117 research outputs found
Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both boundary
and initial value problems that returns a joint Gaussian process posterior over
the solution. Such methods have concrete value in the statistics on Riemannian
manifolds, where non-analytic ordinary differential equations are involved in
virtually all computations. The probabilistic formulation permits marginalising
the uncertainty of the numerical solution such that statistics are less
sensitive to inaccuracies. This leads to new Riemannian algorithms for mean
value computations and principal geodesic analysis. Marginalisation also means
results can be less precise than point estimates, enabling a noticeable
speed-up over the state of the art. Our approach is an argument for a wider
point that uncertainty caused by numerical calculations should be tracked
throughout the pipeline of machine learning algorithms.Comment: 11 page (9 page conference paper, plus supplements
Isometric Gaussian Process Latent Variable Model for Dissimilarity Data
We present a probabilistic model where the latent variable respects both the
distances and the topology of the modeled data. The model leverages the
Riemannian geometry of the generated manifold to endow the latent space with a
well-defined stochastic distance measure, which is modeled locally as Nakagami
distributions. These stochastic distances are sought to be as similar as
possible to observed distances along a neighborhood graph through a censoring
process. The model is inferred by variational inference based on observations
of pairwise distances. We demonstrate how the new model can encode invariances
in the learned manifolds.Comment: ICML 202
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