2,053 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
Riemannian preconditioned algorithms for tensor completion via tensor ring decomposition
We propose Riemannian preconditioned algorithms for the tensor completion
problem via tensor ring decomposition. A new Riemannian metric is developed on
the product space of the mode-2 unfolding matrices of the core tensors in
tensor ring decomposition. The construction of this metric aims to approximate
the Hessian of the cost function by its diagonal blocks, paving the way for
various Riemannian optimization methods. Specifically, we propose the
Riemannian gradient descent and Riemannian conjugate gradient algorithms. We
prove that both algorithms globally converge to a stationary point. In the
implementation, we exploit the tensor structure and adopt an economical
procedure to avoid large matrix formulation and computation in gradients, which
significantly reduces the computational cost. Numerical experiments on various
synthetic and real-world datasets -- movie ratings, hyperspectral images, and
high-dimensional functions -- suggest that the proposed algorithms are more
efficient and have better reconstruction ability than other candidates.Comment: 25 pages, 7 figures, 5 table
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