1,808 research outputs found
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
We review the basic outline of the highly successful diffusion Monte Carlo
technique commonly used in contexts ranging from electronic structure
calculations to rare event simulation and data assimilation, and propose a new
class of randomized iterative algorithms based on similar principles to address
a variety of common tasks in numerical linear algebra. From the point of view
of numerical linear algebra, the main novelty of the Fast Randomized Iteration
schemes described in this article is that they work in either linear or
constant cost per iteration (and in total, under appropriate conditions) and
are rather versatile: we will show how they apply to solution of linear
systems, eigenvalue problems, and matrix exponentiation, in dimensions far
beyond the present limits of numerical linear algebra. While traditional
iterative methods in numerical linear algebra were created in part to deal with
instances where a matrix (of size ) is too big to store, the
algorithms that we propose are effective even in instances where the solution
vector itself (of size ) may be too big to store or manipulate.
In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes
that have been applied to matrices as large as . We
provide basic convergence results, discuss the dependence of these results on
the dimension of the system, and demonstrate dramatic cost savings on a range
of test problems.Comment: 44 pages, 7 figure
A Universal Scheme for WynerâZiv Coding of Discrete Sources
We consider the WynerâZiv (WZ) problem of lossy compression where the decompressor observes a noisy version of the source, whose statistics are unknown. A new family of WZ coding algorithms is proposed and their universal optimality is proven. Compression consists of sliding-window processing followed by LempelâZiv (LZ) compression, while the decompressor is based on a modification of the discrete universal denoiser (DUDE) algorithm to take advantage of side information. The new algorithms not only universally attain the fundamental limits, but also suggest a paradigm for practical WZ coding. The effectiveness of our approach is illustrated with experiments on binary images, and English text using a low complexity algorithm motivated by our class of universally optimal WZ codes
Lossy Compression with Near-uniform Encoder Outputs
It is well known that lossless compression of a discrete memoryless source
with near-uniform encoder output is possible at a rate above its entropy if and
only if the encoder is randomized. This work focuses on deriving conditions for
near-uniform encoder output(s) in the Wyner-Ziv and the distributed lossy
compression problems. We show that in the Wyner-Ziv problem, near-uniform
encoder output and operation close to the WZ-rate limit is simultaneously
possible, whereas in the distributed lossy compression problem, jointly
near-uniform outputs is achievable in the interior of the distributed lossy
compression rate region if the sources share non-trivial G\'{a}cs-K\"{o}rner
common information.Comment: Submitted to the 2016 IEEE International Symposium on Information
Theory (11 Pages, 3 Figures
Iterative Slepian-Wolf Decoding and FEC Decoding for Compress-and-Forward Systems
While many studies have concentrated on providing theoretical analysis for the relay assisted compress-and-forward systems little effort has yet been made to the construction and evaluation of a practical system. In this paper a practical CF system incorporating an error-resilient multilevel Slepian-Wolf decoder is introduced and a novel iterative processing structure which allows information exchanging between the Slepian-Wolf decoder and the forward error correction decoder of the main source message is proposed. In addition, a new quantization scheme is incorporated as well to avoid the complexity of the reconstruction of the relay signal at the final decoder of the destination. The results demonstrate that the iterative structure not only reduces the decoding loss of the Slepian-Wolf decoder, it also improves the decoding performance of the main message from the source
Joint source-channel coding with feedback
This paper quantifies the fundamental limits of variable-length transmission
of a general (possibly analog) source over a memoryless channel with noiseless
feedback, under a distortion constraint. We consider excess distortion, average
distortion and guaranteed distortion (-semifaithful codes). In contrast to
the asymptotic fundamental limit, a general conclusion is that allowing
variable-length codes and feedback leads to a sizable improvement in the
fundamental delay-distortion tradeoff. In addition, we investigate the minimum
energy required to reproduce source samples with a given fidelity after
transmission over a memoryless Gaussian channel, and we show that the required
minimum energy is reduced with feedback and an average (rather than maximal)
power constraint.Comment: To appear in IEEE Transactions on Information Theor
Quantum Monte Carlo with very large multideterminant wavefunctions
An algorithm to compute efficiently the first two derivatives of (very) large
multideterminant wavefunctions for quantum Monte Carlo calculations is
presented. The calculation of determinants and their derivatives is performed
using the Sherman-Morrison formula for updating the inverse Slater matrix. An
improved implementation based on the reduction of the number of column
substitutions and on a very efficient implementation of the calculation of the
scalar products involved is presented. It is emphasized that multideterminant
expansions contain in general a large number of identical spin-specific
determinants: for typical configuration interaction-type wavefunctions the
number of unique spin-specific determinants
() with a non-negligible weight in the expansion is
of order . We show that a careful implementation
of the calculation of the -dependent contributions can make this
step negligible enough so that in practice the algorithm scales as the total
number of unique spin-specific determinants, , over a wide range of total number of determinants (here,
up to about one million), thus greatly reducing the total
computational cost. Finally, a new truncation scheme for the multideterminant
expansion is proposed so that larger expansions can be considered without
increasing the computational time. The algorithm is illustrated with
all-electron Fixed-Node Diffusion Monte Carlo calculations of the total energy
of the chlorine atom. Calculations using a trial wavefunction including about
750 000 determinants with a computational increase of 400 compared to a
single-determinant calculation are shown to be feasible.Comment: 9 pages, 3 figure
First-principles modeling of quantum nuclear effects and atomic interactions in solid He-4 at high pressure
We present a first-principles computational study of solid He-4 at T = 0 K and pressures up to similar to 160 GPa. Our computational strategy consists in using van der Waals density functional theory (DFT-vdW) to describe the electronic degrees of freedom in this material, and the diffusion Monte Carlo (DMC) method to solve the Schrodinger equation describing the behavior of the quantum nuclei. For this, we construct an analytical interaction function based on the pairwise Aziz potential that closely matches the volume variation of the cohesive energy calculated with DFT-vdW in dense helium. Interestingly, we find that the kinetic energy of solid He-4 does not increase appreciably with compression for P >= 85 GPa. Also, we show that the Lindemann ratio in dense solid He-4 amounts to 0.10 almost independently of pressure. The reliability of customary quasiharmonic DFT (QH DFT) approaches in describing quantum nuclear effects in solids is also studied. We find that QH DFT simulations, although provide a reasonable equation of state in agreement with experiments, are not able to reproduce correctly these critical effects in compressed He-4. In particular, we disclose huge discrepancies of at least similar to 50% in the calculated He-4 kinetic energies using both the QH DFT and present DFT-DMC methods.Postprint (published version
Discrete denoising of heterogenous two-dimensional data
We consider discrete denoising of two-dimensional data with characteristics
that may be varying abruptly between regions.
Using a quadtree decomposition technique and space-filling curves, we extend
the recently developed S-DUDE (Shifting Discrete Universal DEnoiser), which was
tailored to one-dimensional data, to the two-dimensional case. Our scheme
competes with a genie that has access, in addition to the noisy data, also to
the underlying noiseless data, and can employ different two-dimensional
sliding window denoisers along distinct regions obtained by a quadtree
decomposition with leaves, in a way that minimizes the overall loss. We
show that, regardless of what the underlying noiseless data may be, the
two-dimensional S-DUDE performs essentially as well as this genie, provided
that the number of distinct regions satisfies , where is the total
size of the data. The resulting algorithm complexity is still linear in both
and , as in the one-dimensional case. Our experimental results show that
the two-dimensional S-DUDE can be effective when the characteristics of the
underlying clean image vary across different regions in the data.Comment: 16 pages, submitted to IEEE Transactions on Information Theor
How to Achieve the Capacity of Asymmetric Channels
We survey coding techniques that enable reliable transmission at rates that
approach the capacity of an arbitrary discrete memoryless channel. In
particular, we take the point of view of modern coding theory and discuss how
recent advances in coding for symmetric channels help provide more efficient
solutions for the asymmetric case. We consider, in more detail, three basic
coding paradigms.
The first one is Gallager's scheme that consists of concatenating a linear
code with a non-linear mapping so that the input distribution can be
appropriately shaped. We explicitly show that both polar codes and spatially
coupled codes can be employed in this scenario. Furthermore, we derive a
scaling law between the gap to capacity, the cardinality of the input and
output alphabets, and the required size of the mapper.
The second one is an integrated scheme in which the code is used both for
source coding, in order to create codewords distributed according to the
capacity-achieving input distribution, and for channel coding, in order to
provide error protection. Such a technique has been recently introduced by
Honda and Yamamoto in the context of polar codes, and we show how to apply it
also to the design of sparse graph codes.
The third paradigm is based on an idea of B\"ocherer and Mathar, and
separates the two tasks of source coding and channel coding by a chaining
construction that binds together several codewords. We present conditions for
the source code and the channel code, and we describe how to combine any source
code with any channel code that fulfill those conditions, in order to provide
capacity-achieving schemes for asymmetric channels. In particular, we show that
polar codes, spatially coupled codes, and homophonic codes are suitable as
basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published
in IEEE Trans. Inform. Theor
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