1,271 research outputs found
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
Distributed Estimation and Inference for the Analysis of Big Biomedical Data
This thesis focuses on developing and implementing new statistical methods to address some of the current difficulties encountered in the analysis of high-dimensional correlated biomedical data. Following the divide-and-conquer paradigm, I develop a theoretically sound and computationally tractable class of distributed statistical methods that are made accessible to practitioners through R statistical software.
This thesis aims to establish a class of distributed statistical methods for regression analyses with very large outcome variables arising in many biomedical fields, such as in metabolomic or imaging research. The general distributed procedure divides data into blocks that are analyzed on a parallelized computational platform and combines these separate results via Hansen’s (1982) generalized method of moments. These new methods provide distributed and efficient statistical inference in many different regression settings. Computational efficiency is achieved by leveraging recent developments in large scale computing, such as the MapReduce paradigm on the Hadoop platform.
In the first project presented in Chapter III, I develop a divide-and-conquer procedure implemented in a parallelized computational scheme for statistical estimation and inference of regression parameters with high-dimensional correlated responses. This project is motivated by an electroencephalography study whose goal is to determine the effect of iron deficiency on infant auditory recognition memory. The proposed method (published as Hector and Song (2020a)), the Distributed and Integrated Method of Moments (DIMM), divides responses into subvectors to be analyzed in parallel using pairwise composite likelihood, and combines results using an optimal one-step meta-estimator.
In the second project presented in Chapter IV, I develop an extended theoretical framework of distributed estimation and inference to incorporate a broad range of classical statistical models and biomedical data types. To reduce computational speed and meet data privacy demands, I propose to divide data by outcomes and subjects, leading to a doubly divide-and-conquer paradigm. I also address parameter heterogeneity explicitly for added flexibility. I establish a new theoretical framework for the analysis of a broad class of big data problems to facilitate valid statistical inference for biomedical researchers. Possible applications include genomic data, metabolomic data, longitudinal and spatial data, and many more.
In the third project presented in Chapter V, I propose a distributed quadratic inference function framework to jointly estimate regression parameters from multiple potentially heterogeneous data sources with correlated vector outcomes. This project is motivated by the analysis of the association between smoking and metabolites in a large cohort study. The primary goal of this joint integrative analysis is to estimate covariate effects on all outcomes through a marginal regression model in a statistically and computationally efficient way. To overcome computational and modeling challenges arising from the high-dimensional likelihood of the correlated vector outcomes, I propose to analyze each data source using Qu et al.’s quadratic inference funtions, and then to jointly reestimate parameters from each data source by accounting for correlation between data sources.PHDBiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163220/1/ehector_1.pd
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
Stable Camera Motion Estimation Using Convex Programming
We study the inverse problem of estimating n locations (up to
global scale, translation and negation) in from noisy measurements of a
subset of the (unsigned) pairwise lines that connect them, that is, from noisy
measurements of for some pairs (i,j) (where the
signs are unknown). This problem is at the core of the structure from motion
(SfM) problem in computer vision, where the 's represent camera locations
in . The noiseless version of the problem, with exact line measurements,
has been considered previously under the general title of parallel rigidity
theory, mainly in order to characterize the conditions for unique realization
of locations. For noisy pairwise line measurements, current methods tend to
produce spurious solutions that are clustered around a few locations. This
sensitivity of the location estimates is a well-known problem in SfM,
especially for large, irregular collections of images.
In this paper we introduce a semidefinite programming (SDP) formulation,
specially tailored to overcome the clustering phenomenon. We further identify
the implications of parallel rigidity theory for the location estimation
problem to be well-posed, and prove exact (in the noiseless case) and stable
location recovery results. We also formulate an alternating direction method to
solve the resulting semidefinite program, and provide a distributed version of
our formulation for large numbers of locations. Specifically for the camera
location estimation problem, we formulate a pairwise line estimation method
based on robust camera orientation and subspace estimation. Lastly, we
demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts
updated, some typos correcte
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