13,979 research outputs found
Limits on Support Recovery with Probabilistic Models: An Information-Theoretic Framework
The support recovery problem consists of determining a sparse subset of a set
of variables that is relevant in generating a set of observations, and arises
in a diverse range of settings such as compressive sensing, and subset
selection in regression, and group testing. In this paper, we take a unified
approach to support recovery problems, considering general probabilistic models
relating a sparse data vector to an observation vector. We study the
information-theoretic limits of both exact and partial support recovery, taking
a novel approach motivated by thresholding techniques in channel coding. We
provide general achievability and converse bounds characterizing the trade-off
between the error probability and number of measurements, and we specialize
these to the linear, 1-bit, and group testing models. In several cases, our
bounds not only provide matching scaling laws in the necessary and sufficient
number of measurements, but also sharp thresholds with matching constant
factors. Our approach has several advantages over previous approaches: For the
achievability part, we obtain sharp thresholds under broader scalings of the
sparsity level and other parameters (e.g., signal-to-noise ratio) compared to
several previous works, and for the converse part, we not only provide
conditions under which the error probability fails to vanish, but also
conditions under which it tends to one.Comment: Accepted to IEEE Transactions on Information Theory; presented in
part at ISIT 2015 and SODA 201
Tight conditions for consistency of variable selection in the context of high dimensionality
We address the issue of variable selection in the regression model with very
high ambient dimension, that is, when the number of variables is very large.
The main focus is on the situation where the number of relevant variables,
called intrinsic dimension, is much smaller than the ambient dimension d.
Without assuming any parametric form of the underlying regression function, we
get tight conditions making it possible to consistently estimate the set of
relevant variables. These conditions relate the intrinsic dimension to the
ambient dimension and to the sample size. The procedure that is provably
consistent under these tight conditions is based on comparing quadratic
functionals of the empirical Fourier coefficients with appropriately chosen
threshold values. The asymptotic analysis reveals the presence of two quite
different re gimes. The first regime is when the intrinsic dimension is fixed.
In this case the situation in nonparametric regression is the same as in linear
regression, that is, consistent variable selection is possible if and only if
log d is small compared to the sample size n. The picture is different in the
second regime, that is, when the number of relevant variables denoted by s
tends to infinity as . Then we prove that consistent variable
selection in nonparametric set-up is possible only if s+loglog d is small
compared to log n. We apply these results to derive minimax separation rates
for the problem of variableComment: arXiv admin note: text overlap with arXiv:1102.3616; Published in at
http://dx.doi.org/10.1214/12-AOS1046 the Annals of Statistics
(http://www.imstat.org/aos/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Sparse Support Vector Infinite Push
In this paper, we address the problem of embedded feature selection for
ranking on top of the list problems. We pose this problem as a regularized
empirical risk minimization with -norm push loss function () and
sparsity inducing regularizers. We leverage the issues related to this
challenging optimization problem by considering an alternating direction method
of multipliers algorithm which is built upon proximal operators of the loss
function and the regularizer. Our main technical contribution is thus to
provide a numerical scheme for computing the infinite push loss function
proximal operator. Experimental results on toy, DNA microarray and BCI problems
show how our novel algorithm compares favorably to competitors for ranking on
top while using fewer variables in the scoring function.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012
A Feature Selection Method for Multivariate Performance Measures
Feature selection with specific multivariate performance measures is the key
to the success of many applications, such as image retrieval and text
classification. The existing feature selection methods are usually designed for
classification error. In this paper, we propose a generalized sparse
regularizer. Based on the proposed regularizer, we present a unified feature
selection framework for general loss functions. In particular, we study the
novel feature selection paradigm by optimizing multivariate performance
measures. The resultant formulation is a challenging problem for
high-dimensional data. Hence, a two-layer cutting plane algorithm is proposed
to solve this problem, and the convergence is presented. In addition, we adapt
the proposed method to optimize multivariate measures for multiple instance
learning problems. The analyses by comparing with the state-of-the-art feature
selection methods show that the proposed method is superior to others.
Extensive experiments on large-scale and high-dimensional real world datasets
show that the proposed method outperforms -SVM and SVM-RFE when choosing a
small subset of features, and achieves significantly improved performances over
SVM in terms of -score
Jointly Sparse Support Recovery via Deep Auto-encoder with Applications in MIMO-based Grant-Free Random Access for mMTC
In this paper, a data-driven approach is proposed to jointly design the
common sensing (measurement) matrix and jointly support recovery method for
complex signals, using a standard deep auto-encoder for real numbers. The
auto-encoder in the proposed approach includes an encoder that mimics the noisy
linear measurement process for jointly sparse signals with a common sensing
matrix, and a decoder that approximately performs jointly sparse support
recovery based on the empirical covariance matrix of noisy linear measurements.
The proposed approach can effectively utilize the feature of common support and
properties of sparsity patterns to achieve high recovery accuracy, and has
significantly shorter computation time than existing methods. We also study an
application example, i.e., device activity detection in Multiple-Input
Multiple-Output (MIMO)-based grant-free random access for massive machine type
communications (mMTC). The numerical results show that the proposed approach
can provide pilot sequences and device activity detection with better detection
accuracy and substantially shorter computation time than well-known recovery
methods.Comment: 5 pages, 8 figures, to be publised in IEEE SPAWC 2020. arXiv admin
note: text overlap with arXiv:2002.0262
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