30,468 research outputs found
Randomized GCUR decompositions
By exploiting the random sampling techniques, this paper derives an efficient
randomized algorithm for computing a generalized CUR decomposition, which
provides low-rank approximations of both matrices simultaneously in terms of
some of their rows and columns. For large-scale data sets that are expensive to
store and manipulate, a new variant of the discrete empirical interpolation
method known as L-DEIM, which needs much lower cost and provides a significant
acceleration in practice, is also combined with the random sampling approach to
further improve the efficiency of our algorithm. Moreover, adopting the
randomized algorithm to implement the truncation process of restricted singular
value decomposition (RSVD), combined with the L-DEIM procedure, we propose a
fast algorithm for computing an RSVD based CUR decomposition, which provides a
coordinated low-rank approximation of the three matrices in a CUR-type format
simultaneously and provides advantages over the standard CUR approximation for
some applications. We establish detailed probabilistic error analysis for the
algorithms and provide numerical results that show the promise of our
approaches
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
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
Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
The global behavior of dynamical systems can be studied by analyzing the
eigenvalues and corresponding eigenfunctions of linear operators associated
with the system. Two important operators which are frequently used to gain
insight into the system's behavior are the Perron-Frobenius operator and the
Koopman operator. Due to the curse of dimensionality, computing the
eigenfunctions of high-dimensional systems is in general infeasible. We will
propose a tensor-based reformulation of two numerical methods for computing
finite-dimensional approximations of the aforementioned infinite-dimensional
operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD).
The aim of the tensor formulation is to approximate the eigenfunctions by
low-rank tensors, potentially resulting in a significant reduction of the time
and memory required to solve the resulting eigenvalue problems, provided that
such a low-rank tensor decomposition exists. Typically, not all variables of a
high-dimensional dynamical system contribute equally to the system's behavior,
often the dynamics can be decomposed into slow and fast processes, which is
also reflected in the eigenfunctions. Thus, the weak coupling between different
variables might be approximated by low-rank tensor cores. We will illustrate
the efficiency of the tensor-based formulation of Ulam's method and EDMD using
simple stochastic differential equations
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
In this paper we review basic and emerging models and associated algorithms
for large-scale tensor networks, especially Tensor Train (TT) decompositions
using novel mathematical and graphical representations. We discus the concept
of tensorization (i.e., creating very high-order tensors from lower-order
original data) and super compression of data achieved via quantized tensor
train (QTT) networks. The purpose of a tensorization and quantization is to
achieve, via low-rank tensor approximations "super" compression, and
meaningful, compact representation of structured data. The main objective of
this paper is to show how tensor networks can be used to solve a wide class of
big data optimization problems (that are far from tractable by classical
numerical methods) by applying tensorization and performing all operations
using relatively small size matrices and tensors and applying iteratively
optimized and approximative tensor contractions.
Keywords: Tensor networks, tensor train (TT) decompositions, matrix product
states (MPS), matrix product operators (MPO), basic tensor operations,
tensorization, distributed representation od data optimization problems for
very large-scale problems: generalized eigenvalue decomposition (GEVD),
PCA/SVD, canonical correlation analysis (CCA).Comment: arXiv admin note: text overlap with arXiv:1403.204
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