and ai,j ≥ 0 give the value of the jth signal at point i. We study the problem of finding norms that are cumulative of the multiple signals in the data stream. For example, consider the max-dominance norm, defined as i maxj{ai,j}. It may be thought as estimating the norm of the “upper envelope

Abstract. Let Pa(Z) = Z n + ∑n j=1 ajZ n−j be a Ck curve of monic polynomials, ai ∈ Ck(I,C) where I ⊂ R is an interval. We show that if k = k(n) is sufficiently large then any choice of continuous roots of Pa is locally absolutely continuous, in a uniform way with respect to maxj ‖aj‖Ck on compact

= heightI. Consider for 1 ≤ i ≤ p the numbers Mi(A/I) = max{j ∈ Z | βi,j(A/I) ̸ = 0} & mi(A/I) = min{j ∈ Z | βi,j(A/I) ̸ = 0}. j∈Z Let e(A/I) denote the multiplicity of A/I. Set L(I) = 1 c∏ mi(A/I) and U(I) = c! 1 c! i=1 c∏ Mi(A/I). The conjecture of Herzog, Huneke and Srinivasan states that

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A b s t r a c t. In this paper, the structure of nitely generated free objects in the variety of three-valued closure Lukasiewicz algebras is determined. We describe their indecomposable factors and we give their cardinality..1 Introduction and Preliminaries A Lukasiewicz algebra of order n, or an n-valued Lukasiewicz algebra, is

Compressive sensing (CS) allows for acquisition of sparse signals at sampling rates significantly lower than the Nyquist rate required for bandlimited signals. Recovery guarantees for CS are generally derived based on the assumption that measurement projections are selected independently at random. However, for many practical signal acquisition applications, including medical imaging and remote sensing, this assumption is violated as the projections must be taken in groups. In this paper, we consider such applications and derive requirements on the number of measurements needed for successful recovery of signals when groups of dependent projections are taken at random. We find a penalty factor on the number of required measurements with respect to the standard CS scheme that employs conventional independent measurement selection and evaluate the accuracy of the predicted penalty through simulations. I.

This article gives theoretical insights into the performance of K-SVD, a dictionary learning algorithm that has gained significant popularity in practical applications. The particular question studied here is when a dictionary Φ ∈ Rd×K can be recovered as local minimum of the minimisation criterion underlying K-SVD from a set of N training signals yn = Φxn. A theoretical analysis of the problem leads to two types of identifiability results assuming the training signals are generated from a tight frame with coefficients drawn from a random symmetric distribution. First asymptotic results showing, that in expectation the generating dictionary can be recovered exactly as a local minimum of the K-SVD criterion if the coefficient distribution exhibits sufficient decay. This decay can be characterised by the coherence of the dictionary and the `1-norm of the coefficients. Based on the asymptotic results it is further demonstrated that given a finite number of training samples N, such that N / logN = O(K3d), except with probability O(N−Kd) there is a local minimum of the K-SVD criterion within distance O(KN−1/4) to the generating dictionary. Index Terms dictionary learning, sparse coding, K-SVD, finite sample size, sampling complexity, dictionary identification, minimisation criterion, sparse representation 1

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