2,768 research outputs found
Polytope of Correct (Linear Programming) Decoding and Low-Weight Pseudo-Codewords
We analyze Linear Programming (LP) decoding of graphical binary codes
operating over soft-output, symmetric and log-concave channels. We show that
the error-surface, separating domain of the correct decoding from domain of the
erroneous decoding, is a polytope. We formulate the problem of finding the
lowest-weight pseudo-codeword as a non-convex optimization (maximization of a
convex function) over a polytope, with the cost function defined by the channel
and the polytope defined by the structure of the code. This formulation
suggests new provably convergent heuristics for finding the lowest weight
pseudo-codewords improving in quality upon previously discussed. The algorithm
performance is tested on the example of the Tanner [155, 64, 20] code over the
Additive White Gaussian Noise (AWGN) channel.Comment: 6 pages, 2 figures, accepted for IEEE ISIT 201
Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences
Questions in computational molecular biology generate various discrete
optimization problems, such as DNA sequence alignment and RNA secondary
structure prediction. However, the optimal solutions are fundamentally
dependent on the parameters used in the objective functions. The goal of a
parametric analysis is to elucidate such dependencies, especially as they
pertain to the accuracy and robustness of the optimal solutions. Techniques
from geometric combinatorics, including polytopes and their normal fans, have
been used previously to give parametric analyses of simple models for DNA
sequence alignment and RNA branching configurations. Here, we present a new
computational framework, and proof-of-principle results, which give the first
complete parametric analysis of the branching portion of the nearest neighbor
thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure
Practical and Optimal LSH for Angular Distance
We show the existence of a Locality-Sensitive Hashing (LSH) family for the
angular distance that yields an approximate Near Neighbor Search algorithm with
the asymptotically optimal running time exponent. Unlike earlier algorithms
with this property (e.g., Spherical LSH [Andoni, Indyk, Nguyen, Razenshteyn
2014], [Andoni, Razenshteyn 2015]), our algorithm is also practical, improving
upon the well-studied hyperplane LSH [Charikar, 2002] in practice. We also
introduce a multiprobe version of this algorithm, and conduct experimental
evaluation on real and synthetic data sets.
We complement the above positive results with a fine-grained lower bound for
the quality of any LSH family for angular distance. Our lower bound implies
that the above LSH family exhibits a trade-off between evaluation time and
quality that is close to optimal for a natural class of LSH functions.Comment: 22 pages, an extended abstract is to appear in the proceedings of the
29th Annual Conference on Neural Information Processing Systems (NIPS 2015
Tradeoffs for nearest neighbors on the sphere
We consider tradeoffs between the query and update complexities for the
(approximate) nearest neighbor problem on the sphere, extending the recent
spherical filters to sparse regimes and generalizing the scheme and analysis to
account for different tradeoffs. In a nutshell, for the sparse regime the
tradeoff between the query complexity and update complexity
for data sets of size is given by the following equation in
terms of the approximation factor and the exponents and :
For small , minimizing the time for updates leads to a linear
space complexity at the cost of a query time complexity .
Balancing the query and update costs leads to optimal complexities
, matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner,
IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn,
STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A
subpolynomial query time complexity can be achieved at the cost of a
space complexity of the order , matching the bound
of [Andoni-Indyk-Patrascu, FOCS'06] and
[Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of
[Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98].
For large , minimizing the update complexity results in a query complexity
of , improving upon the related exponent for large of
[Kapralov, PODS'15] by a factor , and matching the bound
of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal
complexities , while a minimum query time complexity can be
achieved with update complexity , improving upon the
previous best exponents of Kapralov by a factor .Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580
[cs.DS] (along with arXiv:1605.02701 [cs.DS]
A Novel Symmetric Four Dimensional Polytope Found Using Optimization Strategies Inspired by Thomson's Problem of Charges on a Sphere
Inspired by, and using methods of optimization derived from classical three
dimensional electrostatics, we note a novel beautiful symmetric four
dimensional polytope we have found with 80 vertices. We also describe how the
method used to find this symmetric polytope, and related methods can
potentially be used to find good examples for the kissing and packing problems
in D dimensions
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