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

Geometric symmetry in the quadratic Fisher discriminant operating on image pixels

This article examines the design of Quadratic Fisher Discriminants (QFDs) that operate directly on image pixels, when image ensembles are taken to comprise all rotated and reflected versions of distinct sample images. A procedure based on group theory is devised to identify and discard QFD coefficients made redundant by symmetry, for arbitrary sampling lattices. This procedure introduces the concept of a degeneracy matrix. Tensor representations are established for the square lattice point group (8-fold symmetry) and hexagonal lattice point group (12-fold symmetry). The analysis is largely applicable to the symmetrisation of any quadratic filter, and generalises to higher order polynomial (Volterra) filters. Experiments on square lattice sampled synthetic aperture radar (SAR) imagery verify that symmetrisation of QFDs can improve their generalisation and discrimination ability.Comment: Accepted for publication in IEEE Transactions on Information Theor

Parallel-cascade realizations and approximations of truncated volterra systems

Journal ArticleAbstract This paper introduces parallel-cascade realizations of truncated Volterra systems with arbitrary, but finite order of nonlinearity. Parallel-cascade realizations implement higher-order Volterra systems using parallel and multiplicative combinations of lower-order Volterra systems. Such realizations are very modular and therefore well-suited for VLSI implementation. A Systematic way of approximating higher-order Volterra systems in parallel-cascade form using a reduced number of branches and a bound on the mean-square error in the output signals caused by such approximate realizations are derived in this paper. A variation of the parallel-cascade structure in which a pth order Volterra filter is implemented as a parallel combination of linear filters whose outputs are raised to the pth power is also described in this paper

Parallel-cascade realizations and approximations of truncated volterra systems

Journal ArticleThis paper introduces parallel-cascade realizations of truncated Volterra systems with arbitrary, but finite order of nonlinearity. Parallel-cascade realizations implement higher-order Volterra systems using parallel and multiplicative combinations of lower-order Volterra systems. Such realizations are very modular and therefore well-suited for VLSI implementation. A systematic way of approximating higher-order Volterra systems in parallel-cascade form using a reduced number of branches and a bound on the mean-square error in the output signals caused by such approximate realizations are derived in this paper. A variation of the parallel-cascade structure in which a pth order Volterra filter is implemented as a parallel combination of linear filters whose outputs are raised to the pth power is also described in this paper

Black box probabilistic numerics

Peer reviewe

Black box probabilistic numerics

Peer reviewe

A Comparative Analysis of the Complexity/Accuracy Tradeoff in Power Amplifier Behavioral Models

A comparative study of state-of-the-art behavioral models for microwave power amplifiers (PAs) is presented in this paper. After establishing a proper definition for accuracy and complexity for PA behavioral models, a short description on various behavioral models is presented. The main focus of this paper is on the modeling accuracy as a function of computational complexity. Data is collected from measurements on two PAs—a general-purpose amplifier and a Doherty PA designed for WiMAX—for different output power levels. The models are characterized in terms of accuracy and complexity for both in-band and out-of-band error. The results show that, among the models studied, the generalized memory polynomial behavioral model has the best tradeoff for accuracy versus complexity for both PAs, and can obtain high performance at half of the computational cost of all other models analyzed

A Primer on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal Processin