29,875 research outputs found
Optimal low-rank approximations of Bayesian linear inverse problems
In the Bayesian approach to inverse problems, data are often informative,
relative to the prior, only on a low-dimensional subspace of the parameter
space. Significant computational savings can be achieved by using this subspace
to characterize and approximate the posterior distribution of the parameters.
We first investigate approximation of the posterior covariance matrix as a
low-rank update of the prior covariance matrix. We prove optimality of a
particular update, based on the leading eigendirections of the matrix pencil
defined by the Hessian of the negative log-likelihood and the prior precision,
for a broad class of loss functions. This class includes the F\"{o}rstner
metric for symmetric positive definite matrices, as well as the
Kullback-Leibler divergence and the Hellinger distance between the associated
distributions. We also propose two fast approximations of the posterior mean
and prove their optimality with respect to a weighted Bayes risk under
squared-error loss. These approximations are deployed in an offline-online
manner, where a more costly but data-independent offline calculation is
followed by fast online evaluations. As a result, these approximations are
particularly useful when repeated posterior mean evaluations are required for
multiple data sets. We demonstrate our theoretical results with several
numerical examples, including high-dimensional X-ray tomography and an inverse
heat conduction problem. In both of these examples, the intrinsic
low-dimensional structure of the inference problem can be exploited while
producing results that are essentially indistinguishable from solutions
computed in the full space
Uniform Sampling for Matrix Approximation
Random sampling has become a critical tool in solving massive matrix
problems. For linear regression, a small, manageable set of data rows can be
randomly selected to approximate a tall, skinny data matrix, improving
processing time significantly. For theoretical performance guarantees, each row
must be sampled with probability proportional to its statistical leverage
score. Unfortunately, leverage scores are difficult to compute.
A simple alternative is to sample rows uniformly at random. While this often
works, uniform sampling will eliminate critical row information for many
natural instances. We take a fresh look at uniform sampling by examining what
information it does preserve. Specifically, we show that uniform sampling
yields a matrix that, in some sense, well approximates a large fraction of the
original. While this weak form of approximation is not enough for solving
linear regression directly, it is enough to compute a better approximation.
This observation leads to simple iterative row sampling algorithms for matrix
approximation that run in input-sparsity time and preserve row structure and
sparsity at all intermediate steps. In addition to an improved understanding of
uniform sampling, our main proof introduces a structural result of independent
interest: we show that every matrix can be made to have low coherence by
reweighting a small subset of its rows
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
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
- …