11,160 research outputs found
Sampling from large matrices: an approach through geometric functional analysis
We study random submatrices of a large matrix A. We show how to approximately
compute A from its random submatrix of the smallest possible size O(r log r)
with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the
numerical rank of A. The numerical rank is always bounded by, and is a stable
relaxation of, the rank of A. This yields an asymptotically optimal guarantee
in an algorithm for computing low-rank approximations of A. We also prove
asymptotically optimal estimates on the spectral norm and the cut-norm of
random submatrices of A. The result for the cut-norm yields a slight
improvement on the best known sample complexity for an approximation algorithm
for MAX-2CSP problems. We use methods of Probability in Banach spaces, in
particular the law of large numbers for operator-valued random variables.Comment: Our initial claim about Max-2-CSP problems is corrected. We put an
exponential failure probability for the algorithm for low-rank
approximations. Proofs are a little more explaine
Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations
A recurring theme in attempts to break the curse of dimensionality in the
numerical approximations of solutions to high-dimensional partial differential
equations (PDEs) is to employ some form of sparse tensor approximation.
Unfortunately, there are only a few results that quantify the possible
advantages of such an approach. This paper introduces a class of
functions, which can be written as a sum of rank-one tensors using a total of
at most parameters and then uses this notion of sparsity to prove a
regularity theorem for certain high-dimensional elliptic PDEs. It is shown,
among other results, that whenever the right-hand side of the elliptic PDE
can be approximated with a certain rate in the norm of
by elements of , then the solution can be
approximated in from to accuracy
for any . Since these results require
knowledge of the eigenbasis of the elliptic operator considered, we propose a
second "basis-free" model of tensor sparsity and prove a regularity theorem for
this second sparsity model as well. We then proceed to address the important
question of the extent such regularity theorems translate into results on
computational complexity. It is shown how this second model can be used to
derive computational algorithms with performance that breaks the curse of
dimensionality on certain model high-dimensional elliptic PDEs with
tensor-sparse data.Comment: 41 pages, 1 figur
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Blind Multilinear Identification
We discuss a technique that allows blind recovery of signals or blind
identification of mixtures in instances where such recovery or identification
were previously thought to be impossible: (i) closely located or highly
correlated sources in antenna array processing, (ii) highly correlated
spreading codes in CDMA radio communication, (iii) nearly dependent spectra in
fluorescent spectroscopy. This has important implications --- in the case of
antenna array processing, it allows for joint localization and extraction of
multiple sources from the measurement of a noisy mixture recorded on multiple
sensors in an entirely deterministic manner. In the case of CDMA, it allows the
possibility of having a number of users larger than the spreading gain. In the
case of fluorescent spectroscopy, it allows for detection of nearly identical
chemical constituents. The proposed technique involves the solution of a
bounded coherence low-rank multilinear approximation problem. We show that
bounded coherence allows us to establish existence and uniqueness of the
recovered solution. We will provide some statistical motivation for the
approximation problem and discuss greedy approximation bounds. To provide the
theoretical underpinnings for this technique, we develop a corresponding theory
of sparse separable decompositions of functions, including notions of rank and
nuclear norm that specialize to the usual ones for matrices and operators but
apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor
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