7,339 research outputs found
Stable low-rank matrix recovery via null space properties
The problem of recovering a matrix of low rank from an incomplete and
possibly noisy set of linear measurements arises in a number of areas. In order
to derive rigorous recovery results, the measurement map is usually modeled
probabilistically. We derive sufficient conditions on the minimal amount of
measurements ensuring recovery via convex optimization. We establish our
results via certain properties of the null space of the measurement map. In the
setting where the measurements are realized as Frobenius inner products with
independent standard Gaussian random matrices we show that
measurements are enough to uniformly and stably recover an
matrix of rank at most . We then significantly generalize this result by
only requiring independent mean-zero, variance one entries with four finite
moments at the cost of replacing by some universal constant. We also study
the case of recovering Hermitian rank- matrices from measurement matrices
proportional to rank-one projectors. For rank-one projective
measurements onto independent standard Gaussian vectors, we show that nuclear
norm minimization uniformly and stably reconstructs Hermitian rank- matrices
with high probability. Next, we partially de-randomize this by establishing an
analogous statement for projectors onto independent elements of a complex
projective 4-designs at the cost of a slightly higher sampling rate . Moreover, if the Hermitian matrix to be recovered is known to be
positive semidefinite, then we show that the nuclear norm minimization approach
may be replaced by minimizing the -norm of the residual subject to the
positive semidefinite constraint. Then no estimate of the noise level is
required a priori. We discuss applications in quantum physics and the phase
retrieval problem.Comment: 26 page
A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices
This paper investigates the uniqueness of a nonnegative vector solution and
the uniqueness of a positive semidefinite matrix solution to underdetermined
linear systems. A vector solution is the unique solution to an underdetermined
linear system only if the measurement matrix has a row-span intersecting the
positive orthant. Focusing on two types of binary measurement matrices,
Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we
show that, in both cases, the support size of a unique nonnegative solution can
grow linearly, namely O(n), with the problem dimension n. We also provide
closed-form characterizations of the ratio of this support size to the signal
dimension. For the matrix case, we show that under a necessary and sufficient
condition for the linear compressed observations operator, there will be a
unique positive semidefinite matrix solution to the compressed linear
observations. We further show that a randomly generated Gaussian linear
compressed observations operator will satisfy this condition with
overwhelmingly high probability
Complex spherical codes with three inner products
Let be a finite set in a complex sphere of dimension. Let be
the set of usual inner products of two distinct vectors in . A set is
called a complex spherical -code if the cardinality of is and
contains an imaginary number. We would like to classify the largest
possible -codes for given dimension . In this paper, we consider the
problem for the case . Roy and Suda (2014) gave a certain upper bound for
the cardinalities of -codes. A -code is said to be tight if
attains the bound. We show that there exists no tight -code except for
dimensions , . Moreover we make an algorithm to classify the largest
-codes by considering representations of oriented graphs. By this algorithm,
the largest -codes are classified for dimensions , , with a
current computer.Comment: 26 pages, no figur
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