67,950 research outputs found
On the Closest Vector Problem with a Distance Guarantee
We present a substantially more efficient variant, both in terms of running
time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky,
and Micciancio for solving CVPP (the preprocessing version of the Closest
Vector Problem, CVP) with a distance guarantee. For instance, for any , our algorithm finds the (unique) closest lattice point for any target
point whose distance from the lattice is at most times the length of
the shortest nonzero lattice vector, requires as preprocessing advice only vectors, and runs in
time .
As our second main contribution, we present reductions showing that it
suffices to solve CVP, both in its plain and preprocessing versions, when the
input target point is within some bounded distance of the lattice. The
reductions are based on ideas due to Kannan and a recent sparsification
technique due to Dadush and Kun. Combining our reductions with the LLM
algorithm gives an approximation factor of for search
CVPP, improving on the previous best of due to Lagarias, Lenstra,
and Schnorr. When combined with our improved algorithm we obtain, somewhat
surprisingly, that only O(n) vectors of preprocessing advice are sufficient to
solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance
Decoding and the Closest Vector Problem with Preprocessing". Conference on
Computational Complexity (2014
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
We show the first dimension-preserving search-to-decision reductions for
approximate SVP and CVP. In particular, for any ,
we obtain an efficient dimension-preserving reduction from -SVP to -GapSVP and an efficient dimension-preserving reduction
from -CVP to -GapCVP. These results generalize the known
equivalences of the search and decision versions of these problems in the exact
case when . For SVP, we actually obtain something slightly stronger
than a search-to-decision reduction---we reduce -SVP to
-unique SVP, a potentially easier problem than -GapSVP.Comment: Updated to acknowledge additional prior wor
GOGMA: Globally-Optimal Gaussian Mixture Alignment
Gaussian mixture alignment is a family of approaches that are frequently used
for robustly solving the point-set registration problem. However, since they
use local optimisation, they are susceptible to local minima and can only
guarantee local optimality. Consequently, their accuracy is strongly dependent
on the quality of the initialisation. This paper presents the first
globally-optimal solution to the 3D rigid Gaussian mixture alignment problem
under the L2 distance between mixtures. The algorithm, named GOGMA, employs a
branch-and-bound approach to search the space of 3D rigid motions SE(3),
guaranteeing global optimality regardless of the initialisation. The geometry
of SE(3) was used to find novel upper and lower bounds for the objective
function and local optimisation was integrated into the scheme to accelerate
convergence without voiding the optimality guarantee. The evaluation
empirically supported the optimality proof and showed that the method performed
much more robustly on two challenging datasets than an existing
globally-optimal registration solution.Comment: Manuscript in press 2016 IEEE Conference on Computer Vision and
Pattern Recognitio
Solving the Closest Vector Problem in Time--- The Discrete Gaussian Strikes Again!
We give a -time and space randomized algorithm for solving the
exact Closest Vector Problem (CVP) on -dimensional Euclidean lattices. This
improves on the previous fastest algorithm, the deterministic
-time and -space algorithm of
Micciancio and Voulgaris.
We achieve our main result in three steps. First, we show how to modify the
sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian
sampling over lattice shifts, , with very low parameters. While the
actual algorithm is a natural generalization of [ADRS15], the analysis uses
substantial new ideas. This yields a -time algorithm for
approximate CVP for any approximation factor .
Second, we show that the approximate closest vectors to a target vector can
be grouped into "lower-dimensional clusters," and we use this to obtain a
recursive reduction from exact CVP to a variant of approximate CVP that
"behaves well with these clusters." Third, we show that our discrete Gaussian
sampling algorithm can be used to solve this variant of approximate CVP.
The analysis depends crucially on some new properties of the discrete
Gaussian distribution and approximate closest vectors, which might be of
independent interest
- …