439 research outputs found
Concentration of the Frobenius norm of generalized matrix inverses
Revised/condensed/renamed version of preprint "Beyond Moore-Penrose Part II: The Sparse Pseudoinverse"International audienceIn many applications it is useful to replace the Moore-Penrose pseudoinverse (MPP) by a different generalized inverse with more favorable properties. We may want, for example, to have many zero entries, but without giving up too much of the stability of the MPP. One way to quantify stability is by how much the Frobenius norm of a generalized inverse exceeds that of the MPP. In this paper we derive finite-size concentration bounds for the Frobenius norm of -minimal general inverses of iid Gaussian matrices, with . For we prove exponential concentration of the Frobenius norm of the sparse pseudoinverse; for , we get a similar concentration bound for the MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike previous applications which give asymptotic results, we derive finite-size bounds
Beyond Moore-Penrose Part II: The Sparse Pseudoinverse
This is the second part of a two-paper series on generalized inverses that
minimize matrix norms. In Part II we focus on generalized inverses that are
minimizers of entrywise p norms whose main representative is the sparse
pseudoinverse for . We are motivated by the idea to replace the
Moore-Penrose pseudoinverse by a sparser generalized inverse which is in some
sense well-behaved. Sparsity implies that it is faster to apply the resulting
matrix; well-behavedness would imply that we do not lose much in stability with
respect to the least-squares performance of the MPP. We first address questions
of uniqueness and non-zero count of (putative) sparse pseu-doinverses. We show
that a sparse pseudoinverse is generically unique, and that it indeed reaches
optimal sparsity for almost all matrices. We then turn to proving our main
stability result: finite-size concentration bounds for the Frobenius norm of
p-minimal inverses for \le\le. Our proof is based on tools from
convex analysis and random matrix theory, in particular the recently developed
convex Gaussian min-max theorem. Along the way we prove several results about
sparse representations and convex programming that were known folklore, but of
which we could find no proof
Spectral analysis of large reflexive generalized inverse and Moore-Penrose inverse matrices
A reflexive generalized inverse and the Moore-Penrose inverse are often
confused in statistical literature but in fact they have completely different
behaviour in case the population covariance matrix is not a multiple of
identity. In this paper, we study the spectral properties of a reflexive
generalized inverse and of the Moore-Penrose inverse of the sample covariance
matrix. The obtained results are used to assess the difference in the
asymptotic behaviour of their eigenvalues.Comment: 13 pages, 1 figure, a letter/short articl
Iterative Row Sampling
There has been significant interest and progress recently in algorithms that
solve regression problems involving tall and thin matrices in input sparsity
time. These algorithms find shorter equivalent of a n*d matrix where n >> d,
which allows one to solve a poly(d) sized problem instead. In practice, the
best performances are often obtained by invoking these routines in an iterative
fashion. We show these iterative methods can be adapted to give theoretical
guarantees comparable and better than the current state of the art.
Our approaches are based on computing the importances of the rows, known as
leverage scores, in an iterative manner. We show that alternating between
computing a short matrix estimate and finding more accurate approximate
leverage scores leads to a series of geometrically smaller instances. This
gives an algorithm that runs in
time for any , where the term is comparable
to the cost of solving a regression problem on the small approximation. Our
results are built upon the close connection between randomized matrix
algorithms, iterative methods, and graph sparsification.Comment: 26 pages, 2 figure
Nonparametric estimation of covariance functions by model selection
We propose a model selection approach for covariance estimation of a
multi-dimensional stochastic process. Under very general assumptions, observing
i.i.d replications of the process at fixed observation points, we construct an
estimator of the covariance function by expanding the process onto a collection
of basis functions. We study the non asymptotic property of this estimate and
give a tractable way of selecting the best estimator among a possible set of
candidates. The optimality of the procedure is proved via an oracle inequality
which warrants that the best model is selected
Algebraic properties of Manin matrices 1
We study a class of matrices with noncommutative entries, which were first
considered by Yu. I. Manin in 1988 in relation with quantum group theory. They
are defined as "noncommutative endomorphisms" of a polynomial algebra. More
explicitly their defining conditions read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). The basic claim
is that despite noncommutativity many theorems of linear algebra hold true for
Manin matrices in a form identical to that of the commutative case. Moreover in
some examples the converse is also true. The present paper gives a complete
list and detailed proofs of algebraic properties of Manin matrices known up to
the moment; many of them are new. In particular we present the formulation in
terms of matrix (Leningrad) notations; provide complete proofs that an inverse
to a M.m. is again a M.m. and for the Schur formula for the determinant of a
block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered
recently [arXiv:0809.3516], which includes the classical Capelli and related
identities. We also discuss many other properties, such as the Cramer formula
for the inverse matrix, the Cayley-Hamilton theorem, Newton and
MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the
Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some
multiplicativity properties for the determinant, relations with
quasideterminants, calculation of the determinant via Gauss decomposition,
conjugation to the second normal (Frobenius) form, and so on and so forth. We
refer to [arXiv:0711.2236] for some applications.Comment: 80 page
Signals on Graphs: Uncertainty Principle and Sampling
In many applications, the observations can be represented as a signal defined
over the vertices of a graph. The analysis of such signals requires the
extension of standard signal processing tools. In this work, first, we provide
a class of graph signals that are maximally concentrated on the graph domain
and on its dual. Then, building on this framework, we derive an uncertainty
principle for graph signals and illustrate the conditions for the recovery of
band-limited signals from a subset of samples. We show an interesting link
between uncertainty principle and sampling and propose alternative signal
recovery algorithms, including a generalization to frame-based reconstruction
methods. After showing that the performance of signal recovery algorithms is
significantly affected by the location of samples, we suggest and compare a few
alternative sampling strategies. Finally, we provide the conditions for perfect
recovery of a useful signal corrupted by sparse noise, showing that this
problem is also intrinsically related to vertex-frequency localization
properties.Comment: This article is the revised version submitted to the IEEE
Transactions on Signal Processing on May, 2016; first revision was submitted
on January, 2016; original manuscript was submitted on July, 2015. The work
includes 16 pages, 8 figure
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