Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement Vectors

While single measurement vector (SMV) models have been widely studied in signal processing, there is a surging interest in addressing the multiple measurement vectors (MMV) problem. In the MMV setting, more than one measurement vector is available and the multiple signals to be recovered share some commonalities such as a common support. Applications in which MMV is a naturally occurring phenomenon include online streaming, medical imaging, and video recovery. This work presents a stochastic iterative algorithm for the support recovery of jointly sparse corrupted MMV. We present a variant of the Sparse Randomized Kaczmarz algorithm for corrupted MMV and compare our proposed method with an existing Kaczmarz type algorithm for MMV problems. We also showcase the usefulness of our approach in the online (streaming) setting and provide empirical evidence that suggests the robustness of the proposed method to the distribution of the corruption and the number of corruptions occurring.Comment: 13 pages, 6 figure

What is... an Oka Manifold?

Alternative title: What is an Oka Manifold?Finnur Lárusso

Affine simplices in Oka manifolds

We show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces Cn, n ≥ 0, into X. Among such X are all complex Lie groups and their homogeneous spaces. We present generalisations of this result, one of which states that the homotopy type of the space of continuous maps from any smooth manifold to X is given by a simplicial set whose simplices are holomorphic maps into X.Finnur Lárusso

Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds

The basic result of Oka theory, due to Gromov, states that every continuous map f from a Stein manifold S to an elliptic manifold X can be deformed to a holomorphic map. It is natural to ask whether this can be done for all f at once, in a way that depends continuously on f and leaves f fixed if it is holomorphic to begin with. In other words, is O ( S, X ) a deformation retract of C ( S, X )? We prove that it is if S has a strictly plurisubharmonic Morse exhaustion with finitely many critical points, in particular, if S is affine algebraic. The only property of X used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that X is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.Finnur Lárusso

Model structures and the Oka Principle

Copyright © 2004 Elsevier B.V. All rights reserved.We embed the category of complex manifolds into the simplicial category of prestacks on the simplicial site of Stein manifolds, a prestack being a contravariant simplicial functor from the site to the category of simplicial sets. The category of prestacks carries model structures, one of them defined for the first time here, which allow us to develop holomorphic homotopy theory. More specifically, we use homotopical algebra to study lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. We prove that holomorphic maps satisfy certain versions of the Oka Principle if and only if they are fibrations in suitable model structures. We are naturally led to a simplicial, rather than a topological, approach, which is a novelty in analysisFinnur Lárussonhttp://www.elsevier.com/wps/find/journaldescription.cws_home/505614/description#descriptio

Survey of Oka theory

Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open problems. This article is a much expanded version of the lecture given by the first-named author at the conference RAFROT 2010 in Rincón, Puerto Rico, on 22 March 2010, and of a recent survey article by the second-named author, 2010.Franc Forstnerič and Finnur Lárusso

Sufficient conditions for holomorphic linearisation

Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label. The counterexamples to the Linearisation Problem construct an action of G such that QX is not stratified biholomorphic to any QV.Our main theorem shows that, for most X, a stratified biholomorphism of QX to some QV is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to ℂn, only that X is a Stein manifold.Frank Kutzschebauch, Finnur Lárusson, Gerald W. Schwar

The parametric h-principle for minimal surfaces in R(n) and null curves in C(n)

Let M be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion M→R3 is isotopic to the real part of a holomorphic null curve M→C3. In this paper, we prove the following much stronger result in this direction: for any n≥3, the inclusion ι of the space of real parts of nonflat null holomorphic immersions M→Cn into the space of nonflat conformal minimal immersions M→Rn satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. We prove analogous results for several other related maps, and we describe the homotopy type of the space of all holomorphic immersions M→Cn. For an open Riemann surface M of finite topological type, we obtain optimal results by showing that ι and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.Franc Forstneric, Finnur Larusso

Holomorphic Legendrian Curves in Projectivised Cotangent Bundles

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bunWe provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve M -> X from a compact bordered Riemann surface M can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provided that all vertical curves under consideration are nondegenerate. Stronger results are obtained when the base Z is an Oka manifold or a Stein manifold with the density property. Finally, we establish basic and 1-parametric h-principles for holomorphic Legendrian curves in X.Forstneric, Franc, Larusson, Finnu

Chaotic Holomorphic Automorphisms of Stein Manifolds with the Volume Density Property

Let X be a Stein manifold of dimension n >= 2 satisfying the volume density property with respect to an exact holomorphic volume form. For example, X could be C-n, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product Y x C, where Y is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of X. In particular, X has a chaotic holomorphic automorphism. A proof for X = C-n may be found in work of Fornaess and Sibony. We follow their approach closely. Peters, Vivas, and Wold showed that a generic volume-preserving automorphism of C-n, n >= 2, has a hyperbolic fixed point whose stable manifold is dense in C-n. This property can be interpreted as a kind of chaos. We generalise their theorem to a Stein manifold as above