244 research outputs found
A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary
A nonnegative matrix factorization (NMF) can be computed efficiently under
the separability assumption, which asserts that all the columns of the given
input data matrix belong to the cone generated by a (small) subset of them. The
provably most robust methods to identify these conic basis columns are based on
nonnegative sparse regression and self dictionaries, and require the solution
of large-scale convex optimization problems. In this paper we study a
particular nonnegative sparse regression model with self dictionary. As opposed
to previously proposed models, this model yields a smooth optimization problem
where the sparsity is enforced through linear constraints. We show that the
Euclidean projection on the polyhedron defined by these constraints can be
computed efficiently, and propose a fast gradient method to solve our model. We
compare our algorithm with several state-of-the-art methods on synthetic data
sets and real-world hyperspectral images
Generalized Separable Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a linear dimensionality technique
for nonnegative data with applications such as image analysis, text mining,
audio source separation and hyperspectral unmixing. Given a data matrix and
a factorization rank , NMF looks for a nonnegative matrix with
columns and a nonnegative matrix with rows such that .
NMF is NP-hard to solve in general. However, it can be computed efficiently
under the separability assumption which requires that the basis vectors appear
as data points, that is, that there exists an index set such that
. In this paper, we generalize the separability
assumption: We only require that for each rank-one factor for
, either for some or for
some . We refer to the corresponding problem as generalized separable NMF
(GS-NMF). We discuss some properties of GS-NMF and propose a convex
optimization model which we solve using a fast gradient method. We also propose
a heuristic algorithm inspired by the successive projection algorithm. To
verify the effectiveness of our methods, we compare them with several
state-of-the-art separable NMF algorithms on synthetic, document and image data
sets.Comment: 31 pages, 12 figures, 4 tables. We have added discussions about the
identifiability of the model, we have modified the first synthetic
experiment, we have clarified some aspects of the contributio
Spectral Unmixing with Multiple Dictionaries
Spectral unmixing aims at recovering the spectral signatures of materials,
called endmembers, mixed in a hyperspectral or multispectral image, along with
their abundances. A typical assumption is that the image contains one pure
pixel per endmember, in which case spectral unmixing reduces to identifying
these pixels. Many fully automated methods have been proposed in recent years,
but little work has been done to allow users to select areas where pure pixels
are present manually or using a segmentation algorithm. Additionally, in a
non-blind approach, several spectral libraries may be available rather than a
single one, with a fixed number (or an upper or lower bound) of endmembers to
chose from each. In this paper, we propose a multiple-dictionary constrained
low-rank matrix approximation model that address these two problems. We propose
an algorithm to compute this model, dubbed M2PALS, and its performance is
discussed on both synthetic and real hyperspectral images
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
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