Estimating the reach of a manifold via its convexity defect function
Clément Berenfeld, John Harvey, Marc Hoffmann, Krishnan Shankar
备注：
35 pages, 4 figures. Various minor changes in v2 to correct minor errors and/or improve clarity. Thanks to excellent work by peer reviewers, in v3 an error in Lemma 4.9 was rectified, Section 4.2 was substantially revised and other minor changes made throughout and the manuscript was accepted for publication by Discrete & Computational Geometry. Extremely minor changes in v4
The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C^3 and C^4 cases, with a gap given by a logarithmic factor.
Various problems in manifold estimation make use of a quantity called the reach, denoted by $τ\_M$, 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 study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hatτ$ of $τ\_{M}$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}\_{n}$, …
Given a random sample of points from some unknown density, we propose a data-driven method for estimating density level sets under the r-convexity assumption. This shape condition generalizes the convexity property. However, the main problem in practice is that r is an unknown geometric characteristic of the set related to its curvature. A stochastic algorithm is proposed for selecting its optimal value from the data. The resulting reconstruction of the level set is able to achieve minimax rates for Hausdorff metric and distance in measure, up to log factors, uniformly on the level of the set.
In many instances, imposing a constraint on the shape of a density is a reasonable and flexible assumption. It offers an alternative to parametric models which can be too rigid and to other nonparametric methods requiring the choice of tuning parameters. This paper treats the nonparametric estimation of log-concave or log-convex density ratios by means of active set algorithms in a unified framework. In the setting of log-concave densities, the new algorithm is similar to but substantially faster than previously considered active set methods. Log-convexity is a less common shape constraint whi…
State University of New York at Buffalo（纽约州立大学布法罗分校）
Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the \textit{Stochastic Linear Combination of Non-linear Regressions} model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assump…
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