73 research outputs found
Probabilistic Spectral Sparsification In Sublinear Time
In this paper, we introduce a variant of spectral sparsification, called
probabilistic -spectral sparsification. Roughly speaking,
it preserves the cut value of any cut with an
multiplicative error and a additive error. We show how
to produce a probabilistic -spectral sparsifier with
edges in time
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An time -approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A time approximation minimum s-t cut algorithm
with an additive error
Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time
We present the first almost-linear time algorithm for constructing
linear-sized spectral sparsification for graphs. This improves all previous
constructions of linear-sized spectral sparsification, which requires
time.
A key ingredient in our algorithm is a novel combination of two techniques
used in literature for constructing spectral sparsification: Random sampling by
effective resistance, and adaptive constructions based on barrier functions.Comment: 22 pages. A preliminary version of this paper is to appear in
proceedings of the 56th Annual IEEE Symposium on Foundations of Computer
Science (FOCS 2015
Online Row Sampling
Finding a small spectral approximation for a tall matrix is
a fundamental numerical primitive. For a number of reasons, one often seeks an
approximation whose rows are sampled from those of . Row sampling improves
interpretability, saves space when is sparse, and preserves row structure,
which is especially important, for example, when represents a graph.
However, correctly sampling rows from can be costly when the matrix is
large and cannot be stored and processed in memory. Hence, a number of recent
publications focus on row sampling in the streaming setting, using little more
space than what is required to store the outputted approximation [KL13,
KLM+14].
Inspired by a growing body of work on online algorithms for machine learning
and data analysis, we extend this work to a more restrictive online setting: we
read rows of one by one and immediately decide whether each row should be
kept in the spectral approximation or discarded, without ever retracting these
decisions. We present an extremely simple algorithm that approximates up to
multiplicative error and additive error using online samples, with memory overhead
proportional to the cost of storing the spectral approximation. We also present
an algorithm that uses ) memory but only requires
samples, which we show is
optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior
work, and expose new theoretical properties of leverage score based matrix
approximation
An Efficient Parallel Solver for SDD Linear Systems
We present the first parallel algorithm for solving systems of linear
equations in symmetric, diagonally dominant (SDD) matrices that runs in
polylogarithmic time and nearly-linear work. The heart of our algorithm is a
construction of a sparse approximate inverse chain for the input matrix: a
sequence of sparse matrices whose product approximates its inverse. Whereas
other fast algorithms for solving systems of equations in SDD matrices exploit
low-stretch spanning trees, our algorithm only requires spectral graph
sparsifiers
Large-scale semi-supervised learning with online spectral graph sparsification
International audienceWe introduce Sparse-HFS, a scalable algorithm that can compute solutions to SSL problems using only O(n polylog(n)) space and O(m polylog(n)) time
- …