Multiresolution kernel matrix algebra

We propose a sparse arithmetic for kernel matrices, enabling efficient scattered data analysis. The compression of kernel matrices by means of samplets yields sparse matrices such that assembly, addition, and multiplication of these matrices can be performed with essentially linear cost. Since the inverse of a kernel matrix is compressible, too, we have also fast access to the inverse kernel matrix by employing exact sparse selected inversion techniques. As a consequence, we can rapidly evaluate series expansions and contour integrals to access, numerically and approximately in a data-sparse format, more complicated matrix functions such as $A^\alpha$ and $\exp(A)$. By exploiting the matrix arithmetic, also efficient Gaussian process learning algorithms for spatial statistics can be realized. Numerical results are presented to quantify and quality our findings

Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections

Block projections have been used, in [Eberly et al. 2006], to obtain an efficient algorithm to find solutions for sparse systems of linear equations. A bound of softO(n^(2.5)) machine operations is obtained assuming that the input matrix can be multiplied by a vector with constant-sized entries in softO(n) machine operations. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections, and this has been conjectured. In this paper we establish the correctness of the algorithm from [Eberly et al. 2006] by proving the existence of efficient block projections over sufficiently large fields. We demonstrate the usefulness of these projections by deriving improved bounds for the cost of several matrix problems, considering, in particular, ``sparse'' matrices that can be be multiplied by a vector using softO(n) field operations. We show how to compute the inverse of a sparse matrix over a field F using an expected number of softO(n^(2.27)) operations in F. A basis for the null space of a sparse matrix, and a certification of its rank, are obtained at the same cost. An application to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix yields algorithms requiring softO(n^(2.66)) machine operations. The derived algorithms are all probabilistic of the Las Vegas type

Fast and Accurate Camera Covariance Computation for Large 3D Reconstruction

Estimating uncertainty of camera parameters computed in Structure from Motion (SfM) is an important tool for evaluating the quality of the reconstruction and guiding the reconstruction process. Yet, the quality of the estimated parameters of large reconstructions has been rarely evaluated due to the computational challenges. We present a new algorithm which employs the sparsity of the uncertainty propagation and speeds the computation up about ten times \wrt previous approaches. Our computation is accurate and does not use any approximations. We can compute uncertainties of thousands of cameras in tens of seconds on a standard PC. We also demonstrate that our approach can be effectively used for reconstructions of any size by applying it to smaller sub-reconstructions.Comment: ECCV 201

Feasibility and performances of compressed-sensing and sparse map-making with Herschel/PACS data

The Herschel Space Observatory of ESA was launched in May 2009 and is in operation since. From its distant orbit around L2 it needs to transmit a huge quantity of information through a very limited bandwidth. This is especially true for the PACS imaging camera which needs to compress its data far more than what can be achieved with lossless compression. This is currently solved by including lossy averaging and rounding steps on board. Recently, a new theory called compressed-sensing emerged from the statistics community. This theory makes use of the sparsity of natural (or astrophysical) images to optimize the acquisition scheme of the data needed to estimate those images. Thus, it can lead to high compression factors. A previous article by Bobin et al. (2008) showed how the new theory could be applied to simulated Herschel/PACS data to solve the compression requirement of the instrument. In this article, we show that compressed-sensing theory can indeed be successfully applied to actual Herschel/PACS data and give significant improvements over the standard pipeline. In order to fully use the redundancy present in the data, we perform full sky map estimation and decompression at the same time, which cannot be done in most other compression methods. We also demonstrate that the various artifacts affecting the data (pink noise, glitches, whose behavior is a priori not well compatible with compressed-sensing) can be handled as well in this new framework. Finally, we make a comparison between the methods from the compressed-sensing scheme and data acquired with the standard compression scheme. We discuss improvements that can be made on ground for the creation of sky maps from the data.Comment: 11 pages, 6 figures, 5 tables, peer-reviewed articl

Selected inversion as key to a stable Langevin evolution across the QCD phase boundary

We present new results of full QCD at nonzero chemical potential. In PRD 92, 094516 (2015) the complex Langevin method was shown to break down when the inverse coupling decreases and enters the transition region from the deconfined to the confined phase. We found that the stochastic technique used to estimate the drift term can be very unstable for indefinite matrices. This may be avoided by using the full inverse of the Dirac operator, which is, however, too costly for four-dimensional lattices. The major breakthrough in this work was achieved by realizing that the inverse elements necessary for the drift term can be computed efficiently using the selected inversion technique provided by the parallel sparse direct solver package PARDISO. In our new study we show that no breakdown of the complex Langevin method is encountered and that simulations can be performed across the phase boundary.Comment: 8 pages, 6 figures, Proceedings of the 35th International Symposium on Lattice Field Theory, Granada, Spai

Spin states of asteroids in the Eos collisional family

Eos family was created during a catastrophic impact about 1.3 Gyr ago. Rotation states of individual family members contain information about the history of the whole population. We aim to increase the number of asteroid shape models and rotation states within the Eos collision family, as well as to revise previously published shape models from the literature. Such results can be used to constrain theoretical collisional and evolution models of the family, or to estimate other physical parameters by a thermophysical modeling of the thermal infrared data. We use all available disk-integrated optical data (i.e., classical dense-in-time photometry obtained from public databases and through a large collaboration network as well as sparse-in-time individual measurements from a few sky surveys) as input for the convex inversion method, and derive 3D shape models of asteroids together with their rotation periods and orientations of rotation axes. We present updated shape models for 15 asteroids and new shape model determinations for 16 asteroids. Together with the already published models from the publicly available DAMIT database, we compiled a sample of 56 Eos family members with known shape models that we used in our analysis of physical properties within the family. Rotation states of asteroids smaller than ~20 km are heavily influenced by the YORP effect, whilst the large objects more or less retained their rotation state properties since the family creation. Moreover, we also present a shape model and bulk density of asteroid (423) Diotima, an interloper in the Eos family, based on the disk-resolved data obtained by the Near InfraRed Camera (Nirc2) mounted on the W.M. Keck II telescope.Comment: Accepted for publication in ICARUS Special Issue - Asteroids: Origin, Evolution & Characterizatio