12,237 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 Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
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