2,065 research outputs found
Similarity-Aware Spectral Sparsification by Edge Filtering
In recent years, spectral graph sparsification techniques that can compute
ultra-sparse graph proxies have been extensively studied for accelerating
various numerical and graph-related applications. Prior nearly-linear-time
spectral sparsification methods first extract low-stretch spanning tree from
the original graph to form the backbone of the sparsifier, and then recover
small portions of spectrally-critical off-tree edges to the spanning tree to
significantly improve the approximation quality. However, it is not clear how
many off-tree edges should be recovered for achieving a desired spectral
similarity level within the sparsifier. Motivated by recent graph signal
processing techniques, this paper proposes a similarity-aware spectral graph
sparsification framework that leverages efficient spectral off-tree edge
embedding and filtering schemes to construct spectral sparsifiers with
guaranteed spectral similarity (relative condition number) level. An iterative
graph densification scheme is introduced to facilitate efficient and effective
filtering of off-tree edges for highly ill-conditioned problems. The proposed
method has been validated using various kinds of graphs obtained from public
domain sparse matrix collections relevant to VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications
Iterative solutions to the steady state density matrix for optomechanical systems
We present a sparse matrix permutation from graph theory that gives stable
incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions
to the steady state density matrix for quantum optomechanical systems. This
reordering is efficient, adding little overhead to the computation, and results
in a marked reduction in both memory and runtime requirements compared to other
solution methods, with performance gains increasing with system size. Either of
these benchmarks can be tuned via the preconditioner accuracy and solution
tolerance. This reordering optimizes the condition number of the approximate
inverse, and is the only method found to be stable at large Hilbert space
dimensions. This allows for steady state solutions to otherwise intractable
quantum optomechanical systems.Comment: 10 pages, 5 figure
Matrix Factorization at Scale: a Comparison of Scientific Data Analytics in Spark and C+MPI Using Three Case Studies
We explore the trade-offs of performing linear algebra using Apache Spark,
compared to traditional C and MPI implementations on HPC platforms. Spark is
designed for data analytics on cluster computing platforms with access to local
disks and is optimized for data-parallel tasks. We examine three widely-used
and important matrix factorizations: NMF (for physical plausability), PCA (for
its ubiquity) and CX (for data interpretability). We apply these methods to
TB-sized problems in particle physics, climate modeling and bioimaging. The
data matrices are tall-and-skinny which enable the algorithms to map
conveniently into Spark's data-parallel model. We perform scaling experiments
on up to 1600 Cray XC40 nodes, describe the sources of slowdowns, and provide
tuning guidance to obtain high performance
Regularity Properties for Sparse Regression
Statistical and machine learning theory has developed several conditions
ensuring that popular estimators such as the Lasso or the Dantzig selector
perform well in high-dimensional sparse regression, including the restricted
eigenvalue, compatibility, and sensitivity properties. However, some
of the central aspects of these conditions are not well understood. For
instance, it is unknown if these conditions can be checked efficiently on any
given data set. This is problematic, because they are at the core of the theory
of sparse regression.
Here we provide a rigorous proof that these conditions are NP-hard to check.
This shows that the conditions are computationally infeasible to verify, and
raises some questions about their practical applications.
However, by taking an average-case perspective instead of the worst-case view
of NP-hardness, we show that a particular condition, sensitivity, has
certain desirable properties. This condition is weaker and more general than
the others. We show that it holds with high probability in models where the
parent population is well behaved, and that it is robust to certain data
processing steps. These results are desirable, as they provide guidance about
when the condition, and more generally the theory of sparse regression, may be
relevant in the analysis of high-dimensional correlated observational data.Comment: Manuscript shortened and more motivation added. To appear in
Communications in Mathematics and Statistic
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