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We study the problem of modeling data by several affine subspaces, which generalizes the common modeling by a single subspace. It arises, for example, in object tracking and structure from motion. One of the simplest methods for such modeling is based on energy minimization, where the energy involves p-th powers of distances of data points from subspaces. We first analyze under certain assumptions (e.g., spherically symmetric outliers) the effectiveness of such energy minimization for recovering all subspaces simultaneously and also recovering the most significant subspace. We reveal the following phase transition in our setting: when p<=1 the underlying subspaces can be recovered by such energy minimization; whereas when p>1 the underlying subspaces are sufficiently far from the global minimizer. Nevertheless, for more general settings (i.e., outliers which are not spherically symmetric) we can point at some disadvantages of the energy minimization strategy. In order to practically solve the problem, we present a simple and fast geometric method for multiple subspaces modeling. It forms a collection of local best fit affine subspaces, where the size of the local neighborhoods is determined automatically by the Peter Jones’ beta numbers. This collection of subspaces can then be further processed in various ways. For example, greedy selection procedure according to an appropriate energy or a spectral method to generate the final model. We demonstrate the state of the art accuracy and speed of the suggested procedure on applications for several applications.

Over the last two decades, the world has witnessed an enormous growth in data sets that are complex, high-dimensional, and massive. Traditional techniques for analyzing data have become inadequate. In this talk we will discuss how some classical mathematics created for completely different purposes (e.g. algebraic and differential topology, combinatorial differential geometry), could nonetheless provide powerful new tools for exploring these modern data sets. In particular we will focus on some novel applications of Hodge Theory, a bridge over topology and geometry, in statistical data analysis which have never been exploited in such a perspective.

We consider moderately large matrices (millions or billions of entries) of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion finds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items. The aim is to predict the unobserved ratings. This problem is commonly stated in a constrained optimization framework. We follow an approach that exploits the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on the Grassmann manifold. We then apply a superlinear Riemannian trust-region method to solve it. We improve on key aspects of existing methods such as Admira, FPCA, OptSpace, SET and others. I will demonstrate the performance of our algorithm in terms of accuracy and speed on synthetic data, and discuss real data in the conclusions.

In this talk, we discuss some large time behaviors of a quantum Markov semigroup of bounded linear transformations acting on a von Neumann algebra on a complex Hilbert space. Large deviation principle, Lyapunov stability as well as necessary and sufficient conditions for existence of an invariant state are established using integral representation of normal state. The results are directly applicable to open quantum systems.

In this talk, we create links between the properties of diffusion of the Riemannian manifold and its geometry. We embedd a family of Riemannian manifolds whose metric is time dependent, into a Hilbert space with its diffusion properties. Namely, via the eigenfunctions of the corresponding laplacian or its heat kernel. We prove that we can construct embeddings via a finite number of eigenfunctions for all families of Riemannian manifolds (M,g(t)) such that g(t) is analytic in t. Then, we construct the fundamental solution P for the non-linear heat equation acting on (M,g(t)), such that the volume (M,g(t)) is constant, and we prove that, under certain constraints, we can embed (M,g(t)) into a Hilbert space via P.

Efficient data processing algorithms may be realized by exploiting the low-dimensional manifold structure often inherent in a data set. We seek efficient parameterizations of such data sets via projection into appropriate manifold tangent planes. Parameterizing a data set thus becomes a problem of estimating local tangent planes from noisy manifold samples. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise. We present our approach that uses the geometry of the data to guide our definition of locality, discovering the optimal balance of this noise-curvature trade-off. Using perturbation theory of eigenspaces, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals the optimal scale for local tangent plane recovery. Applications are discussed, with a focus on image processing.