5,106 research outputs found
Bayesian Inference of Log Determinants
The log-determinant of a kernel matrix appears in a variety of machine
learning problems, ranging from determinantal point processes and generalized
Markov random fields, through to the training of Gaussian processes. Exact
calculation of this term is often intractable when the size of the kernel
matrix exceeds a few thousand. In the spirit of probabilistic numerics, we
reinterpret the problem of computing the log-determinant as a Bayesian
inference problem. In particular, we combine prior knowledge in the form of
bounds from matrix theory and evidence derived from stochastic trace estimation
to obtain probabilistic estimates for the log-determinant and its associated
uncertainty within a given computational budget. Beyond its novelty and
theoretic appeal, the performance of our proposal is competitive with
state-of-the-art approaches to approximating the log-determinant, while also
quantifying the uncertainty due to budget-constrained evidence.Comment: 12 pages, 3 figure
Deterministic Annealing and Nonlinear Assignment
For combinatorial optimization problems that can be formulated as Ising or
Potts spin systems, the Mean Field (MF) approximation yields a versatile and
simple ANN heuristic, Deterministic Annealing. For assignment problems the
situation is more complex -- the natural analog of the MF approximation lacks
the simplicity present in the Potts and Ising cases. In this article the
difficulties associated with this issue are investigated, and the options for
solving them discussed. Improvements to existing Potts-based MF-inspired
heuristics are suggested, and the possibilities for defining a proper
variational approach are scrutinized.Comment: 15 pages, 3 figure
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
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