Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

ℓ-Independence for a system of motivic representations

Let M be a motive over a number field F and v a non-archimedean valuation of F with residual characteristic p. Let \({\rho_{M,\ell} : \Gamma_{F} \rightarrow G_{M}(\mathbb{Q}_{\ell})}\) be the canonical system of ?-adic Galois representations associated to M, with values in the motivic Galois group G _M of M. Let \({\Phi_{v} \in \Gamma_{F}}\) be an arithmetic Frobenius element. When M belongs to a particular family of motives, we show the following (under certain hypotheses): (i) if M has good reduction at v, then for \({\ell \neq p}\) , the conjugacy class of \({\rho_{M,\ell}(\Phi_{v})}\) in G _M is rational over \({\mathbb{Q}}\) and is independent of ?, thus giving a partial answer to a question of Serre; (ii) if M has semistable reduction at v, then the system of representations of the Weil–Deligne group \({'W_{v}}\) , associated to \({\rho_{M,\ell}}\) for \({\ell \neq p}\) , is a compatible system of representations of \({'W_{v}}\) with values in G _M .     

A Projection Proximal-Point Algorithm for ℓ1 Minimization

The problem of the minimization of least squares functionals with ?¹ penalties is considered in an infinite dimensional Hilbert space setting. Though there are several algorithms available in the finite dimensional setting there are only a few of them that come with a proper convergence analysis in the infinite dimensional setting.

In this work we provide an algorithm from a class that has not been considered for ?¹ minimization before, namely, a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy-to-implement algorithm. We present experiments that indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of ?¹ minimization.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

Let $b_{\ell }(n)$ be the number of ℓ-regular partitions of n. We show that the generating functions of $b_{\ell }(n)$ with $\ell =3,5,6,7$ and 10 are congruent to the products of two items of Ramanujan's theta functions $\psi (q)$, $f(-q)$ and ${(q;q)}_{\infty }^3$ modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for $b_{\ell }(n)$ modulo 3, 5 and 7.     

ℓ-Divisibility of ℓ-regular partition functions

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b _ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏ _m=1^∞(1−q ^m ) ^ℓ−1.      

Primal and dual alternating direction algorithms for ℓ 1-ℓ 1-norm minimization problems in compressive sensing

In this paper, we propose, analyze and test primal and dual versions of the alternating direction algorithm for the sparse signal reconstruction from its major noise contained observation data. The algorithm minimizes a convex non-smooth function consisting of the sum of ? ₁-norm regularization term and ? ₁-norm data fidelity term. We minimize the corresponding augmented Lagrangian function alternatively from either primal or dual forms. Both of the resulting subproblems admit explicit solutions either by using a one-dimensional shrinkage or by an efficient Euclidean projection. The algorithm is easily implementable and it requires only two matrix-vector multiplications per-iteration. The global convergence of the proposed algorithm is established under some technical conditions. The extensions to the non-negative signal recovery problem and the weighted regularization minimization problem are also discussed and tested. Numerical results illustrate that the proposed algorithm performs better than the state-of-the-art algorithm YALL1.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

Three-Dimensional ℓ-Algebras

The family of all three-dimensional almost f-algebras, d-algebras, and f-algebras is constructed. It is shown that it contains all three-dimensional directly indecomposable ?-algebras. Also, a list of representatives of (algebraic) isomorphism classes of all three-dimensional algebras that can be ordered as an almost f-algebra, a d-algebra, or an f-algebra is given.     

Lamron ℓ-groups

The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)^?1 is a Hausdorff topological space, where Min(G)^?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups.     

Low noise regimes for ℓ regularization : continuous and discrete settings

We focus on support recovery for signal deconvolution with sparsity assumption. We adopt the continuous setting defined by several recent works and we try to reconstruct a sum of Dirac masses from its low frequencies (possibly perturbed by some noise), by using a total variation prior for Radon measures (i.e. the generalization to measures of the ℓ¹ norm). We show that, under a non degenerate source condition, there exists a small noise regime in which the model recovers exactly the same number of spikes as the original signal, and the spikes converge to those of the original signal as the noise vanishes. This continuous setting, by allowing the spikes to “move”, provides robust support recovery for signals composed of well separated spikes. In a discrete setting, where the spikes are reconstructed on a grid, similar low noise regimes which guarantee the exact recovery of the support also exist (see [3]). Yet, this property only concerns a small class of signals. Considering the asymptotics of the discrete problems as the size of the grid tends to zero, we show that the support of the original signal cannot be stable on thin grids, and that the discrete models actually reconstruct pairs of spikes near each original spike. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

Feebly projectable ℓ-groups

In the article [17], we introduced and investigated feebly and flatly projectable frames. In this article, we apply these two properties to lattice-ordered groups. An example is constructed to illustrate that the two properties are distinct, which solves a question from [17]. We also investigate these properties with respect to archimedean ℓ-groups with weak order unit, as well as commutative semiprime f-rings.     

Codegree Threshold for Tiling k-graphs with Two Edges Sharing Exactly ℓ Vertices

Given integer k and a k-graph F,let t(k-1)(n,F)be the minimum integer t such that every k-graph H on n vertices with codegree at least t contains an F-factor.For integers k>3 and 0≤l≤k-1,let y(k,l)be a k-graph with two edges that shares exactly l vertices.Han and Zhao(J.Combin.Theory Ser.A,(2015))asked the following question:For all k≥3,0≤l≤k-1 and sufficiently large n divisible by 2 k-l,determine the exact value of t_k-1(n,y(k,l)).In this paper,we show that t(k-1)(n,y(k,l))=n/(2 k-l)for k>3 and 1≤l≤k-2,combining with two previously known results of R?dl,Rucinski and Szemeredi(J.Combin.Theory Ser.A,(2009))and Gao,Han and Zhao(Combinatorics,Probability and Computing,(2019)),the question of Han and Zhao is solved completely.     

The geometric algebraCℓ 3 as a model for a projective plane

We show that the geometric algebraCℓ ₃ can be used as a model for the real projective plane, in the sense that the axioms defining the plane and their duals can be proved as theorems. However, it seems that there is some difficulty in using a geometric algebra to model a projective space over a noncommutative division ring.     

Clean unital ℓ-groups

A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital ?-groups. We mention the relationship of clean unital ?-groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean ?-groups with distinguished weak order unit.     

A topological duality for some lattice ordered algebraic structures including ℓ-groups

A topological duality is developed for a wide class of lattice ordered algebraic structures by introducing in an ordered Stone space a natural binary and continuous function. In particular, duality theorems are obtained for -groups and for abelian -groups. Dedicated to my wife Eugenia Presented by W. Taylor.This research is a part of the Doctorial Thesis that the author presented at the Universidad de Buenos Aires, under the supervision of Prof. R. Cignoli. The author wishes to thank Prof. Cignoli for his guidance. The author also wishes to thank Keith Kearnes for his careful reading of this paper and for his deep suggestions that led to a substantial improvement of the first version.