69 research outputs found
Convolutional Dictionary Learning through Tensor Factorization
Tensor methods have emerged as a powerful paradigm for consistent learning of
many latent variable models such as topic models, independent component
analysis and dictionary learning. Model parameters are estimated via CP
decomposition of the observed higher order input moments. However, in many
domains, additional invariances such as shift invariances exist, enforced via
models such as convolutional dictionary learning. In this paper, we develop
novel tensor decomposition algorithms for parameter estimation of convolutional
models. Our algorithm is based on the popular alternating least squares method,
but with efficient projections onto the space of stacked circulant matrices.
Our method is embarrassingly parallel and consists of simple operations such as
fast Fourier transforms and matrix multiplications. Our algorithm converges to
the dictionary much faster and more accurately compared to the alternating
minimization over filters and activation maps
-K-SVD: A Robust Dictionary Learning Algorithm With Simultaneous Update
We develop a dictionary learning algorithm by minimizing the
distortion metric on the data term, which is known to be robust for
non-Gaussian noise contamination. The proposed algorithm exploits the idea of
iterative minimization of weighted error. We refer to this algorithm
as -K-SVD, where the dictionary atoms and the corresponding sparse
coefficients are simultaneously updated to minimize the objective,
resulting in noise-robustness. We demonstrate through experiments that the
-K-SVD algorithm results in higher atom recovery rate compared with the
K-SVD and the robust dictionary learning (RDL) algorithm proposed by Lu et al.,
both in Gaussian and non-Gaussian noise conditions. We also show that, for
fixed values of sparsity, number of dictionary atoms, and data-dimension, the
-K-SVD algorithm outperforms the K-SVD and RDL algorithms when the
training set available is small. We apply the proposed algorithm for denoising
natural images corrupted by additive Gaussian and Laplacian noise. The images
denoised using -K-SVD are observed to have slightly higher peak
signal-to-noise ratio (PSNR) over K-SVD for Laplacian noise, but the
improvement in structural similarity index (SSIM) is significant (approximately
) for lower values of input PSNR, indicating the efficacy of the
metric
New Guarantees for Blind Compressed Sensing
Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS)
where the optimal sparsifying dictionary is assumed to be unknown and subject
to estimation (in addition to the CS sparse coefficients). Since the emergence
of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix
factorization problem where its sample complexity, uniqueness and
identifiability have been addressed thoroughly. However, in spite of the strong
connections between BCS and sparse coding, recent results from the sparse
coding problem area have not been exploited within the context of BCS. In
particular, prior BCS efforts have focused on learning constrained and complete
dictionaries that limit the scope and utility of these efforts. In this paper,
we develop new theoretical bounds for perfect recovery for the general
unconstrained BCS problem. These unconstrained BCS bounds cover the case of
overcomplete dictionaries, and hence, they go well beyond the existing BCS
theory. Our perfect recovery results integrate the combinatorial theories of
sparse coding with some of the recent results from low-rank matrix recovery. In
particular, we propose an efficient CS measurement scheme that results in
practical recovery bounds for BCS. Moreover, we discuss the performance of BCS
under polynomial-time sparse coding algorithms.Comment: To appear in the 53rd Annual Allerton Conference on Communication,
Control and Computing, University of Illinois at Urbana-Champaign, IL, USA,
201
Sparse Matrix Factorization
We investigate the problem of factorizing a matrix into several sparse
matrices and propose an algorithm for this under randomness and sparsity
assumptions. This problem can be viewed as a simplification of the deep
learning problem where finding a factorization corresponds to finding edges in
different layers and values of hidden units. We prove that under certain
assumptions for a sparse linear deep network with nodes in each layer, our
algorithm is able to recover the structure of the network and values of top
layer hidden units for depths up to . We further discuss the
relation among sparse matrix factorization, deep learning, sparse recovery and
dictionary learning.Comment: 20 page
Multi-modal dictionary learning for image separation with application in art investigation
In support of art investigation, we propose a new source separation method
that unmixes a single X-ray scan acquired from double-sided paintings. In this
problem, the X-ray signals to be separated have similar morphological
characteristics, which brings previous source separation methods to their
limits. Our solution is to use photographs taken from the front and back-side
of the panel to drive the separation process. The crux of our approach relies
on the coupling of the two imaging modalities (photographs and X-rays) using a
novel coupled dictionary learning framework able to capture both common and
disparate features across the modalities using parsimonious representations;
the common component models features shared by the multi-modal images, whereas
the innovation component captures modality-specific information. As such, our
model enables the formulation of appropriately regularized convex optimization
procedures that lead to the accurate separation of the X-rays. Our dictionary
learning framework can be tailored both to a single- and a multi-scale
framework, with the latter leading to a significant performance improvement.
Moreover, to improve further on the visual quality of the separated images, we
propose to train coupled dictionaries that ignore certain parts of the painting
corresponding to craquelure. Experimentation on synthetic and real data - taken
from digital acquisition of the Ghent Altarpiece (1432) - confirms the
superiority of our method against the state-of-the-art morphological component
analysis technique that uses either fixed or trained dictionaries to perform
image separation.Comment: submitted to IEEE Transactions on Images Processin
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