The Wasserstein distance between two probability measures on a metric space
is a measure of closeness with applications in statistics, probability, and
machine learning. In this work, we consider the fundamental question of how
quickly the empirical measure obtained from $n$ independent samples from $\mu$
approaches $\mu$ in the Wasserstein distance of any order. We prove sharp
asymptotic and finite-sample results for this rate of convergence for general
measures on general compact metric spaces. Our finite-sample results show the
existence of multi-scale behavior, where measures can exhibit radically
different rates of convergence as $n$ grows.

Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance

We present a multiscale method for the determination of collective reaction coordinates for macromolecular dynamics based on two recently developed mathematical techniques: diffusion map and the determination of local intrinsic dimensionality of large datasets. Our method accounts for the local variation of molecular configuration space, and the resulting global coordinates are correlated with the time scales of the molecular motion. To illustrate the approach, we present results for two model systems: all-atom alanine dipeptide and coarse-grained src homology 3 protein domain. We provide clear physical interpretation for the emerging coordinates and use them to calculate transition rates. The technique is general enough to be applied to any system for which a Boltzmann-sampled set of molecular configurations is available.

Determination of reaction coordinates via locally scaled diffusion map

The long-timescale dynamics of macromolecular systems can be oftentimes viewed as a reaction connecting metastable states of the system. In the past decade, various approaches have been developed to discover the collective motions associated with this dynamics. The corresponding collective variables are used in many applications, e.g., to understand the reaction mechanism, to quantify the system's free energy landscape, to enhance the sampling of the reaction path, and to determine the reaction rate. In this review we focus on a number of key developments in this field, providing an overview of several methods along with their relative regimes of applicability.

Discovering Mountain Passes via Torchlight: Methods for the Definition of Reaction Coordinates and Pathways in Complex Macromolecular Reactions

We introduce vector diffusion maps (VDM), a new mathematical framework for organizing and analyzing massive high-dimensional data sets, images, and shapes. VDMis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods, such as LLE, ISOMAP, and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low-dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the vector diffusion distance. In the manifold learning setup, where the data set is distributed on a low-dimensional manifold embedded in , we prove the relation between VDM and the connection Laplacian operator for vector fields over the manifold. © 2012 Wiley Periodicals, Inc.

/pdf/vector-diffusion-maps-and-the-connection-laplacian-2dm25h2vhi.pdf

Vector diffusion maps and the connection Laplacian

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jones' β 2 numbers (we prove under certain geometric conditions that our method finds the optimal local neighborhoods). The collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model. We discuss applications to tracking-based motion segmentation and clustering of faces under different illuminating conditions. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the MNIST handwritten digits data; and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces.

/pdf/hybrid-linear-modeling-via-local-best-fit-flats-4x7xmoqmi9.pdf

Hybrid Linear Modeling via Local Best-Fit Flats

http://cbcl.mit.edu/publications/ps/MultiscaleSVD_TR_rev.pdf

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

The problem of estimating the intrinsic dimensionality of certain point clouds is of interest in many applications in statistics and analysis of high-dimensional data sets. Our setting is the following: the points are sampled from a manifold M of dimension k, embedded in ℝD, with k ≪ D, and corrupted by D-dimensional noise. When M is a linear manifold (hyperplane), one may analyse this situation by SVD, hoping the noise would perturb the rank k covariance matrix. When M is a nonlinear manifold, SVD performed globally may dramatically overestimate the intrinsic dimensionality. We discuss a multiscale version SVD that is useful in estimating the intrinsic dimensionality of nonlinear manifolds.

Estimation of intrinsic dimensionality of samples from noisy low-dimensional manifolds in high dimensions with multiscale SVD

We present a novel approach for estimating the intrinsic dimensionality of certain point clouds: we assume that the points are sampled from a manifold M of dimension k , with k << D, and corrupted by D -dimensional noise. When M is linear, one may analyze this situation by SVD: with no noise one would obtain a rank k matrix, and noise may be treated as a perturbation of the covariance matrix. When M is a nonlinear manifold, global SVD may dramatically overestimate the intrinsic dimensionality. We introduce a multiscale version SVD and discuss how one can extract estimators for the intrinsic dimensionality that are highly robust to noise, while require a smaller sample size than current estimators.

/pdf/multiscale-estimation-of-intrinsic-dimensionality-of-data-4orjsoy866.pdf

Multiscale Estimation of Intrinsic Dimensionality of Data Sets

We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points.

/pdf/path-based-spectral-clustering-guarantees-robustness-to-6f1pna1la0.pdf

Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms

We discuss recent work based on multiscale geometric analyis for the study of large data sets that lie in high-dimensional spaces but have low-dimensional structure. We present three applications: the first one to the estimation of intrinsic dimension of sampled manifolds, the second one to the construction of multiscale dictionaries, called Geometric Wavelets, for the analysis of point clouds, and the third one to the inference of point clouds modeled as unions of multiple planes of varying dimensions.

/pdf/some-recent-advances-in-multiscale-geometric-analysis-of-5c8vltdteb.pdf

Anna Little

Papers

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

Estimation of intrinsic dimensionality of samples from noisy low-dimensional manifolds in high dimensions with multiscale SVD

Multiscale Estimation of Intrinsic Dimensionality of Data Sets

Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms

Some Recent Advances in Multiscale Geometric Analysis of Point Clouds