95 research outputs found
A Matrix Hyperbolic Cosine Algorithm and Applications
In this paper, we generalize Spencer's hyperbolic cosine algorithm to the
matrix-valued setting. We apply the proposed algorithm to several problems by
analyzing its computational efficiency under two special cases of matrices; one
in which the matrices have a group structure and an other in which they have
rank-one. As an application of the former case, we present a deterministic
algorithm that, given the multiplication table of a finite group of size ,
it constructs an expanding Cayley graph of logarithmic degree in near-optimal
O(n^2 log^3 n) time. For the latter case, we present a fast deterministic
algorithm for spectral sparsification of positive semi-definite matrices, which
implies an improved deterministic algorithm for spectral graph sparsification
of dense graphs. In addition, we give an elementary connection between spectral
sparsification of positive semi-definite matrices and element-wise matrix
sparsification. As a consequence, we obtain improved element-wise
sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work
in (current) Section
Improved Practical Matrix Sketching with Guarantees
Matrices have become essential data representations for many large-scale
problems in data analytics, and hence matrix sketching is a critical task.
Although much research has focused on improving the error/size tradeoff under
various sketching paradigms, the many forms of error bounds make these
approaches hard to compare in theory and in practice. This paper attempts to
categorize and compare most known methods under row-wise streaming updates with
provable guarantees, and then to tweak some of these methods to gain practical
improvements while retaining guarantees.
For instance, we observe that a simple heuristic iSVD, with no guarantees,
tends to outperform all known approaches in terms of size/error trade-off. We
modify the best performing method with guarantees FrequentDirections under the
size/error trade-off to match the performance of iSVD and retain its
guarantees. We also demonstrate some adversarial datasets where iSVD performs
quite poorly. In comparing techniques in the time/error trade-off, techniques
based on hashing or sampling tend to perform better. In this setting we modify
the most studied sampling regime to retain error guarantee but obtain dramatic
improvements in the time/error trade-off.
Finally, we provide easy replication of our studies on APT, a new testbed
which makes available not only code and datasets, but also a computing platform
with fixed environmental settings.Comment: 27 page
Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication
In this paper we develop algorithms for approximating matrix multiplication
with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n
\times p} be two matrices and \eps>0. We approximate the product A^\top B using
two down-sampled sketches, \tilde{A}\in\RR^{t\times m} and
\tilde{B}\in\RR^{t\times p}, where t\ll n such that \norm{\tilde{A}^\top
\tilde{B} - A^\top B} \leq \eps \norm{A}\norm{B} with high probability. We use
two different sampling procedures for constructing \tilde{A} and \tilde{B}; one
of them is done by i.i.d. non-uniform sampling rows from A and B and the other
is done by taking random linear combinations of their rows. We prove bounds
that depend only on the intrinsic dimensionality of A and B, that is their rank
and their stable rank; namely the squared ratio between their Frobenius and
operator norm. For achieving bounds that depend on rank we employ standard
tools from high-dimensional geometry such as concentration of measure arguments
combined with elaborate \eps-net constructions. For bounds that depend on the
smaller parameter of stable rank this technology itself seems weak. However, we
show that in combination with a simple truncation argument is amenable to
provide such bounds. To handle similar bounds for row sampling, we develop a
novel matrix-valued Chernoff bound inequality which we call low rank
matrix-valued Chernoff bound. Thanks to this inequality, we are able to give
bounds that depend only on the stable rank of the input matrices...Comment: 15 pages, To appear in 22nd ACM-SIAM Symposium on Discrete Algorithms
(SODA 2011
Pipage Rounding, Pessimistic Estimators and Matrix Concentration
Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting.
We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and conc
Spectral Sparsification via Bounded-Independence Sampling
We give a deterministic, nearly logarithmic-space algorithm for mild spectral
sparsification of undirected graphs. Given a weighted, undirected graph on
vertices described by a binary string of length , an integer , and an error parameter , our algorithm runs in space
where
and are the maximum and minimum edge
weights in , and produces a weighted graph with
edges that spectrally approximates , in
the sense of Spielmen and Teng [ST04], up to an error of .
Our algorithm is based on a new bounded-independence analysis of Spielman and
Srivastava's effective resistance based edge sampling algorithm [SS08] and uses
results from recent work on space-bounded Laplacian solvers [MRSV17]. In
particular, we demonstrate an inherent tradeoff (via upper and lower bounds)
between the amount of (bounded) independence used in the edge sampling
algorithm, denoted by above, and the resulting sparsity that can be
achieved.Comment: 37 page
On Weighted Graph Sparsification by Linear Sketching
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a
cut sparsifier of an unweighted undirected graph by taking a near-linear number
of linear measurements on the graph. Subsequent works also studied computing
other graph sparsifiers using linear sketching, and obtained near-linear upper
bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and
first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21].
All these linear sketching algorithms, however, only work on unweighted graphs.
In this paper, we initiate the study of weighted graph sparsification by
linear sketching by investigating a natural class of linear sketches that we
call incidence sketches, in which each measurement is a linear combination of
the weights of edges incident on a single vertex. Our results are:
1. Weighted cut sparsification: We give an algorithm that computes a -cut sparsifier using linear
measurements, which is nearly optimal.
2. Weighted spectral sparsification: We give an algorithm that computes a -spectral sparsifier using
linear measurements. Complementing our algorithm, we then prove a superlinear
lower bound of measurements for computing some
-spectral sparsifier using incidence sketches.
3. Weighted spanner computation: We focus on graphs whose largest/smallest
edge weights differ by an factor, and prove that, for incidence
sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are
optimal up to an factor
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