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Abstract: Recovering missing data from its partial samples is a fundamental problem in mathematics and it has wide range of applications in image and signal processing. While many such algorithms have been developed recently, there are very few papers available on their error estimations. This paper is to analyze the error of a frame based data recovery approach from random samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal norm of the canonical frame coefficients among all the possible solutions. [Copyright &y& Elsevier]

Abstract: The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. [Copyright &y& Elsevier]

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 2 2 In Victor Pan's own words: “Our algorithms are quite involved, and their implementation would require a non-trivial work, incorporating numerous known implementation techniques and tricks”. In fact, we are not aware of any implementation of Pan's method. and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted A New Dsc , which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. A New Dsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method for solving feasibility problems has attracted a lot of attention due to its good performance even in nonconvex settings. In this paper, we consider the Douglas-Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinite-dimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns systems with small coherence parameter. Simple greedy-type algorithms perform well on these systems, which are also useful in the construction of compressed sensing matrices. We discuss the following problems for both and . How large can a dictionary be, if we prescribe the coherence parameter? How small could the resulting coherence parameter be, if we impose a size on the dictionary? How could we construct such a system? Several fundamental results from different areas of mathematics shed light on these important problems with far-reaching implications in approximation theory. [Copyright &y& Elsevier]

Abstract: In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Gröbner bases via Buchberger’s Algorithm. In a nutshell, the idea is to extend the advantages of computing with homogeneous polynomials or vectors to the general case. When the input data are not homogeneous, we use as a main tool the procedure of a self-saturating Buchberger’s Algorithm. Another strictly related topic is treated later when a mathematical foundation is given to the sugar trick which is nowadays widely used in most of the implementations of Buchberger’s Algorithm. A special emphasis is also given to the case of a single grading, and subsequently some timings and indicators showing the practical merits of our approach. [Copyright &y& Elsevier]

Abstract: In this paper a novel region-merging image segmentation approach is presented. This approach is based on a two-step procedure: a distance metric is learned from some features on the image, then a piecewise approximation function for the Mumford–Shah model is optimized by this metric. The global optimum of the approximation function is inductively achieved under high polynomial terms of the Mahalanobis distance, extracting the nonlinear features of the pattern distributions into topological maps. The penalizer terms of the Mumford–Shah equation are based on new similarity criteria, computed from the topological maps and the class label information. The results we obtained show a better discrimination of object boundaries and the location of regions when compared with the conventional Mumford–Shah algorithm, even when supplied with other well-known similarity functions. A quantitative objective evaluation of the proposed approach was performed in order to compute the quality of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: The problem of recognizing approximate graph products arises in theoretical biology. This paper presents an algorithm that recognizes a large class of approximate graph products. The main part of this contribution is concerned with a new, local prime factorization algorithm that factorizes all strong products on an extensive class of graphs that contains, in particular, all products of triangle-free graphs on at least three vertices. The local approach is linear for graph with fixed maximal degree. [Copyright &y& Elsevier]