149 research outputs found
Bayesian Methods in Tensor Analysis
Tensors, also known as multidimensional arrays, are useful data structures in
machine learning and statistics. In recent years, Bayesian methods have emerged
as a popular direction for analyzing tensor-valued data since they provide a
convenient way to introduce sparsity into the model and conduct uncertainty
quantification. In this article, we provide an overview of frequentist and
Bayesian methods for solving tensor completion and regression problems, with a
focus on Bayesian methods. We review common Bayesian tensor approaches
including model formulation, prior assignment, posterior computation, and
theoretical properties. We also discuss potential future directions in this
field.Comment: 32 pages, 8 figures, 2 table
Machine-learning of atomic-scale properties based on physical principles
We briefly summarize the kernel regression approach, as used recently in
materials modelling, to fitting functions, particularly potential energy
surfaces, and highlight how the linear algebra framework can be used to both
predict and train from linear functionals of the potential energy, such as the
total energy and atomic forces. We then give a detailed account of the Smooth
Overlap of Atomic Positions (SOAP) representation and kernel, showing how it
arises from an abstract representation of smooth atomic densities, and how it
is related to several popular density-based representations of atomic
structure. We also discuss recent generalisations that allow fine control of
correlations between different atomic species, prediction and fitting of
tensorial properties, and also how to construct structural kernels---applicable
to comparing entire molecules or periodic systems---that go beyond an additive
combination of local environments
Bayesian Nonlinear Tensor Regression with Functional Fused Elastic Net Prior
Tensor regression methods have been widely used to predict a scalar response
from covariates in the form of a multiway array. In many applications, the
regions of tensor covariates used for prediction are often spatially connected
with unknown shapes and discontinuous jumps on the boundaries. Moreover, the
relationship between the response and the tensor covariates can be nonlinear.
In this article, we develop a nonlinear Bayesian tensor additive regression
model to accommodate such spatial structure. A functional fused elastic net
prior is proposed over the additive component functions to comprehensively
model the nonlinearity and spatial smoothness, detect the discontinuous jumps,
and simultaneously identify the active regions. The great flexibility and
interpretability of the proposed method against the alternatives are
demonstrated by a simulation study and an analysis on facial feature data
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