178 research outputs found
Symmetry-based matrix factorization
AbstractWe present a method for factoring a given matrix M into a short product of sparse matrices, provided that M has a suitable āsymmetryā. This sparse factorization represents a fast algorithm for the matrixāvector multiplication with M. The factorization method consists of two essential steps. First, a combinatorial search is used to compute a suitable symmetry of M in the form of a pair of group representations. Second, the group representations are decomposed stepwise, which yields factorized decomposition matrices and determines a sparse factorization of M. The focus of this article is the first step, finding the symmetries. All algorithms described have been implemented in the library AREP. We present examples for automatically generated sparse factorizationsāand hence fast algorithmsāfor a class of matrices corresponding to digital signal processing transforms including the discrete Fourier, cosine, Hartley, and Haar transforms
Causal Fourier Analysis on Directed Acyclic Graphs and Posets
We present a novel form of Fourier analysis, and associated signal processing
concepts, for signals (or data) indexed by edge-weighted directed acyclic
graphs (DAGs). This means that our Fourier basis yields an eigendecomposition
of a suitable notion of shift and convolution operators that we define. DAGs
are the common model to capture causal relationships between data values and in
this case our proposed Fourier analysis relates data with its causes under a
linearity assumption that we define. The definition of the Fourier transform
requires the transitive closure of the weighted DAG for which several forms are
possible depending on the interpretation of the edge weights. Examples include
level of influence, distance, or pollution distribution. Our framework is
different from prior GSP: it is specific to DAGs and leverages, and extends,
the classical theory of Moebius inversion from combinatorics. For a
prototypical application we consider DAGs modeling dynamic networks in which
edges change over time. Specifically, we model the spread of an infection on
such a DAG obtained from real-world contact tracing data and learn the
infection signal from samples assuming sparsity in the Fourier domain.Comment: 13 pages, 11 figure
Fast M\"obius and Zeta Transforms
M\"obius inversion of functions on partially ordered sets (posets)
is a classical tool in combinatorics. For finite posets it
consists of two, mutually inverse, linear transformations called zeta and
M\"obius transform, respectively. In this paper we provide novel fast
algorithms for both that require time and space, where and is the width (length of longest antichain) of
, compared to for a direct computation. Our approach
assumes that is given as directed acyclic graph (DAG)
. The algorithms are then constructed using a chain
decomposition for a one time cost of , where is the number of
edges in the DAG's transitive reduction. We show benchmarks with
implementations of all algorithms including parallelized versions. The results
show that our algorithms enable M\"obius inversion on posets with millions of
nodes in seconds if the defining DAGs are sufficiently sparse.Comment: 16 pages, 7 figures, submitted for revie
D-ADMM: A Communication-Efficient Distributed Algorithm For Separable Optimization
We propose a distributed algorithm, named Distributed Alternating Direction
Method of Multipliers (D-ADMM), for solving separable optimization problems in
networks of interconnected nodes or agents. In a separable optimization problem
there is a private cost function and a private constraint set at each node. The
goal is to minimize the sum of all the cost functions, constraining the
solution to be in the intersection of all the constraint sets. D-ADMM is proven
to converge when the network is bipartite or when all the functions are
strongly convex, although in practice, convergence is observed even when these
conditions are not met. We use D-ADMM to solve the following problems from
signal processing and control: average consensus, compressed sensing, and
support vector machines. Our simulations show that D-ADMM requires less
communications than state-of-the-art algorithms to achieve a given accuracy
level. Algorithms with low communication requirements are important, for
example, in sensor networks, where sensors are typically battery-operated and
communicating is the most energy consuming operation.Comment: To appear in IEEE Transactions on Signal Processin
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