Multi-resolution Low-rank Tensor Formats

We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors.Comment: 29 pages, 9 figure

On Algorithms for and Computing with the Tensor Ring Decomposition

Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage-cost representations in the tensor ring format, which is an extension of the tensor train format with variable end-ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage-cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph-based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.Comment: 24 pages, 3 figures, 6 tables, implementation of algorithms available at https://github.com/oscarmickelin/tensor-ring-decompositio

Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms

The number of noisy images required for molecular reconstruction in single-particle cryo-electron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly-oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain new theoretical and computational contributions for this challenging non-linear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the non-sparse case, where the third-order autocorrelation is required for uniformly-oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.Comment: 31 pages, 5 figures, 1 movi

Moment-based metrics for molecules computable from cryo-EM images

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.Comment: 21 Pages, 9 Figures, 2 Algorithms, and 3 Table

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Taha Toros Arşivi, Dosya No: 262-Tarık Buğr

Spektralolikheter inom Kvantmekanik och Konform Fältteori

Following Exner et al. (Commun. Math. Phys. 26 (2014), no. 2, 531–541), we prove new Lieb-Thirring inequalities for a general class of self-adjoint, second order differential operators with matrix-valued potentials, acting in one space-dimension. This class contains, but is not restricted to, the magnetic and non-magnetic Schrödinger operators. We consider the three cases of functions defined on all reals, all positive reals, and an interval, respectively, and acquire three different kinds of bounds. We also investigate the spectral properties of a family of operators from conformal field theory, by proving an asymptotic phase-space bound on the eigenvalue counting function and establishing a number of spectral inequalities. These bound the Riesz-means of eigenvalues for these operators, together with each individual eigenvalue, and are applied to a few physically interesting examples.Vi följer Exner et al. (Commun. Math. Phys. 26 (2014), nr. 2, 531–541) och bevisar nya Lieb-Thirring-olikheter för generella, andra gradens självadjungerade differentialoperatorer med matrisvärda potentialfunktioner, verkandes i en rumsdimension. Dessa innefattar och generaliserar de magnetiska och icke-magnetiska Schrödingeroperatorerna. Vi betraktar tre olika fall, med funktioner definierade på hela reella axeln, på den positiva reella axeln, samt på ett interval. Detta resulterar i tre sorters olikheter.  Vidare undersöker vi spektralegenskaperna för en klass operatorer från konform fältteori, genom att asymptotiskt begränsa antalet egenvärden med ett fasrymdsuttryck, samt genom att bevisa ett antal spektralolikheter. Dessa begränsar Riesz-medelvärdena för operatorerna, samt varje enskilt egenvärde, och tillämpas på ett par fysikaliskt intressanta exempel

Themes in numerical tensor calculus

This thesis studies several distinct, but related, aspects of numerical tensor calculus. First, we introduce a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, the format captures low-rank structures on each grid-scale, which leads to an increase in compression for a fixed accuracy. Secondly, we consider phase retrieval problems for signals that exhibit a low-rank tensor structure. This class of signals naturally includes a wide set of multidimensional spatial and temporal signals, as well as one- or two-dimensional signals that can be reshaped to higher-dimensional tensors. For a tensor of order , dimension and rank , we present a provably correct, polynomial-time algorithm that can recover the tensor-structured signals using a total of () measurements, far lower than the ( ) measurements required by dense methods. Thirdly, we consider the problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude. We focus on the particular case where each mode in the decomposition is corrupted by noise vectors with components that are correlated locally, i.e., with nearby components. This deterministic tensor completion problem has the unusual property that it can be solved in polynomial time if the rank of the tensor is sufficiently large. This is the polar opposite of the low-rank assumptions of typical low-rank tensor and matrix completion settings. Our approach enables recovery even with a substantial number of missing entries, for instance for -dimensional tensors of rank with up to 40% missing entries. Lastly, we study properties and algorithms for low storage-cost representations in two constrained tensor formats. We study algorithms for computing with the tensor ring format, which is an extension of the tensor train format with variable end-ranks, as well as properties of orthogonally decomposable symmetric tensors.Ph.D

Correlation between human natural killer cell migration and cytotoxicity.

Natural killer cells constitute part of the innate immune system, defending against cancer tumours and infections. Ongoing research has shown a diering e-ciency to kill target cells among individual cells in natural killer cell populations, and new tools allow for in-depth studies of large cell numbers over an extended period of time. In this thesis, the killing e-ciency of natural killer cells is correlated with their migration behaviour. Migratory properties are found to be of either of two essentially dierent forms, being active or inactive, and killing e-ciency is demonstrated to not be strongly related to migration behaviour. Further, natural killer cell populations are shown to exhibit additional heterogeneity as cells inducing fast death of target cells are shown to dier in migration compared to cells inducing slow death. Lastly, cells showing exhaustion in cytotoxicity during the assay are demonstrated to also experience migratory exhaustion