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Manifold hypotheses are typically used for tasks such as dimensionality reduction, interpolation, or improving classification performance. In the less common problem of manifold estimation, the task is to characterize the geometric structure of the manifold in the original ambient space from a sample. We focus on the role that tangent bundle learners (TBL) can play in estimating the underlying manifold from which data is assumed to be sampled. Since the unbounded tangent spaces natively represent a poor manifold estimate, the problem reduces to one of estimating regions in the tangent space where it acts as a relatively faithful linear approximator to the surface of the manifold. Local PCA methods, such as the Mixtures of Probabilistic Principal Component Analyzers method of Tipping and Bishop produce a subset of the tangent bundle of the manifold along with an assignment function that assigns points in the training data used by the TBL to elements of the estimated tangent bundle. We formulate three methods that use the data assigned to each tangent space to estimate the underlying bounded subspaces for which the tangent space is a faithful estimate of the manifold and offer thoughts on how this perspective is theoretically grounded in the manifold assumption. We seek to explore the conceptual and technical challenges that arise in trying to utilize simple TBL methods to arrive at reliable estimates of the underlying manifold.
Pure combinatorial models for BPL_n and Gauss map of a combinatorial manifold are described.
A common problem in Bayesian inference is the sampling of target probability distributions at sufficient resolution and accuracy to estimate the probability density, and to compute credible regions. Often by construction, many target distributions ca
Continuous state spaces and stochastic, switching dynamics characterize a number of rich, realworld domains, such as robot navigation across varying terrain. We describe a reinforcementlearning algorithm for learning in these domains and prove for ce
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kahlerian Ricci-flat manifolds in four real dimensions.
Various problems in manifold estimation make use of a quantity called the reach, denoted by $tau_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we