, low-rankness is explicitly imposed on the sought
subspace data matrix by exploiting sparse Bayesian learning principles.
Moreover, sparsity, {\it simultaneously} to low-rankness, is favored on the
subspace matrix by the sophisticated hierarchical Bayesian scheme that is
adopted. In doing so, the proposed algorithm becomes adept in dealing with
applications whereby the underlying subspace may be also sparse, as, e.g., in
sparse dictionary learning problems. As shown, the new subspace tracking scheme
outperforms its state-of-the-art counterparts in terms of estimation accuracy,
in a variety of experiments conducted on simulated and real data
An Adaptive Markov Random Field for Structured Compressive Sensing
Exploiting intrinsic structures in sparse signals underpins the recent
progress in compressive sensing (CS). The key for exploiting such structures is
to achieve two desirable properties: generality (\ie, the ability to fit a wide
range of signals with diverse structures) and adaptability (\ie, being adaptive
to a specific signal). Most existing approaches, however, often only achieve
one of these two properties. In this study, we propose a novel adaptive Markov
random field sparsity prior for CS, which not only is able to capture a broad
range of sparsity structures, but also can adapt to each sparse signal through
refining the parameters of the sparsity prior with respect to the compressed
measurements. To maximize the adaptability, we also propose a new sparse signal
estimation where the sparse signals, support, noise and signal parameter
estimation are unified into a variational optimization problem, which can be
effectively solved with an alternative minimization scheme. Extensive
experiments on three real-world datasets demonstrate the effectiveness of the
proposed method in recovery accuracy, noise tolerance, and runtime.Comment: 13 pages, submitted to IEEE Transactions on Image Processin
Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset
Recent research on problem formulations based on decomposition into low-rank
plus sparse matrices shows a suitable framework to separate moving objects from
the background. The most representative problem formulation is the Robust
Principal Component Analysis (RPCA) solved via Principal Component Pursuit
(PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix.
However, similar robust implicit or explicit decompositions can be made in the
following problem formulations: Robust Non-negative Matrix Factorization
(RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust
Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal
of these similar problem formulations is to obtain explicitly or implicitly a
decomposition into low-rank matrix plus additive matrices. In this context,
this work aims to initiate a rigorous and comprehensive review of the similar
problem formulations in robust subspace learning and tracking based on
decomposition into low-rank plus additive matrices for testing and ranking
existing algorithms for background/foreground separation. For this, we first
provide a preliminary review of the recent developments in the different
problem formulations which allows us to define a unified view that we called
Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine
carefully each method in each robust subspace learning/tracking frameworks with
their decomposition, their loss functions, their optimization problem and their
solvers. Furthermore, we investigate if incremental algorithms and real-time
implementations can be achieved for background/foreground separation. Finally,
experimental results on a large-scale dataset called Background Models
Challenge (BMC 2012) show the comparative performance of 32 different robust
subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv
admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297,
arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805,
arXiv:1403.8067 by other authors, Computer Science Review, November 201
Lasso Meets Horseshoe : A Survey
The goal of this paper is to contrast and survey the major advances in two of
the most commonly used high-dimensional techniques, namely, the Lasso and
horseshoe regularization. Lasso is a gold standard for predictor selection
while horseshoe is a state-of-the-art Bayesian estimator for sparse signals.
Lasso is fast and scalable and uses convex optimization whilst the horseshoe is
non-convex. Our novel perspective focuses on three aspects: (i) theoretical
optimality in high dimensional inference for the Gaussian sparse model and
beyond, (ii) efficiency and scalability of computation and (iii) methodological
development and performance.Comment: 32 pages, 4 figure
Bayesian inference in high-dimensional models
Models with dimension more than the available sample size are now commonly
used in various applications. A sensible inference is possible using a
lower-dimensional structure. In regression problems with a large number of
predictors, the model is often assumed to be sparse, with only a few predictors
active. Interdependence between a large number of variables is succinctly
described by a graphical model, where variables are represented by nodes on a
graph and an edge between two nodes is used to indicate their conditional
dependence given other variables. Many procedures for making inferences in the
high-dimensional setting, typically using penalty functions to induce sparsity
in the solution obtained by minimizing a loss function, were developed.
Bayesian methods have been proposed for such problems more recently, where the
prior takes care of the sparsity structure. These methods have the natural
ability to also automatically quantify the uncertainty of the inference through
the posterior distribution. Theoretical studies of Bayesian procedures in
high-dimension have been carried out recently. Questions that arise are,
whether the posterior distribution contracts near the true value of the
parameter at the minimax optimal rate, whether the correct lower-dimensional
structure is discovered with high posterior probability, and whether a credible
region has adequate frequentist coverage. In this paper, we review these
properties of Bayesian and related methods for several high-dimensional models
such as many normal means problem, linear regression, generalized linear
models, Gaussian and non-Gaussian graphical models. Effective computational
approaches are also discussed.Comment: Review chapter, 42 page
On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models
We study high-dimensional Bayesian linear regression with a general beta
prime distribution for the scale parameter. Under the assumption of sparsity,
we show that appropriate selection of the hyperparameters in the beta prime
prior leads to the (near) minimax posterior contraction rate when .
For finite samples, we propose a data-adaptive method for estimating the
hyperparameters based on marginal maximum likelihood (MML). This enables our
prior to adapt to both sparse and dense settings, and under our proposed
empirical Bayes procedure, the MML estimates are never at risk of collapsing to
zero. We derive efficient Monte Carlo EM and variational EM algorithms for
implementing our model, which are available in the R package NormalBetaPrime.
Simulations and analysis of a gene expression data set illustrate our model's
self-adaptivity to varying levels of sparsity and signal strengths.Comment: 37 pages, 4 figures, 3 tables. We have added a section on posterior
computation and corrected the theoretical results. Sections on normal means
estimation were removed in this updated technical repor
Iteratively Reweighted Approaches to Sparse Composite Regularization
Motivated by the observation that a given signal admits
sparse representations in multiple dictionaries but with
varying levels of sparsity across dictionaries, we propose two new algorithms
for the reconstruction of (approximately) sparse signals from noisy linear
measurements. Our first algorithm, Co-L1, extends the well-known lasso
algorithm from the L1 regularizer to composite
regularizers of the form while self-adjusting the regularization weights
. Our second algorithm, Co-IRW-L1, extends the well-known
iteratively reweighted L1 algorithm to the same family of composite
regularizers. We provide several interpretations of both algorithms: i)
majorization-minimization (MM) applied to a non-convex log-sum-type penalty,
ii) MM applied to an approximate -type penalty, iii) MM applied to
Bayesian MAP inference under a particular hierarchical prior, and iv)
variational expectation-maximization (VEM) under a particular prior with
deterministic unknown parameters. A detailed numerical study suggests that our
proposed algorithms yield significantly improved recovery SNR when compared to
their non-composite L1 and IRW-L1 counterparts
Supervised Multiscale Dimension Reduction for Spatial Interaction Networks
We introduce a multiscale supervised dimension reduction method for SPatial
Interaction Network (SPIN) data, which consist of a collection of spatially
coordinated interactions. This type of predictor arises when the sampling unit
of data is composed of a collection of primitive variables, each of them being
essentially unique, so that it becomes necessary to group the variables in
order to simplify the representation and enhance interpretability. In this
paper, we introduce an empirical Bayes approach called spinlets, which first
constructs a partitioning tree to guide the reduction over multiple spatial
granularities, and then refines the representation of predictors according to
the relevance to the response. We consider an inverse Poisson regression model
and propose a new multiscale generalized double Pareto prior, which is induced
via a tree-structured parameter expansion scheme. Our approach is motivated by
an application in soccer analytics, in which we obtain compact vectorial
representations and readily interpretable visualizations of the complex network
objects, supervised by the response of interest.Comment: 30 pages, 12 figures, revised for clarity and concisenes
Spike-and-Slab Meets LASSO: A Review of the Spike-and-Slab LASSO
High-dimensional data sets have become ubiquitous in the past few decades,
often with many more covariates than observations. In the frequentist setting,
penalized likelihood methods are the most popular approach for variable
selection and estimation in high-dimensional data. In the Bayesian framework,
spike-and-slab methods are commonly used as probabilistic constructs for
high-dimensional modeling. Within the context of linear regression, Rockova and
George (2018) introduced the spike-and-slab LASSO (SSL), an approach based on a
prior which provides a continuum between the penalized likelihood LASSO and the
Bayesian point-mass spike-and-slab formulations. Since its inception, the
spike-and-slab LASSO has been extended to a variety of contexts, including
generalized linear models, factor analysis, graphical models, and nonparametric
regression. The goal of this paper is to survey the landscape surrounding
spike-and-slab LASSO methodology. First we elucidate the attractive properties
and the computational tractability of SSL priors in high dimensions. We then
review methodological developments of the SSL and outline several theoretical
developments. We illustrate the methodology on both simulated and real
datasets.Comment: 34 pages, 2 tables, 3 figures. Section 3.3 was added to illustrate
the metho
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