8,933 research outputs found
A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing
Subspace-based signal processing traditionally focuses on problems involving
a few subspaces. Recently, a number of problems in different application areas
have emerged that involve a significantly larger number of subspaces relative
to the ambient dimension. It becomes imperative in such settings to first
identify a smaller set of active subspaces that contribute to the observation
before further processing can be carried out. This problem of identification of
a small set of active subspaces among a huge collection of subspaces from a
single (noisy) observation in the ambient space is termed subspace unmixing.
This paper formally poses the subspace unmixing problem under the parsimonious
subspace-sum (PS3) model, discusses connections of the PS3 model to problems in
wireless communications, hyperspectral imaging, high-dimensional statistics and
compressed sensing, and proposes a low-complexity algorithm, termed marginal
subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the
subspace unmixing problem for the PS3 model into a multiple hypothesis testing
(MHT) problem and its analysis in the paper helps control the family-wise error
rate of this MHT problem at any level under two random
signal generation models. Some other highlights of the analysis of the MSD
algorithm include: (i) it is applicable to an arbitrary collection of subspaces
on the Grassmann manifold; (ii) it relies on properties of the collection of
subspaces that are computable in polynomial time; and () it allows for
linear scaling of the number of active subspaces as a function of the ambient
dimension. Finally, numerical results are presented in the paper to better
understand the performance of the MSD algorithm.Comment: Submitted for journal publication; 33 pages, 14 figure
Block-Sparse Recovery via Convex Optimization
Given a dictionary that consists of multiple blocks and a signal that lives
in the range space of only a few blocks, we study the problem of finding a
block-sparse representation of the signal, i.e., a representation that uses the
minimum number of blocks. Motivated by signal/image processing and computer
vision applications, such as face recognition, we consider the block-sparse
recovery problem in the case where the number of atoms in each block is
arbitrary, possibly much larger than the dimension of the underlying subspace.
To find a block-sparse representation of a signal, we propose two classes of
non-convex optimization programs, which aim to minimize the number of nonzero
coefficient blocks and the number of nonzero reconstructed vectors from the
blocks, respectively. Since both classes of problems are NP-hard, we propose
convex relaxations and derive conditions under which each class of the convex
programs is equivalent to the original non-convex formulation. Our conditions
depend on the notions of mutual and cumulative subspace coherence of a
dictionary, which are natural generalizations of existing notions of mutual and
cumulative coherence. We evaluate the performance of the proposed convex
programs through simulations as well as real experiments on face recognition.
We show that treating the face recognition problem as a block-sparse recovery
problem improves the state-of-the-art results by 10% with only 25% of the
training data.Comment: IEEE Transactions on Signal Processin
Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
In this paper we establish links between, and new results for, three problems
that are not usually considered together. The first is a matrix decomposition
problem that arises in areas such as statistical modeling and signal
processing: given a matrix formed as the sum of an unknown diagonal matrix
and an unknown low rank positive semidefinite matrix, decompose into these
constituents. The second problem we consider is to determine the facial
structure of the set of correlation matrices, a convex set also known as the
elliptope. This convex body, and particularly its facial structure, plays a
role in applications from combinatorial optimization to mathematical finance.
The third problem is a basic geometric question: given points
(where ) determine whether there is a centered
ellipsoid passing \emph{exactly} through all of the points.
We show that in a precise sense these three problems are equivalent.
Furthermore we establish a simple sufficient condition on a subspace that
ensures any positive semidefinite matrix with column space can be
recovered from for any diagonal matrix using a convex
optimization-based heuristic known as minimum trace factor analysis. This
result leads to a new understanding of the structure of rank-deficient
correlation matrices and a simple condition on a set of points that ensures
there is a centered ellipsoid passing through them.Comment: 20 page
Eutactic quantum codes
We consider sets of quantum observables corresponding to eutactic stars.
Eutactic stars are systems of vectors which are the lower dimensional
``shadow'' image, the orthogonal view, of higher dimensional orthonormal bases.
Although these vector systems are not comeasurable, they represent redundant
coordinate bases with remarkable properties. One application is quantum secret
sharing.Comment: 6 page
Generation of Hyperentangled Photons Pairs
We experimentally demonstrate the first quantum system entangled in every
degree of freedom (hyperentangled). Using pairs of photons produced in
spontaneous parametric downconversion, we verify entanglement by observing a
Bell-type inequality violation in each degree of freedom: polarization, spatial
mode and time-energy. We also produce and characterize maximally hyperentangled
states and novel states simultaneously exhibiting both quantum and classical
correlations. Finally, we report the tomography of a 2x2x3x3 system
(36-dimensional Hilbert space), which we believe is the first reported photonic
entangled system of this size to be so characterized.Comment: 5 pages, 3 figures, 1 table, published versio
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