Connect the dots: how many random points can a regular curve pass through?

Abstract

Suppose n points are scattered uniformly at random in the unit square [0, 1] 2 . Question: How many of these points can possibly lie on some curve of length λ? Answer, proved here:We consider a general class of such questions; in each case, we are given a class Γ of curves in the square, and we ask: in a cloud of n uniform random points, how many can lie on some curve γ ∈ Γ? Classes of interest include (in addition to the rectifiable curves mentioned above): Lipschitz graphs, monotone graphs, twice-differentiable curves, graphs of smooth functions with m-bounded derivatives. In each case we get order-of-magnitude estimates; for example, there are twice-differentiable curves containing as many as O P (n 1/3 ) uniform random points, but not essentially more than this.We also consider generalizations to higher dimensions and to hypersurfaces of various codimensions. Thus, twice-differentiable k-dimensional hypersurfaces in R d may contain as many as O P (n k/(2d−k) ) uniform random points. We also consider other notions of 'passing through' such as passing through given space/direction pairs. Thus, twice-differentiable curves in R 2 may pass through at most O P (n 1/4 ) uniform random location/direction pairs.We give both concrete approaches to our results, based on geometric multiscale analysis, and abstract approaches, based on ε-entropy.Stylized applications in image processing and perceptual psychophysics are described.

Manifold estimation and singular deconvolution under Hausdorff loss

“…This is similar to estimating manifolds but, to the best of our knowledge, this literature does not establish minimax bounds for estimation in Hausdorff distance. Finally we would like to mention the related problem of testing for a set of points on a surface in a field of uniform noise [Arias-Castro et al (2005)], but, despite some similarity, this problem is quite different.…”

An Overview of Robust Subspace Recovery

“…This statistic measures the maximal number of occurrences in a sliding window of a fixed length. Arias-Castro et al [3] proposed using the scan statistic in a multi-scale, multi-orientation fashion for the more general problem of robust manifold recovery. In this problem, inliers are uniformly sampled from a sufficiently smooth surface in [0, 1] D , outliers are uniformly distributed in [0, 1] D and one needs to recover the underlying manifold.…”

“…Arias-Castro et al [3] proved that their mutli-scale, multiorientation scan statistics may recover inliers sampled uniformly from a d-dimensional graph in [0, 1] D of an mdifferentiable function, when the outliers are uniform in [0, 1] D and the SNR is Ω(N −m(D−d)/(d+m(D−d)) ). They also mention results for other kinds of surfaces.…”

$${l_p}$$ l p -Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers

“…There are non-convex methods for removing outliers (or detecting the hidden lowdimensional structures) that can handle arbitrarily large fraction of outliers. For example, Arias-Castro et al [2] proved that the scan statistics may detect points sampled uniformly from a d-dimensional graph in R D of an m-differentiable function among uniform outliers in a cube in R D with fraction of order 1−O(N −m(D−d)/(d+m(D−d)) ). Arias-Castro et al [1] used higher order spectral clustering affinities to remove outliers and thus detect differentiable surfaces (or certain unions of such surfaces) among uniform outliers, whose maximal fraction can be of a similar asymptotic order as that of the scan statistics.…”