Abstract:
We study greedy-type algorithms such that at a greedy step we pick several dictionary elements contrary to a single dictionary element in standard greedy-type algorithms. We call such greedy algorithms {\it super greedy algorithms}. The idea of picking several elements at a greedy step of the algorithm is not new. Recently, we observed the following new phenomenon. For incoherent dictionaries these new type of algorithms (super greedy algorithms) provide the same (in the sense of order) upper bound for the error as their analogues from the standard greedy algorithms. The super greedy algorithms are computationally simpler than their analogues from the standard greedy algorithms. We continue to study this phenomenon.

Abstract:
For compressed sensing over arbitrarily connected networks, we consider the problem of estimating underlying sparse signals in a distributed manner. We introduce a new signal model that helps to describe inter-signal correlation among connected nodes. Based on this signal model along with a brief survey of existing greedy algorithms, we develop distributed greedy algorithms with low communication overhead. Incorporating appropriate modifications, we design two new distributed algorithms where the local algorithms are based on appropriately modified existing orthogonal matching pursuit and subspace pursuit. Further, by combining advantages of these two local algorithms, we design a new greedy algorithm that is well suited for a distributed scenario. By extensive simulations we demonstrate that the new algorithms in a sparsely connected network provide good performance, close to the performance of a centralized greedy solution.

Abstract:
We consider the problem of approximating a given element $f$ from a Hilbert space $\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118--2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries.

Abstract:
We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes.

Abstract:
We discuss the upper and lower estimates for the rate of convergence of Pure and Orthogonal Greedy Algorithms for dictionary with bounded cumulative coherence.

Abstract:
It is a survey on recent results in constructive sparse approximation. Three directions are discussed here: (1) Lebesgue-type inequalities for greedy algorithms with respect to a special class of dictionaries, (2) constructive sparse approximation with respect to the trigonometric system, (3) sparse approximation with respect to dictionaries with tensor product structure. In all three cases constructive ways are provided for sparse approximation. The technique used is based on fundamental results from the theory of greedy approximation. In particular, results in the direction (1) are based on deep methods developed recently in compressed sensing. We present some of these results with detailed proofs.

Abstract:
In the design of algorithms, the greedy paradigm provides a powerful tool for solving efficiently classical computational problems, within the framework of procedural languages. However, expressing these algorithms within the declarative framework of logic-based languages has proven a difficult research challenge. In this paper, we extend the framework of Datalog-like languages to obtain simple declarative formulations for such problems, and propose effective implementation techniques to ensure computational complexities comparable to those of procedural formulations. These advances are achieved through the use of the "choice" construct, extended with preference annotations to effect the selection of alternative stable-models and nondeterministic fixpoints. We show that, with suitable storage structures, the differential fixpoint computation of our programs matches the complexity of procedural algorithms in classical search and optimization problems.

Abstract:
In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest.

Abstract:
In this paper we present an original approach for community detection in complex networks. The approach belongs to the family of seed-centric algorithms. However, instead of expanding communities around selected seeds as most of existing approaches do, we explore here applying an ensemble clustering approach to different network partitions derived from ego-centered communities computed for each selected seed. Ego-centered communities are themselves computed applying a recently proposed ensemble ranking based approach that allow to efficiently combine various local modularities used to guide a greedy optimisation process. Results of first experiments on real world networks for which a ground truth decomposition into communities are known, argue for the validity of our approach.

Abstract:
In a standard NP-complete optimization problem we introduce an interpolating algorithm between the quick decrease along the gradient (greedy dynamics) and a slow decrease close to the level curves (reluctant dynamics). We find that for a fixed elapsed computer time the best performance of the optimization is reached at a special value of the interpolation parameter, considerably improving the results of the pure cases greedy and reluctant.