Parallel geostatistics for sparse and dense datasets

Very large spatially-referenced datasets, for example, those derived from satellite-based sensors which sample across the globe or large monitoring networks of individual sensors, are becoming increasingly common and more widely available for use in environmental decision making. In large or dense sensor networks, huge quantities of data can be collected over small time periods. In many applications the generation of maps, or predictions at specific locations, from the data in (near) real-time is crucial. Geostatistical operations such as interpolation are vital in this map-generation process and in emergency situations, the resulting predictions need to be available almost instantly, so that decision makers can make informed decisions and define risk and evacuation zones. It is also helpful when analysing data in less time critical applications, for example when interacting directly with the data for exploratory analysis, that the algorithms are responsive within a reasonable time frame. Performing geostatistical analysis on such large spatial datasets can present a number of problems, particularly in the case where maximum likelihood. Although the storage requirements only scale linearly with the number of observations in the dataset, the computational complexity in terms of memory and speed, scale quadratically and cubically respectively. Most modern commodity hardware has at least 2 processor cores if not more. Other mechanisms for allowing parallel computation such as Grid based systems are also becoming increasingly commonly available. However, currently there seems to be little interest in exploiting this extra processing power within the context of geostatistics. In this paper we review the existing parallel approaches for geostatistics. By recognising that diffeerent natural parallelisms exist and can be exploited depending on whether the dataset is sparsely or densely sampled with respect to the range of variation, we introduce two contrasting novel implementations of parallel algorithms based on approximating the data likelihood extending the methods of Vecchia [1988] and Tresp [2000]. Using parallel maximum likelihood variogram estimation and parallel prediction algorithms we show that computational time can be significantly reduced. We demonstrate this with both sparsely sampled data and densely sampled data on a variety of architectures ranging from the common dual core processor, found in many modern desktop computers, to large multi-node super computers. To highlight the strengths and weaknesses of the diffeerent methods we employ synthetic data sets and go on to show how the methods allow maximum likelihood based inference on the exhaustive Walker Lake data set.

Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Properties of sparse random matrices over finite fields

Typical properties of sparse random matrices over finite (Galois) fields are studied, in the limit of large matrices, using techniques from the physics of disordered systems. For the case of a finite field GF(q) with prime order q, we present results for the average kernel dimension, average dimension of the eigenvector spaces and the distribution of the eigenvalues. The number of matrices for a given distribution of entries is also calculated for the general case. The significance of these results to error-correcting codes and random graphs is also discussed.

Inference and optimization of real edges on sparse graphs:A statistical physics perspective

Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real edge variables that interact at the network nodes are obtained in various cases. When applied to the representative problem of network resource allocation, efficient distributed algorithms are also devised. Scaling properties with respect to the network connectivity and the resource availability are found, and links to probabilistic Bayesian approximation methods are established. Different cost measures are considered and algorithmic solutions in the various cases are devised and examined numerically. Simulation results are in full agreement with the theory. © 2007 The American Physical Society.

Binary sparse nonnegative matrix factorization

This paper presents a fast part-based subspace selection algorithm, termed the binary sparse nonnegative matrix factorization (B-SNMF). Both the training process and the testing process of B-SNMF are much faster than those of binary principal component analysis (B-PCA). Besides, B-SNMF is more robust to occlusions in images. Experimental results on face images demonstrate the effectiveness and the efficiency of the proposed B-SNMF.

Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval

Non-uniform B-spline dictionaries on a compact interval are discussed in the context of sparse signal representation. For each given partition, dictionaries of B-spline functions for the corresponding spline space are built up by dividing the partition into subpartitions and joining together the bases for the concomitant subspaces. The resulting slightly redundant dictionaries are composed of B-spline functions of broader support than those corresponding to the B-spline basis for the identical space. Such dictionaries are meant to assist in the construction of adaptive sparse signal representation through a combination of stepwise optimal greedy techniques.

Competition for shortest paths on sparse graphs

Optimal paths connecting randomly selected network nodes and fixed routers are studied analytically in the presence of a nonlinear overlap cost that penalizes congestion. Routing becomes more difficult as the number of selected nodes increases and exhibits ergodicity breaking in the case of multiple routers. The ground state of such systems reveals nonmonotonic complex behaviors in average path length and algorithmic convergence, depending on the network topology, and densities of communicating nodes and routers. A distributed linearly scalable routing algorithm is also devised. © 2012 American Physical Society.

Sparse representation of astronomical images

Sparse representation of astronomical images is discussed. It is shown that a significant gain in sparsity is achieved when particular mixed dictionaries are used for approximating these types of images with greedy selection strategies. Experiments are conducted to confirm (i) the effectiveness at producing sparse representations and (ii) competitiveness, with respect to the time required to process large images. The latter is a consequence of the suitability of the proposed dictionaries for approximating images in partitions of small blocks. This feature makes it possible to apply the effective greedy selection technique called orthogonal matching pursuit, up to some block size. For blocks exceeding that size, a refinement of the original matching pursuit approach is considered. The resulting method is termed "self-projected matching pursuit," because it is shown to be effective for implementing, via matching pursuit itself, the optional backprojection intermediate steps in that approach. © 2013 Optical Society of America.

Sparse image representation with encryption

In this thesis we present an overview of sparse approximations of grey level images. The sparse representations are realized by classic, Matching Pursuit (MP) based, greedy selection strategies. One such technique, termed Orthogonal Matching Pursuit (OMP), is shown to be suitable for producing sparse approximations of images, if they are processed in small blocks. When the blocks are enlarged, the proposed Self Projected Matching Pursuit (SPMP) algorithm, successfully renders equivalent results to OMP. A simple coding algorithm is then proposed to store these sparse approximations. This is shown, under certain conditions, to be competitive with JPEG2000 image compression standard. An application termed image folding, which partially secures the approximated images is then proposed. This is extended to produce a self contained folded image, containing all the information required to perform image recovery. Finally a modified OMP selection technique is applied to produce sparse approximations of Red Green Blue (RGB) images. These RGB approximations are then folded with the self contained approach.

Research allocation on sparse graphs

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Message-passing for inference and optimization of real variables on sparse graphs

The inference and optimization in sparse graphs with real variables is studied using methods of statistical mechanics. Efficient distributed algorithms for the resource allocation problem are devised. Numerical simulations show excellent performance and full agreement with the theoretical results. © Springer-Verlag Berlin Heidelberg 2006.

A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations

2000 Mathematics Subject Classification: 65H10.

100-Gb/s point-to-point solutions for long-reach passive optical networks in sparse rural and urban areas

Long reach-passive optical networks (LR-PON) are being proposed as a means of enabling ubiquitous fiber-to-the-home (FTTH) by massive sharing of network resources and therefore reducing per customer costs to affordable levels. In this paper, we analyze the chain solutions for LR-PON deployment in urban and rural areas at 100-Gb/s point-to-point transmission using dual polarization-quaternary phase shift-keying (DP-QPSK) modulation. The numerical analysis shows that with appropriate finite impulse response (FIR) filter designs, 100-Gb/s transmission can be achieved with at least 512 way split and up to 160 km total distance, which is sufficient for many of the optical paths in a practical situation, for point-to-point link from one LR-PON to another LR-PON through the optical switch at the metro nodes and across a core light path through the core network without regeneration.

A template model predicts detection of sparse stimul

Contemporary models of contrast integration across space assume that pooling operates uniformly over the target region. For sparse stimuli, where high contrast regions are separated by areas containing no signal, this strategy may be sub-optimal because it pools more noise than signal as area increases. Little is known about the behaviour of human observers for detecting such stimuli. We performed an experiment in which three observers detected regular textures of various areas, and six levels of sparseness. Stimuli were regular grids of horizontal grating micropatches, each 1 cycle wide. We varied the ratio of signals (marks) to gaps (spaces), with mark:space ratios ranging from 1 : 0 (a dense texture with no spaces) to 1 : 24. To compensate for the decline in sensitivity with increasing distance from fixation, we adjusted the stimulus contrast as a function of eccentricity based on previous measurements [Baldwin, Meese & Baker, 2012, J Vis, 12(11):23]. We used the resulting area summation functions and psychometric slopes to test several filter-based models of signal combination. A MAX model failed to predict the thresholds, but did a good job on the slopes. Blanket summation of stimulus energy improved the threshold fit, but did not predict an observed slope increase with mark:space ratio. Our best model used a template matched to the sparseness of the stimulus, and pooled the squared contrast signal over space. Templates for regular patterns have also recently been proposed to explain the regular appearance of slightly irregular textures (Morgan et al, 2012, Proc R Soc B, 279, 2754–2760)

Cooperative greedy pursuit strategies for sparse signal representation by partitioning

Cooperative Greedy Pursuit Strategies are considered for approximating a signal partition subjected to a global constraint on sparsity. The approach aims at producing a high quality sparse approximation of the whole signal, using highly coherent redundant dictionaries. The cooperation takes place by ranking the partition units for their sequential stepwise approximation, and is realized by means of i)forward steps for the upgrading of an approximation and/or ii) backward steps for the corresponding downgrading. The advantage of the strategy is illustrated by approximation of music signals using redundant trigonometric dictionaries. In addition to rendering stunning improvements in sparsity with respect to the concomitant trigonometric basis, these dictionaries enable a fast implementation of the approach via the Fast Fourier Transform