17,685 research outputs found
Rank-1 Tensor Approximation Methods and Application to Deflation
Because of the attractiveness of the canonical polyadic (CP) tensor
decomposition in various applications, several algorithms have been designed to
compute it, but efficient ones are still lacking. Iterative deflation
algorithms based on successive rank-1 approximations can be used to perform
this task, since the latter are rather easy to compute. We first present an
algebraic rank-1 approximation method that performs better than the standard
higher-order singular value decomposition (HOSVD) for three-way tensors.
Second, we propose a new iterative rank-1 approximation algorithm that improves
any other rank-1 approximation method. Third, we describe a probabilistic
framework allowing to study the convergence of deflation CP decomposition
(DCPD) algorithms based on successive rank-1 approximations. A set of computer
experiments then validates theoretical results and demonstrates the efficiency
of DCPD algorithms compared to other ones
A stochastic algorithm for probabilistic independent component analysis
The decomposition of a sample of images on a relevant subspace is a recurrent
problem in many different fields from Computer Vision to medical image
analysis. We propose in this paper a new learning principle and implementation
of the generative decomposition model generally known as noisy ICA (for
independent component analysis) based on the SAEM algorithm, which is a
versatile stochastic approximation of the standard EM algorithm. We demonstrate
the applicability of the method on a large range of decomposition models and
illustrate the developments with experimental results on various data sets.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS499 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
Decoding the Encoding of Functional Brain Networks: an fMRI Classification Comparison of Non-negative Matrix Factorization (NMF), Independent Component Analysis (ICA), and Sparse Coding Algorithms
Brain networks in fMRI are typically identified using spatial independent
component analysis (ICA), yet mathematical constraints such as sparse coding
and positivity both provide alternate biologically-plausible frameworks for
generating brain networks. Non-negative Matrix Factorization (NMF) would
suppress negative BOLD signal by enforcing positivity. Spatial sparse coding
algorithms ( Regularized Learning and K-SVD) would impose local
specialization and a discouragement of multitasking, where the total observed
activity in a single voxel originates from a restricted number of possible
brain networks.
The assumptions of independence, positivity, and sparsity to encode
task-related brain networks are compared; the resulting brain networks for
different constraints are used as basis functions to encode the observed
functional activity at a given time point. These encodings are decoded using
machine learning to compare both the algorithms and their assumptions, using
the time series weights to predict whether a subject is viewing a video,
listening to an audio cue, or at rest, in 304 fMRI scans from 51 subjects.
For classifying cognitive activity, the sparse coding algorithm of
Regularized Learning consistently outperformed 4 variations of ICA across
different numbers of networks and noise levels (p0.001). The NMF algorithms,
which suppressed negative BOLD signal, had the poorest accuracy. Within each
algorithm, encodings using sparser spatial networks (containing more
zero-valued voxels) had higher classification accuracy (p0.001). The success
of sparse coding algorithms may suggest that algorithms which enforce sparse
coding, discourage multitasking, and promote local specialization may capture
better the underlying source processes than those which allow inexhaustible
local processes such as ICA
Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization
The l1/l2 ratio regularization function has shown good performance for
retrieving sparse signals in a number of recent works, in the context of blind
deconvolution. Indeed, it benefits from a scale invariance property much
desirable in the blind context. However, the l1/l2 function raises some
difficulties when solving the nonconvex and nonsmooth minimization problems
resulting from the use of such a penalty term in current restoration methods.
In this paper, we propose a new penalty based on a smooth approximation to the
l1/l2 function. In addition, we develop a proximal-based algorithm to solve
variational problems involving this function and we derive theoretical
convergence results. We demonstrate the effectiveness of our method through a
comparison with a recent alternating optimization strategy dealing with the
exact l1/l2 term, on an application to seismic data blind deconvolution.Comment: 5 page
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