519 research outputs found
Sublinear-Time Algorithms for Compressive Phase Retrieval
In the compressive phase retrieval problem, or phaseless compressed sensing,
or compressed sensing from intensity only measurements, the goal is to
reconstruct a sparse or approximately -sparse vector
given access to , where denotes the vector obtained from
taking the absolute value of coordinate-wise. In this paper
we present sublinear-time algorithms for different variants of the compressive
phase retrieval problem which are akin to the variants considered for the
classical compressive sensing problem in theoretical computer science. Our
algorithms use pure combinatorial techniques and near-optimal number of
measurements.Comment: The ell_2/ell_2 algorithm was substituted by a modification of the
ell_infty/ell_2 algorithm which strictly subsumes i
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity
Sublinear time algorithms for earth mover's distance
We study the problem of estimating the Earth Mover’s Distance (EMD) between probability distributions
when given access only to samples of the distributions. We give closeness testers and additive-error
estimators over domains in [0, 1][superscript d], with sample complexities independent of domain size – permitting
the testability even of continuous distributions over infinite domains. Instead, our algorithms depend on
other parameters, such as the diameter of the domain space, which may be significantly smaller. We also
prove lower bounds showing the dependencies on these parameters to be essentially optimal. Additionally,
we consider whether natural classes of distributions exist for which there are algorithms with better
dependence on the dimension, and show that for highly clusterable data, this is indeed the case. Lastly,
we consider a variant of the EMD, defined over tree metrics instead of the usual l 1 metric, and give tight
upper and lower bounds
Linear and sublinear time algorithms for the basis of abelian groups
AbstractIt is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G≅G1×G2×⋯×Gt, where each Gi is a cyclic group of order pj for some prime p and integer j≥1. If ai generates the cyclic group of Gi, i=1,2,…,t, then the elements a1,a2,…,at are called a basis of G. We show a randomized algorithm such that given a set of generators M={x1,…,xk} for an abelian group G and the prime factorization of order ord(xi)(i=1,…,k), it computes a basis of G in O(|M|(logn)2+∑i=1tnipini/2) time, where n=|G| has prime factorization p1n1p2n2⋯ptnt (which is not a part of input). This generalizes Buchmann and Schmidt’s algorithm that takes O(|M||G|) time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O(n) time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group
Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs
While algorithms for planar graphs have received a lot of attention, few
papers have focused on the additional power that one gets from assuming an
embedding of the graph is available. While in the classic sequential setting,
this assumption gives no additional power (as a planar graph can be embedded in
linear time), we show that this is far from being the case in other settings.
We assume that the embedding is straight-line, but our methods also generalize
to non-straight-line embeddings. Specifically, we focus on sublinear-time
computation and massively parallel computation (MPC).
Our main technical contribution is a sublinear-time algorithm for computing a
relaxed version of an -division. We then show how this can be used to
estimate Lipschitz additive graph parameters. This includes, for example, the
maximum matching, maximum independent set, or the minimum dominating set. We
also show how this can be used to solve some property testing problems with
respect to the vertex edit distance.
In the second part of our paper, we show an MPC algorithm that computes an
-division of the input graph. We show how this can be used to solve various
classical graph problems with space per machine of for
some , and while performing rounds. This includes for
example approximate shortest paths or the minimum spanning tree. Our results
also imply an improved MPC algorithm for Euclidean minimum spanning tree
Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning
In this paper, we study several important geometric optimization problems arising in machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space ?^d. The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms. Motivated by the recent developments on beyond worst-case analysis, we introduce the notion of stability for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points n. In particular, the second algorithm has the sample complexity even independent of the dimensionality d. We also consider the general case without the stability assumption. We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB, which improves the running time and the number of passes for the previous sublinear MEB algorithms. Further, we extend our proposed notion of stability and design sublinear time algorithms for other geometric optimization problems including MEB with outliers, polytope distance, one-class and two-class linear SVMs (without or with outliers). Our proposed algorithms also work fine for kernels
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