Norm Ball Research Articles

Hessian Schatten-Norm Regularization for Linear Inverse Problems

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto lq norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.

The Convex Geometry of Linear Inverse Problems

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors and low-rank matrices, as well as several others including sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery. Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming.

Twisted Alexander polynomials detect fibered 3-manifolds

A classical result in knot theory says that for a bered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and bered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sucient for a 3-manifold to be bered. By previous work of the authors this result implies that if a manifold of the form S 1 N 3 admits a symplectic structure, then N bers over S 1 . In fact we will completely determine the symplectic cone of S 1 N in terms of the bered faces of the Thurston norm ball of N.

Path-connectedness of frequency-domain uncertainty sets in the graph topology⋆

AbstractPath-connectedness with respect to the topology induced by the gap metric underpins a recent robustness result for uncertain feedback interconnections of transfer functions in the Callier-Desoer algebra; i.e. possibly distributed-parameter with only finitely many unstable poles. In this paper, we establish path-connectedness of the following uncertainty sets of such transfer functions: diagonally structured sets with path-connected entries; arbitrarily large H∞-norm balls; and sufficiently small gap metric balls. Proof of the latter relies on the existence of a certain J-spectral factorisation, which is also established herein.

Splitting fields of elements in arithmetic groups

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive. If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.

Farthest points and strong convexity of sets

We consider the existence and uniqueness of the farthest point of a given set A in a Banach space E from a given point x in the space E. It is assumed that A is a convex, closed, and bounded set in a uniformly convex Banach space E with Frechet differentiable norm. It is shown that, for any point x sufficiently far from the set A, the point of the set A which is farthest from x exists, is unique, and depends continuously on the point x if and only if the set A in the Minkowski sum with some other set yields a ball. Moreover, the farthest (from x) point of the set A also depends continuously on the set A in the sense of the Hausdorff metric. If the norm ball of the space E is a generating set, these conditions on the set A are equivalent to its strong convexity.

Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold $M$ is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration $M\to S^1$ is the modular elliptic curve $E=X_0(49)$, which admits multiplication by the ring of integers of ${\Bbb Q}[\sqrt{-7}]$. We first base change the holomorphic differential on $E$ to a cusp form on ${{\rm GL}(2)}$ over $K={\Bbb Q}[\sqrt{-3}]$, and then transfer over to a quaternion algebra $D/K$ ramified only at the primes above $7$; the fundamental group of $M$ is a quotient of the principal congruence subgroup of ${\cal O}_D^\ast$ of level $7$. To analyze the topological properties of $M$, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.

Faces of the scl norm ball

Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) The unique homogeneous quasimorphism on F dual to this face (up to scale and elements of H^1) is the rotation quasimorphism associated to the action of F on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1-chain in S rationally cobounds an immersed surface with a sufficiently large multiple of the boundary. This is true even if S has no boundary.

Hedlund metrics and the stable norm

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H 1 ( M , R ) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H 1 ( M , R ) are stable norms of a Riemannian metric on M. If the dimension of M is at least three, I. Babenko and F. Balacheff proved in [I. Babenko, F. Balacheff, Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119 (3) (2006) 347–358] that every polyhedral norm ball in H 1 ( M , R ) , whose vertices are rational with respect to the lattice of integer classes in H 1 ( M , R ) , is the stable norm ball of a Riemannian metric on M. This metric can even be chosen to be conformally equivalent to any given metric. In [I. Babenko, F. Balacheff, Sur la forme de la boule unité de la norme stable unidimensionnelle, Manuscripta Math. 119 (3) (2006) 347–358], the stable norm induced by the constructed metric is computed by comparing the metric with a polyhedral one. Here we present an alternative construction for the metric, which remains in the geometric framework of smooth Riemannian metrics.

Integral points on symmetric varieties and Satake compatifications

Let $V$ be an affine symmetric variety defined over $\Bbb Q$. We compute the asymptotic distribution of the angular components of the integral points in $V$. This distribution is described by a family of invariant measures concentrated on the Satake boundary of $V$. In the course of the proof, we describe the structure of the Satake compactifications for general affine symmetric varieties and compute the asymptotic of the volumes of norm balls.

A New Class of Minimum Norm Duality Theorems

In this paper we derive new duality results on the width and the length of symmetrical convex bodies. Let K be a symmetrical convex body in $R^m$ and let $\|\cdot\|$ be some (arbitrary) norm on $R^m$. The width of K is obtained by searching for the smallest “sandwich” (or “slab”) that contains K. It is shown that the dual problem has the following form: Find the largest diameter of a norm ball that is contained in K. Indeed, the diameter of a maximal norm ball equals the width of the smallest sandwich. Moreover, the solutions of the two problems obey certain alignment relations. The length of K is found by searching the largest “sandwich” that contains K. The last problem is closely related to the “maximal chord problem” whose optimal value is called the “diameter” of K. In this case the dual problem is to find the smallest norm ball that contains K. It is proved that the diameter of the smallest norm ball equals the diameter (the length) of K, and that primal and dual solutions satisfy certain alignment relations. Part of the results remain valid for more general convex sets.

Control of Systems With Uncertain Initial Conditions

This note deals with the control of linear discrete-time systems with uncertain initial conditions. Specifically, we consider the problem where the initial condition is known to reside in a norm ball of some radius, and the input disturbance is constrained to satisfy an independent norm condition. The paper focuses on eventually periodic systems; these include both finite horizon and periodic systems as special cases. The main theorem provides exact synthesis conditions for the existence of eventually periodic controllers which both stabilize and provide performance in closed-loop control systems. These conditions are given in terms of a finite-dimensional semidefinite programming problem. We also give a version of the main result for the special case of linear time-invariant systems with uncertain initial states, and conclude with an illustrative example.

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.

Polynomial-Time Random Generation of Uniform Real Matrices in the Spectral Norm Ball

This paper follows the line of research aimed to develop randomized algorithms for probabilistic analysis and design of control systems. In particular, a result for the generation of real matrix samples uniformly distributed in the spectral norm ball is presented. To this end, the distribution of the singular values of random uniform matrices is first studied. The sample generation is then based on the conditional densities method. A key result of this paper is the computation in closed form of the marginal densities of the singular values, leading to a polynomial-time method for the real matrix sample generation problem.

Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations

Let F⊂H 1(M 3, R) be a fibered face of the Thurston norm ball for a hyperbolic 3-manifold M. Any φ∈ R +·F determines a measured foliation F of M. Generalizing the case of Teichmüller geodesics and fibrations, we show F carries a canonical Riemann surface structure on its leaves, and a transverse Teichmüller flow with pseudo-Anosov expansion factor K( φ)>1. We introduce a polynomial invariant Θ F∈ Z[H 1(M, Z)/ torsion] whose roots determine K( φ). The Newton polygon of Θ F allows one to compute fibered faces in practice, as we illustrate for closed braids in S 3. Using fibrations we also obtain a simple proof that the shortest geodesic on moduli space M g has length O(1/ g).

Carathéodory balls and norm balls in a class of convex bounded Reinhardt domains

LetD be the class of domains\(D_{a,p} = \left\{ {z \in \mathbb{C}^n |\left| {z_1 } \right|^{2_{p1} } + 2a\left| {z_1 } \right|^{P1} \left| {z_2 } \right|^{_{P2} } + \left| {z_2 } \right|^{2_{P2} } + \sum\limits_{j = 3}^n {\left| {z_j } \right|^{2_{Pj} }< 1} } \right\}\) forn≥2,a≥0 and p=(p1,...,n) ∈ (ℝ+) n such thatD a,p is convex. The classD is a class of convex bounded Reinhardt domains of ℂ n which are a generalization of complex ellipsoids. In this paper we compare Caratheodory balls and norm balls of the domainsD∈D. We prove that in this case a Caratheodory ball inD∈D is a norm ball if, and only if,D is a complex ellipsoid\(\left\{ {z \in \mathbb{C}\left| {\Sigma _{j = 1}^n \left| {z_j } \right|^{2_{Pj} }< 1} \right.} \right\}\) such thatp k=1 for exactly onek∈{1,…,n},p j=1/2 for allj≠k and the centre lies on thez k-axis.

Geometrical characterizations of constant modulus receivers

Convergence properties of the constant modulus (CM) and the Shalvi-Weinstein (SW) algorithms in the presence of noise remain largely unknown. A new geometrical approach to the analysis of constant modulus and Shalvi-Weinstein receivers is proposed by considering a special constrained optimization involving norms of the combined channel-receiver response. This approach provides a unified framework within which various blind and (nonblind) Wiener receivers can all be analyzed by circumscribing an ellipsoid by norm balls of different types, A necessary and sufficient condition for the equivalence among constant modulus. Shalvi-Weinstein, zero forcing, and Wiener receivers are obtained. Answers to open questions with regard to CM and SW receivers, including their locations and their relationship with Wiener receivers, are provided for the special orthogonal channel and the general two-dimensional (2-D) channel-receiver impulse response. It is also shown that in two dimensions, each CM or SW receiver is associated with one and only one Wiener receiver.

Regular norm balls can be closed in the strong operator topology

The main results obtained are: – A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T ∈ Lr(X,Y): ||T||r ≤ 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y). – A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y). – A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm. – A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm. Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance: – If X = Lp(μ) and Y = Lq(ν), where 1 < p,q < ∞, then Lr(X,Y) is a first category subset of L(X,Y).

Carathéodory balls and norm balls of the domainsH n ={z ∈ ℂ n : |z 1|+…+|z n |&lt;1}

In this short note we prove that the only Caratheodory balls in domains given in the title (forn≥2) which are the norm balls are the ones with the center at 0. It is a generalization of the result of B. Schwarz, who proved this theorem in casen=2.