1,446 research outputs found
Dual-Space Analysis of the Sparse Linear Model
Sparse linear (or generalized linear) models combine a standard likelihood
function with a sparse prior on the unknown coefficients. These priors can
conveniently be expressed as a maximization over zero-mean Gaussians with
different variance hyperparameters. Standard MAP estimation (Type I) involves
maximizing over both the hyperparameters and coefficients, while an empirical
Bayesian alternative (Type II) first marginalizes the coefficients and then
maximizes over the hyperparameters, leading to a tractable posterior
approximation. The underlying cost functions can be related via a dual-space
framework from Wipf et al. (2011), which allows both the Type I or Type II
objectives to be expressed in either coefficient or hyperparmeter space. This
perspective is useful because some analyses or extensions are more conducive to
development in one space or the other. Herein we consider the estimation of a
trade-off parameter balancing sparsity and data fit. As this parameter is
effectively a variance, natural estimators exist by assessing the problem in
hyperparameter (variance) space, transitioning natural ideas from Type II to
solve what is much less intuitive for Type I. In contrast, for analyses of
update rules and sparsity properties of local and global solutions, as well as
extensions to more general likelihood models, we can leverage coefficient-space
techniques developed for Type I and apply them to Type II. For example, this
allows us to prove that Type II-inspired techniques can be successful
recovering sparse coefficients when unfavorable restricted isometry properties
(RIP) lead to failure of popular L1 reconstructions. It also facilitates the
analysis of Type II when non-Gaussian likelihood models lead to intractable
integrations.Comment: 9 pages, 2 figures, submission to NIPS 201
A Control Approach to Robust Utility Maximization with Logarithmic Utility and Time-Consistent Penalties
We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions is important for the use of numerical algorithms, whose applicability is demonstrated in examples.Optimal investment, model uncertainty, incomplete markets, stochastic volatility, coherent risk measure, convex risk measure, optimal control, convex duality
Estimating Infection Sources in Networks Using Partial Timestamps
We study the problem of identifying infection sources in a network based on
the network topology, and a subset of infection timestamps. In the case of a
single infection source in a tree network, we derive the maximum likelihood
estimator of the source and the unknown diffusion parameters. We then introduce
a new heuristic involving an optimization over a parametrized family of Gromov
matrices to develop a single source estimation algorithm for general graphs.
Compared with the breadth-first search tree heuristic commonly adopted in the
literature, simulations demonstrate that our approach achieves better
estimation accuracy than several other benchmark algorithms, even though these
require more information like the diffusion parameters. We next develop a
multiple sources estimation algorithm for general graphs, which first
partitions the graph into source candidate clusters, and then applies our
single source estimation algorithm to each cluster. We show that if the graph
is a tree, then each source candidate cluster contains at least one source.
Simulations using synthetic and real networks, and experiments using real-world
data suggest that our proposed algorithms are able to estimate the true
infection source(s) to within a small number of hops with a small portion of
the infection timestamps being observed.Comment: 15 pages, 15 figures, accepted by IEEE Transactions on Information
Forensics and Securit
Virtual cardiac monolayers for electrical wave propagation
The complex structure of cardiac tissue is considered to be one of the main determinants of an arrhythmogenic substrate. This study is aimed at developing the first mathematical model to describe the formation of cardiac tissue, using a joint in silico-in vitro approach. First, we performed experiments under various conditions to carefully characterise the morphology of cardiac tissue in a culture of neonatal rat ventricular cells. We considered two cell types, namely, cardiomyocytes and fibroblasts. Next, we proposed a mathematical model, based on the Glazier-Graner-Hogeweg model, which is widely used in tissue growth studies. The resultant tissue morphology was coupled to the detailed electrophysiological Korhonen-Majumder model for neonatal rat ventricular cardiomyocytes, in order to study wave propagation. The simulated waves had the same anisotropy ratio and wavefront complexity as those in the experiment. Thus, we conclude that our approach allows us to reproduce the morphological and physiological properties of cardiac tissue
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