Randomized Riemannian Preconditioning for Orthogonality Constrained Problems

Optimization problems with (generalized) orthogonality constraints are prevalent across science and engineering. For example, in computational science they arise in the symmetric (generalized) eigenvalue problem, in nonlinear eigenvalue problems, and in electronic structures computations, to name a few problems. In statistics and machine learning, they arise, for example, in canonical correlation analysis and in linear discriminant analysis. In this article, we consider using randomized preconditioning in the context of optimization problems with generalized orthogonality constraints. Our proposed algorithms are based on Riemannian optimization on the generalized Stiefel manifold equipped with a non-standard preconditioned geometry, which necessitates development of the geometric components necessary for developing algorithms based on this approach. Furthermore, we perform asymptotic convergence analysis of the preconditioned algorithms which help to characterize the quality of a given preconditioner using second-order information. Finally, for the problems of canonical correlation analysis and linear discriminant analysis, we develop randomized preconditioners along with corresponding bounds on the relevant condition number

An algorithm for transistor sizing in CMOS circuits

This paper describes a novel algorithm for automatic transistor sizing which is one technique for improving timing performance in CMOS circuits. The sizing algorithm is used to minimize area and power subject to timing constraints. We define the transistor sizing problem as a graph problem and use a non-linear optimization technique. The algorithm consists of three separate tasks: critical path analysis, transistor sizing and transistor desizing. The main contribution of the presented algorithm is that the delays of all paths in a given design can be tuned simultaneously to satisfy timing constraints. Furthermore, the minimal transistor area and minimal power dissipation under giving timing constraints can be achieved. Experimental results show that this approach has greater control over area/time tradeoffs than traditional sizing algorithms