504,497 research outputs found
Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels
In this paper, we investigate power-constrained sensing matrix design in a
sparse Gaussian linear dimensionality reduction framework. Our study is carried
out in a single--terminal setup as well as in a multi--terminal setup
consisting of orthogonal or coherent multiple access channels (MAC). We adopt
the mean square error (MSE) performance criterion for sparse source
reconstruction in a system where source-to-sensor channel(s) and
sensor-to-decoder communication channel(s) are noisy. Our proposed sensing
matrix design procedure relies upon minimizing a lower-bound on the MSE in
single-- and multiple--terminal setups. We propose a three-stage sensing matrix
optimization scheme that combines semi-definite relaxation (SDR) programming, a
low-rank approximation problem and power-rescaling. Under certain conditions,
we derive closed-form solutions to the proposed optimization procedure. Through
numerical experiments, by applying practical sparse reconstruction algorithms,
we show the superiority of the proposed scheme by comparing it with other
relevant methods. This performance improvement is achieved at the price of
higher computational complexity. Hence, in order to address the complexity
burden, we present an equivalent stochastic optimization method to the problem
of interest that can be solved approximately, while still providing a superior
performance over the popular methods.Comment: Accepted for publication in IEEE Transactions on Signal Processing
(16 pages
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Quench Dynamics in Randomly Generated Extended Quantum Models
We analyze the thermalization properties and the validity of the Eigenstate
Thermalization Hypothesis in a generic class of quantum Hamiltonians where the
quench parameter explicitly breaks a Z_2 symmetry. Natural realizations of such
systems are given by random matrices expressed in a block form where the terms
responsible for the quench dynamics are the off-diagonal blocks. Our analysis
examines both dense and sparse random matrix realizations of the Hamiltonians
and the observables. Sparse random matrices may be associated with local
quantum Hamiltonians and they show a different spread of the observables on the
energy eigenstates with respect to the dense ones. In particular, the numerical
data seems to support the existence of rare states, i.e. states where the
observables take expectation values which are different compared to the typical
ones sampled by the micro-canonical distribution. In the case of sparse random
matrices we also extract the finite size behavior of two different time scales
associated with the thermalization process.Comment: 30 pages, 44 figure
Large deviations of spread measures for Gaussian matrices
For a large Gaussian matrix, we compute the joint statistics,
including large deviation tails, of generalized and total variance - the scaled
log-determinant and trace of the corresponding covariance
matrix. Using a Coulomb gas technique, we find that the Laplace transform of
their joint distribution decays for large (with
fixed) as , where is the Dyson index of the ensemble and
is a -independent large deviation function, which we compute exactly for
any . The corresponding large deviation functions in real space are worked
out and checked with extensive numerical simulations. The results are
complemented with a finite treatment based on the Laguerre-Selberg
integral. The statistics of atypically small log-determinants is shown to be
driven by the split-off of the smallest eigenvalue, leading to an abrupt change
in the large deviation speed.Comment: 20 pages, 3 figures. v4: final versio
A fast algorithm for matrix balancing
As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
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