Maximal operators and differentiation theorems for sparse sets

"Vegeu el resum a l'inici del document del fitxer adjunt."

Spectral analysis of random sparse matrices

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Implementation of zonal control algorithms in adaptive optical systems with sparse control matrices

Miralls deformables més i més grans, amb cada cop més actuadors estan sent utilitzats actualment en aplicacions d'òptica adaptativa. El control dels miralls amb centenars d'actuadors és un tema de gran interès, ja que les tècniques de control clàssiques basades en la seudoinversa de la matriu de control del sistema es tornen massa lentes quan es tracta de matrius de dimensions tan grans. En aquesta tesi doctoral es proposa un mètode per l'acceleració i la paral.lelitzacó dels algoritmes de control d'aquests miralls, a través de l'aplicació d'una tècnica de control basada en la reducció a zero del components més petits de la matriu de control (sparsification), seguida de l'optimització de l'ordenació dels accionadors de comandament atenent d'acord a la forma de la matriu, i finalment de la seva posterior divisió en petits blocs tridiagonals. Aquests blocs són molt més petits i més fàcils de fer servir en els càlculs, el que permet velocitats de càlcul molt superiors per l'eliminació dels components nuls en la matriu de control. A més, aquest enfocament permet la paral.lelització del càlcul, donant una com0onent de velocitat addicional al sistema. Fins i tot sense paral. lelització, s'ha obtingut un augment de gairebé un 40% de la velocitat de convergència dels miralls amb només 37 actuadors, mitjançant la tècnica proposada. Per validar això, s'ha implementat un muntatge experimental nou complet , que inclou un modulador de fase programable per a la generació de turbulència mitjançant pantalles de fase, i s'ha desenvolupat un model complert del bucle de control per investigar el rendiment de l'algorisme proposat. Els resultats, tant en la simulació com experimentalment, mostren l'equivalència total en els valors de desviació després de la compensació dels diferents tipus d'aberracions per als diferents algoritmes utilitzats, encara que el mètode proposat aquí permet una càrrega computacional molt menor. El procediment s'espera que sigui molt exitós quan s'aplica a miralls molt grans.

Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.

Inventory of sparse forest populations using adaptive cluster sampling

Abstract

Sparse image reconstruction on the sphere: implications of a new sampling theorem.

We study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere. We discuss how a reduction in the number of samples required to represent all information content of a band-limited signal acts to improve the fidelity of sparse image reconstruction, through both the dimensionality and sparsity of signals. To demonstrate this result, we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient. We develop a framework for total variation inpainting on the sphere, including fast methods to render the inpainting problem computationally feasible at high resolution. Recently a new sampling theorem on the sphere was developed, reducing the required number of samples by a factor of two for equiangular sampling schemes. Through numerical simulations, we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.

A Sparse Bayesian Approach for Joint Feature Selection and Classifier Learning

Peer-reviewed

Solving large sparse linear systems over the field GF(2)

Nowadays problem of solving sparse linear systems over the field GF(2) remain as a challenge. The popular approach is to improve existing methods such as the block Lanczos method (the Montgomery method) and the Wiedemann-Coppersmith method. Both these methods are considered in the thesis in details: there are their modifications and computational estimation for each process. It demonstrates the most complicated parts of these methods and gives the idea how to improve computations in software point of view. The research provides the implementation of accelerated binary matrix operations computer library which helps to make the progress steps in the Montgomery and in the Wiedemann-Coppersmith methods faster.

Learning Linear, Sparse, Factorial Codes

In previous work (Olshausen & Field 1996), an algorithm was described for learning linear sparse codes which, when trained on natural images, produces a set of basis functions that are spatially localized, oriented, and bandpass (i.e., wavelet-like). This note shows how the algorithm may be interpreted within a maximum-likelihood framework. Several useful insights emerge from this connection: it makes explicit the relation to statistical independence (i.e., factorial coding), it shows a formal relationship to the algorithm of Bell and Sejnowski (1995), and it suggests how to adapt parameters that were previously fixed.

Sparse Representations of Multiple Signals

We discuss the problem of finding sparse representations of a class of signals. We formalize the problem and prove it is NP-complete both in the case of a single signal and that of multiple ones. Next we develop a simple approximation method to the problem and we show experimental results using artificially generated signals. Furthermore,we use our approximation method to find sparse representations of classes of real signals, specifically of images of pedestrians. We discuss the relation between our formulation of the sparsity problem and the problem of finding representations of objects that are compact and appropriate for detection and classification.

Sparse Correlation Kernel Analysis and Reconstruction

This paper presents a new paradigm for signal reconstruction and superresolution, Correlation Kernel Analysis (CKA), that is based on the selection of a sparse set of bases from a large dictionary of class- specific basis functions. The basis functions that we use are the correlation functions of the class of signals we are analyzing. To choose the appropriate features from this large dictionary, we use Support Vector Machine (SVM) regression and compare this to traditional Principal Component Analysis (PCA) for the tasks of signal reconstruction, superresolution, and compression. The testbed we use in this paper is a set of images of pedestrians. This paper also presents results of experiments in which we use a dictionary of multiscale basis functions and then use Basis Pursuit De-Noising to obtain a sparse, multiscale approximation of a signal. The results are analyzed and we conclude that 1) when used with a sparse representation technique, the correlation function is an effective kernel for image reconstruction and superresolution, 2) for image compression, PCA and SVM have different tradeoffs, depending on the particular metric that is used to evaluate the results, 3) in sparse representation techniques, L_1 is not a good proxy for the true measure of sparsity, L_0, and 4) the L_epsilon norm may be a better error metric for image reconstruction and compression than the L_2 norm, though the exact psychophysical metric should take into account high order structure in images.

An Equivalence Between Sparse Approximation and Support Vector Machines

In the first part of this paper we show a similarity between the principle of Structural Risk Minimization Principle (SRM) (Vapnik, 1982) and the idea of Sparse Approximation, as defined in (Chen, Donoho and Saunders, 1995) and Olshausen and Field (1996). Then we focus on two specific (approximate) implementations of SRM and Sparse Approximation, which have been used to solve the problem of function approximation. For SRM we consider the Support Vector Machine technique proposed by V. Vapnik and his team at AT&T Bell Labs, and for Sparse Approximation we consider a modification of the Basis Pursuit De-Noising algorithm proposed by Chen, Donoho and Saunders (1995). We show that, under certain conditions, these two techniques are equivalent: they give the same solution and they require the solution of the same quadratic programming problem.

Big Decisions and Sparse Data: Adapting Scientific Publishing to the Needs of Practical Conservation

The biggest challenge in conservation biology is breaking down the gap between research and practical management. A major obstacle is the fact that many researchers are unwilling to tackle projects likely to produce sparse or messy data because the results would be difficult to publish in refereed journals. The obvious solution to sparse data is to build up results from multiple studies. Consequently, we suggest that there needs to be greater emphasis in conservation biology on publishing papers that can be built on by subsequent research rather than on papers that produce clear results individually. This building approach requires: (1) a stronger theoretical framework, in which researchers attempt to anticipate models that will be relevant in future studies and incorporate expected differences among studies into those models; (2) use of modern methods for model selection and multi-model inference, and publication of parameter estimates under a range of plausible models; (3) explicit incorporation of prior information into each case study; and (4) planning management treatments in an adaptive framework that considers treatments applied in other studies. We encourage journals to publish papers that promote this building approach rather than expecting papers to conform to traditional standards of rigor as stand-alone papers, and believe that this shift in publishing philosophy would better encourage researchers to tackle the most urgent conservation problems.