824 research outputs found
Quadratically-Regularized Optimal Transport on Graphs
Optimal transportation provides a means of lifting distances between points
on a geometric domain to distances between signals over the domain, expressed
as probability distributions. On a graph, transportation problems can be used
to express challenging tasks involving matching supply to demand with minimal
shipment expense; in discrete language, these become minimum-cost network flow
problems. Regularization typically is needed to ensure uniqueness for the
linear ground distance case and to improve optimization convergence;
state-of-the-art techniques employ entropic regularization on the
transportation matrix. In this paper, we explore a quadratic alternative to
entropic regularization for transport over a graph. We theoretically analyze
the behavior of quadratically-regularized graph transport, characterizing how
regularization affects the structure of flows in the regime of small but
nonzero regularization. We further exploit elegant second-order structure in
the dual of this problem to derive an easily-implemented Newton-type
optimization algorithm.Comment: 27 page
Conditional Spectral Analysis of Replicated Multiple Time Series with Application to Nocturnal Physiology
This article considers the problem of analyzing associations between power
spectra of multiple time series and cross-sectional outcomes when data are
observed from multiple subjects. The motivating application comes from sleep
medicine, where researchers are able to non-invasively record physiological
time series signals during sleep. The frequency patterns of these signals,
which can be quantified through the power spectrum, contain interpretable
information about biological processes. An important problem in sleep research
is drawing connections between power spectra of time series signals and
clinical characteristics; these connections are key to understanding biological
pathways through which sleep affects, and can be treated to improve, health.
Such analyses are challenging as they must overcome the complicated structure
of a power spectrum from multiple time series as a complex positive-definite
matrix-valued function. This article proposes a new approach to such analyses
based on a tensor-product spline model of Cholesky components of
outcome-dependent power spectra. The approach flexibly models power spectra as
nonparametric functions of frequency and outcome while preserving geometric
constraints. Formulated in a fully Bayesian framework, a Whittle likelihood
based Markov chain Monte Carlo (MCMC) algorithm is developed for automated
model fitting and for conducting inference on associations between outcomes and
spectral measures. The method is used to analyze data from a study of sleep in
older adults and uncovers new insights into how stress and arousal are
connected to the amount of time one spends in bed
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
A Hierarchical Spatio-Temporal Statistical Model Motivated by Glaciology
In this paper, we extend and analyze a Bayesian hierarchical spatio-temporal
model for physical systems. A novelty is to model the discrepancy between the
output of a computer simulator for a physical process and the actual process
values with a multivariate random walk. For computational efficiency, linear
algebra for bandwidth limited matrices is utilized, and first-order emulator
inference allows for the fast emulation of a numerical partial differential
equation (PDE) solver. A test scenario from a physical system motivated by
glaciology is used to examine the speed and accuracy of the computational
methods used, in addition to the viability of modeling assumptions. We conclude
by discussing how the model and associated methodology can be applied in other
physical contexts besides glaciology.Comment: Revision accepted for publication by the Journal of Agricultural,
Biological, and Environmental Statistic
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