33 research outputs found
A geometric approach to archetypal analysis and non-negative matrix factorization
Archetypal analysis and non-negative matrix factorization (NMF) are staples
in a statisticians toolbox for dimension reduction and exploratory data
analysis. We describe a geometric approach to both NMF and archetypal analysis
by interpreting both problems as finding extreme points of the data cloud. We
also develop and analyze an efficient approach to finding extreme points in
high dimensions. For modern massive datasets that are too large to fit on a
single machine and must be stored in a distributed setting, our approach makes
only a small number of passes over the data. In fact, it is possible to obtain
the NMF or perform archetypal analysis with just two passes over the data.Comment: 36 pages, 13 figure
SCDM-k: Localized orbitals for solids via selected columns of the density matrix
The recently developed selected columns of the density matrix (SCDM) method
[J. Chem. Theory Comput. 11, 1463, 2015] is a simple, robust, efficient and
highly parallelizable method for constructing localized orbitals from a set of
delocalized Kohn-Sham orbitals for insulators and semiconductors with
point sampling of the Brillouin zone. In this work we generalize the SCDM
method to Kohn-Sham density functional theory calculations with k-point
sampling of the Brillouin zone, which is needed for more general electronic
structure calculations for solids. We demonstrate that our new method, called
SCDM-k, is by construction gauge independent and is a natural way to describe
localized orbitals. SCDM-k computes localized orbitals without the use of an
optimization procedure, and thus does not suffer from the possibility of being
trapped in a local minimum. Furthermore, the computational complexity of using
SCDM-k to construct orthogonal and localized orbitals scales as O(N log N )
where N is the total number of k-points in the Brillouin zone. SCDM-k is
therefore efficient even when a large number of k-points are used for Brillouin
zone sampling. We demonstrate the numerical performance of SCDM-k using systems
with model potentials in two and three dimensions.Comment: 25 pages, 7 figures; added more background sections, clarified
presentation of the algorithm, revised the presentation of previous work,
added a more high level overview of the new algorithm, and mildly clarified
the presentation of the results (there were no changes to the numerical
results themselves
Compressed representation of Kohn-Sham orbitals via selected columns of the density matrix
Given a set of Kohn-Sham orbitals from an insulating system, we present a
simple, robust, efficient and highly parallelizable method to construct a set
of, optionally orthogonal, localized basis functions for the associated
subspace. Our method explicitly uses the fact that density matrices associated
with insulating systems decay exponentially along the off-diagonal direction in
the real space representation. Our method avoids the usage of an optimization
procedure, and the localized basis functions are constructed directly from a
set of selected columns of the density matrix (SCDM). Consequently, the only
adjustable parameter in our method is the truncation threshold of the localized
basis functions. Our method can be used in any electronic structure software
package with an arbitrary basis set. We demonstrate the numerical accuracy and
parallel scalability of the SCDM procedure using orbitals generated by the
Quantum ESPRESSO software package. We also demonstrate a procedure for
combining SCDM with Hockney's algorithm to efficiently perform Hartree-Fock
exchange energy calculations with near linear scaling.Comment: 7 pages, 4 figures; short example code for computing the SCDM;
parallel scaling results; slightly restructured introduction and
clarification of the input needed to compute the SCD
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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Near Optimal Rational Approximations of Large Data Sets
We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large data sets. In contrast to wavelet-type approximations often used for the same purpose, these new approximations are effectively shift invariant. On the other hand, when dealing with large data sets the complexity of our current non-linear algorithms for computing near optimal rational approximations prevents their direct use. By using an intermediate representation of the data via B-splines, followed by a rational approximation of the B-splines themselves, we obtain a suboptimal rational approximation of data segments. Then, using reduction and merging algorithms for data segments, we arrive at an efficient procedure for computing near optimal rational approximations for large data sets. A motivating example is the compression of audio signals and we provide several examples of compressed representations produced by the algorithm