8,819 research outputs found
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
Derandomized Graph Product Results using the Low Degree Long Code
In this paper, we address the question of whether the recent derandomization
results obtained by the use of the low-degree long code can be extended to
other product settings. We consider two settings: (1) the graph product results
of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is
stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and
Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of
approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of which exhibits the following property (shown for
by Alon et al.): independent sets close in size to the
maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur and
Shinkar shows that if there exist two sets of vertices and in
with very few edges with one endpoint in and another in
, then it must be the case that the two sets and share a single
influential coordinate. In our second result, we show that a similar "majority
is stablest" statement holds good for a considerably smaller subgraph of
. Furthermore using this result, we give a more efficient
reduction from Unique Games to the graph coloring problem, leading to improved
hardness of approximation results for coloring
Short rainbow cycles in graphs and matroids
Let be a simple -vertex graph and be a colouring of with
colours, where each colour class has size at least . We prove that
contains a rainbow cycle of length at most ,
which is best possible. Our result settles a special case of a strengthening of
the Caccetta-H\"aggkvist conjecture, due to Aharoni. We also show that the
matroid generalization of our main result also holds for cographic matroids,
but fails for binary matroids.Comment: 9 pages, 2 figure
Phase Transitions in Sparse PCA
We study optimal estimation for sparse principal component analysis when the
number of non-zero elements is small but on the same order as the dimension of
the data. We employ approximate message passing (AMP) algorithm and its state
evolution to analyze what is the information theoretically minimal mean-squared
error and the one achieved by AMP in the limit of large sizes. For a special
case of rank one and large enough density of non-zeros Deshpande and Montanari
[1] proved that AMP is asymptotically optimal. We show that both for low
density and for large rank the problem undergoes a series of phase transitions
suggesting existence of a region of parameters where estimation is information
theoretically possible, but AMP (and presumably every other polynomial
algorithm) fails. The analysis of the large rank limit is particularly
instructive.Comment: 6 pages, 3 figure
Report on "Geometry and representation theory of tensors for computer science, statistics and other areas."
This is a technical report on the proceedings of the workshop held July 21 to
July 25, 2008 at the American Institute of Mathematics, Palo Alto, California,
organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We
include a list of open problems coming from applications in 4 different areas:
signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and
holographic algorithms, and entanglement and quantum information theory. We
emphasize the interactions between geometry and representation theory and these
applied areas
From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians
In this work, we make a connection between two seemingly different problems.
The first problem involves characterizing the properties of entanglement in the
ground state of gapped local Hamiltonians, which is a central topic in quantum
many-body physics. The second problem is on the quantum communication
complexity of testing bipartite states with EPR assistance, a well-known
question in quantum information theory. We construct a communication protocol
for testing (or measuring) the ground state and use its communication
complexity to reveal a new structural property for the ground state
entanglement. This property, known as the entanglement spread, roughly measures
the ratio between the largest and the smallest Schmidt coefficients across a
cut in the ground state. Our main result shows that gapped ground states
possess limited entanglement spread across any cut, exhibiting an "area law"
behavior. Our result quite generally applies to any interaction graph with an
improved bound for the special case of lattices. This entanglement spread area
law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that
violate a generalized area law for the entanglement entropy. Our construction
also provides evidence for a conjecture in physics by Li and Haldane on the
entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the
technical side, we use recent advances in Hamiltonian simulation algorithms
along with quantum phase estimation to give a new construction for an
approximate ground space projector (AGSP) over arbitrary interaction graphs.Comment: 29 pages, 1 figur
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