2,494 research outputs found
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
Low rank tensor recovery via iterative hard thresholding
We study extensions of compressive sensing and low rank matrix recovery
(matrix completion) to the recovery of low rank tensors of higher order from a
small number of linear measurements. While the theoretical understanding of low
rank matrix recovery is already well-developed, only few contributions on the
low rank tensor recovery problem are available so far. In this paper, we
introduce versions of the iterative hard thresholding algorithm for several
tensor decompositions, namely the higher order singular value decomposition
(HOSVD), the tensor train format (TT), and the general hierarchical Tucker
decomposition (HT). We provide a partial convergence result for these
algorithms which is based on a variant of the restricted isometry property of
the measurement operator adapted to the tensor decomposition at hand that
induces a corresponding notion of tensor rank. We show that subgaussian
measurement ensembles satisfy the tensor restricted isometry property with high
probability under a certain almost optimal bound on the number of measurements
which depends on the corresponding tensor format. These bounds are extended to
partial Fourier maps combined with random sign flips of the tensor entries.
Finally, we illustrate the performance of iterative hard thresholding methods
for tensor recovery via numerical experiments where we consider recovery from
Gaussian random measurements, tensor completion (recovery of missing entries),
and Fourier measurements for third order tensors.Comment: 34 page
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