Spectral Norm of Symmetric Functions

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions

Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

A set in a domain in has the norm-preserving extension property if every bounded holomorphic function on has a holomorphic extension to with the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton, itself, a complex geodesic of , or the union of the set and a complex geodesic of degree in . We also prove that the complex geodesics in coincide with the nontrivial holomorphic retracts in . Thus, in contrast to the case of the ball or the bidisc, there are sets in which have the norm-preserving extension property but are not holomorphic retracts of . In the course of the proof we obtain a detailed classification of the complex geodesics in modulo automorphisms of . We give applications to von Neumann-type inequalities for -contractions (that is, commuting pairs of operators for which the closure of is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions

The Eigenvalue Problem for Linear and Affine Iterated Function Systems

The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite number of linear or affine functions. The eigen-equation becomes F(X) = rX, where r>0 is real, X is a compact set, and F(X)is the union of f(X), for f in F. The main result is that an irreducible, linear iterated function system F has a unique eigenvalue r equal to the joint spectral radius of the functions in F and a corresponding eigenset S that is centrally symmetric, star-shaped, and full dimensional. Results of Barabanov and of Dranishnikov-Konyagin-Protasov on the joint spectral radius follow as corollaries.Comment: 18 pages, 3 figure

Low-Rank Inducing Norms with Optimality Interpretations

Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing with these types of problem. This paper introduces a family of low-rank inducing norms and regularizers which includes the nuclear norm as a special case. A posteriori guarantees on solving an underlying rank constrained optimization problem with these convex relaxations are provided. We evaluate the performance of the low-rank inducing norms on three matrix completion problems. In all examples, the nuclear norm heuristic is outperformed by convex relaxations based on other low-rank inducing norms. For two of the problems there exist low-rank inducing norms that succeed in recovering the partially unknown matrix, while the nuclear norm fails. These low-rank inducing norms are shown to be representable as semi-definite programs. Moreover, these norms have cheaply computable proximal mappings, which makes it possible to also solve problems of large size using first-order methods

Time-parallel iterative solvers for parabolic evolution equations

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties