18,362 research outputs found
Curvature and complexity: Better lower bounds for geodesically convex optimization
We study the query complexity of geodesically convex (g-convex) optimization
on a manifold. To isolate the effect of that manifold's curvature, we primarily
focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly
g-convex or not; high- or low-dimensional), known upper bounds worsen with
curvature. It is natural to ask whether this is warranted, or an artifact.
For many such settings, we propose a first set of lower bounds which indeed
confirm that (negative) curvature is detrimental to complexity. To do so, we
build on recent lower bounds (Hamilton and Moitra, 2021; Criscitiello and
Boumal, 2022) for the particular case of smooth, strongly g-convex
optimization. Using a number of techniques, we also secure lower bounds which
capture dependence on condition number and optimality gap, which was not
previously the case.
We suspect these bounds are not optimal. We conjecture optimal ones, and
support them with a matching lower bound for a class of algorithms which
includes subgradient descent, and a lower bound for a related game. Lastly, to
pinpoint the difficulty of proving lower bounds, we study how negative
curvature influences (and sometimes obstructs) interpolation with g-convex
functions.Comment: v1 to v2: Renamed the method of Rusciano 2019 from "center-of-gravity
method" to "centerpoint method
Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes
For the majority of the applications of Reed-Solomon (RS) codes, hard
decision decoding is based on syndromes. Recently, there has been renewed
interest in decoding RS codes without using syndromes. In this paper, we
investigate the complexity of syndromeless decoding for RS codes, and compare
it to that of syndrome-based decoding. Aiming to provide guidelines to
practical applications, our complexity analysis differs in several aspects from
existing asymptotic complexity analysis, which is typically based on
multiplicative fast Fourier transform (FFT) techniques and is usually in big O
notation. First, we focus on RS codes over characteristic-2 fields, over which
some multiplicative FFT techniques are not applicable. Secondly, due to
moderate block lengths of RS codes in practice, our analysis is complete since
all terms in the complexities are accounted for. Finally, in addition to fast
implementation using additive FFT techniques, we also consider direct
implementation, which is still relevant for RS codes with moderate lengths.
Comparing the complexities of both syndromeless and syndrome-based decoding
algorithms based on direct and fast implementations, we show that syndromeless
decoding algorithms have higher complexities than syndrome-based ones for high
rate RS codes regardless of the implementation. Both errors-only and
errors-and-erasures decoding are considered in this paper. We also derive
tighter bounds on the complexities of fast polynomial multiplications based on
Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and
Networkin
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
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