4,699 research outputs found
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
Multi-consensus Decentralized Accelerated Gradient Descent
This paper considers the decentralized optimization problem, which has
applications in large scale machine learning, sensor networks, and control
theory. We propose a novel algorithm that can achieve near optimal
communication complexity, matching the known lower bound up to a logarithmic
factor of the condition number of the problem. Our theoretical results give
affirmative answers to the open problem on whether there exists an algorithm
that can achieve a communication complexity (nearly) matching the lower bound
depending on the global condition number instead of the local one. Moreover,
the proposed algorithm achieves the optimal computation complexity matching the
lower bound up to universal constants. Furthermore, to achieve a linear
convergence rate, our algorithm \emph{doesn't} require the individual functions
to be (strongly) convex. Our method relies on a novel combination of known
techniques including Nesterov's accelerated gradient descent, multi-consensus
and gradient-tracking. The analysis is new, and may be applied to other related
problems. Empirical studies demonstrate the effectiveness of our method for
machine learning applications
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Hardness of Approximation for Morse Matching
Discrete Morse theory has emerged as a powerful tool for a wide range of
problems, including the computation of (persistent) homology. In this context,
discrete Morse theory is used to reduce the problem of computing a topological
invariant of an input simplicial complex to computing the same topological
invariant of a (significantly smaller) collapsed cell or chain complex.
Consequently, devising methods for obtaining gradient vector fields on
complexes to reduce the size of the problem instance has become an emerging
theme over the last decade. While computing the optimal gradient vector field
on a simplicial complex is NP-hard, several heuristics have been observed to
compute near-optimal gradient vector fields on a wide variety of datasets.
Understanding the theoretical limits of these strategies is therefore a
fundamental problem in computational topology. In this paper, we consider the
approximability of maximization and minimization variants of the Morse matching
problem, posed as open problems by Joswig and Pfetsch. We establish hardness
results for Max-Morse matching and Min-Morse matching. In particular, we show
that, for a simplicial complex with n simplices and dimension , it is
NP-hard to approximate Min-Morse matching within a factor of
, for any . Moreover, using an L-reduction
from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is
both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial
complexes of dimension within certain explicit constant factors.Comment: 20 pages, 1 figur
Persistence barcodes and Laplace eigenfunctions on surfaces
We obtain restrictions on the persistence barcodes of Laplace-Beltrami
eigenfunctions and their linear combinations on compact surfaces with
Riemannian metrics. Some applications to uniform approximation by linear
combinations of Laplace eigenfunctions are also discussed.Comment: Revised version; some references adde
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