465 research outputs found
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Improving Distributed Gradient Descent Using Reed-Solomon Codes
Today's massively-sized datasets have made it necessary to often perform
computations on them in a distributed manner. In principle, a computational
task is divided into subtasks which are distributed over a cluster operated by
a taskmaster. One issue faced in practice is the delay incurred due to the
presence of slow machines, known as \emph{stragglers}. Several schemes,
including those based on replication, have been proposed in the literature to
mitigate the effects of stragglers and more recently, those inspired by coding
theory have begun to gain traction. In this work, we consider a distributed
gradient descent setting suitable for a wide class of machine learning
problems. We adapt the framework of Tandon et al. (arXiv:1612.03301) and
present a deterministic scheme that, for a prescribed per-machine computational
effort, recovers the gradient from the least number of machines
theoretically permissible, via an decoding algorithm. We also provide
a theoretical delay model which can be used to minimize the expected waiting
time per computation by optimally choosing the parameters of the scheme.
Finally, we supplement our theoretical findings with numerical results that
demonstrate the efficacy of the method and its advantages over competing
schemes
Faster Sparse Matrix Inversion and Rank Computation in Finite Fields
We improve the current best running time value to invert sparse matrices over
finite fields, lowering it to an expected time for the
current values of fast rectangular matrix multiplication. We achieve the same
running time for the computation of the rank and nullspace of a sparse matrix
over a finite field. This improvement relies on two key techniques. First, we
adopt the decomposition of an arbitrary matrix into block Krylov and Hankel
matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the
explicit inverse of a block Hankel matrix using low displacement rank
techniques for structured matrices and fast rectangular matrix multiplication
algorithms. We generalize our inversion method to block structured matrices
with other displacement operators and strengthen the best known upper bounds
for explicit inversion of block Toeplitz-like and block Hankel-like matrices,
as well as for explicit inversion of block Vandermonde-like matrices with
structured blocks. As a further application, we improve the complexity of
several algorithms in topological data analysis and in finite group theory
Bit Complexity of Jordan Normal Form and Spectral Factorization
We study the bit complexity of two related fundamental computational problems
in linear algebra and control theory. Our results are: (1) An
time algorithm for
finding an approximation to the Jordan Normal form of an integer
matrix with bit entries, where is the exponent of matrix
multiplication. (2) An
time algorithm for -approximately computing the spectral
factorization of a given monic rational matrix
polynomial of degree with rational bit coefficients having bit
common denominators, which satisfies for all real . The
first algorithm is used as a subroutine in the second one.
Despite its being of central importance, polynomial complexity bounds were
not previously known for spectral factorization, and for Jordan form the best
previous best running time was an unspecified polynomial in of degree at
least twelve \cite{cai1994computing}. Our algorithms are simple and judiciously
combine techniques from numerical and symbolic computation, yielding
significant advantages over either approach by itself.Comment: 19p
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