Let K be a simplicial complex with a height function. Persistent homology provides a measure of the size of some, but not all, of the topological features of the embedded complex. In the special case that K triangulates a manifold, the theory of extended persistence exploits Poincare Duality to measure all of the topological features. When K does not triangulate a manifold, however, Poincare Duality fails to be true and extended persistence fails to be as useful.

Intersection Homology was developed in the early 80's to restore Poincare Duality to stratified spaces, which are a natural generalization of manifolds. In this talk, we define persistence (and extended persistence) for intersection homology on a stratified simplicial complex with a height function and give an algorithm for its computation. We then demonstrate that this theory satisfies the same properties as extended persistence for ordinary homology on a manifold. Finally, we describe some of the topological features measured by persistent intersection homology.

Given two tame functions, $f$ and $g$, on a triangulable topological space $X$, it has been shown that the bottleneck distance between their persistence diagrams is bounded by the sup-norm of their difference, that is, $dist_{B}(D(f), D(g)) \leq sup_{x\in X}|f(x)-g(x)|$. Thus, for each individual persistence pair of $f$ their is a similar persistence pair of $g$. If one sums up persistences of all the persistence pairs of $f$, such a sum, in general, is not close to the corresponding sum for $g$. In this talk I shall consider a question when this stronger measure of similarity between persistence diagrams holds. In particular, I will show that if $f$ and $g$ are Lipschitz functions from the $d$-sphere, $d=1,2$, then the difference between sums of the $(d+1)$-st powers of persistences is bounded by the sup-norm times a constant. Time permitting, I will also present an application of this result to measuring periodicity of gene expressions.

Given a point set $P$ of points in the plane, a subset $Q\subseteq P$ is an $\eps$-kernel of $P$ if for every slab $W$ containing $Q$, the $(1+\eps)$-expansion of $W$ also contains $P$. We present the first space-optimal data-stream algorithm for maintaining an $\eps$-kernel of a stream of points in the plane. The algorithm uses $O(1/\sqrt{\eps})$ space and takes $O(\log (1/\eps))$ time to process each point. It immediately implies other space-optimal data-stream algorithms such as maintaining the width and robust kernels. Joint work with Pankaj Agarwal.

We describe a variant of Edelsbrunner's WRAP algorithm for surface reconstruction, for which we can prove geometric and topological guarantees within the epsilon-sampling model. The WRAP algorithm is based on ideas from Morse theory applied to the flow map induced by a certain distance function. The variant is made possible by a previous result on the "separation" of critical points for a related distance function that directly applies in this case. Though the variant is easily proposed, in order to prove the quality guarantees for the output, we need to closely investigate the geometric properties of the flow map. This is from a joint work with Edgar Ramos.

While inference from labeled data is one of the traditional problems of machine learning and statistics, it is only recently that we have developed an understanding of how unlabeled data may be helpful in various inferential problems. It turns out that many aspects of the connection between labeled and unlabeled data can be interpreted geometrically.

In this talk I will discuss certain geometric invariants, centered around the notions of the Laplace operator and the heat equation, and their role in machine learning. I will also discuss theoretical results on reconstructing these objects from sampled data, as well as some recent applications of these ideas to computing volumes of convex bodies in polynomial time.

Let P be a convex polygope inside the unit ball in R^3. Its total mean curvature, H(P), is half the sum of edge-lengths times dual dihedral angles, over all edges of P. Its total Gaussian curvature, K(P), is the sum of angle defects, over all vertices of P. I will prove H(P) \leq K(P) = 4 \pi for the case in which the orthogonal projetion of the origin onto the line of every edge of P lands on the edge.

In the first part of the talk I will present some recent work on developing a fast approximation algorithm for fully Bayesian shape matching using geometric hashing. The goal is to allow rapid probabilistic 3D landmark-based shape matching against a database of shapes, motivated by protein structure comparison. This approach also lends itself directly to shape classification.

In the second part of the talk I will give a brief overview of some algorithmic problems in statistics of a combinatorial or geometric flavor, which have arisen in areas of my recent work. I hope to that members of the audience may provide insight or have interest in taking up some of these problems.

When it is difficult to obtain independent samples from a distribution, dependent samples can instead be obtained using a Markov chain, the transition kernel of which is reversible with respect to the distribution. However, if there exists a partition of the state space such that each set in the partition has high probability, but such that the kernel rarely moves between the partition sets, then the chain will take many iterations to converge. This can occur for multimodal distributions.

When such ``bottlenecks'' are suspected to exist in the transition kernel, one can modify the kernel using parallel tempering in hopes of circumventing the bottlenecks. This approach has been shown to be successful for several bimodal examples, including the mean field Ising model. We show that the rapid mixing of parallel tempering in these examples is due to their symmetry, and does not hold in general for asymmetric distributions.

However, we propose a modification of parallel tempering which is guaranteed to be rapidly mixing. The overlap condition is not required. However, an estimate of the normalizing constant must be available. It may be possible to avoid the use of the normalizing constant with a different modification of parallel tempering, and we describe a set of sufficient conditions for such a modification to be rapidly mixing.

We initiate a study of triangulations that generalize the duals of higher order Voronoi diagrams, and show that these can serve as a foundation for a family of multivariate splines that generalize the classic univariate B-splines. This paper focuses on Voronoi diagrams of orders two and three, which produce families of quadratic and cubic bivariate B-splines.

There has been growing interest in algorithms capable of learning models from large volumes of multidimensional data, using statistical, geometrical and dynamical information. There are many domains of application for such algorithms, e. g. in exploratory data analysis, computer vision, system identification, control, computer graphics and multimedia databases. Of particular interest is the problem of motion tracking - for instance, in video sequences - where, under certain conditions, it can be assumed that the whole observation space is not occupied, but rather a manifold embedded in that space.

While the linear case can be solved by the well-known Principal Component Analysis technique, the non-linear case is more complex. Recently, there have been advances in algorithms, such as ISOMAP, Locally Linear Embedding, Gaussian Process Latent Variable Models and others, that approximate the data through manifold learning. These approaches will be briefly reviewed during the talk, and it will be shown that the dynamical case has received relatively little attention.

A recently proposed manifold learning algorithm, named Gaussian Process Tangent Bundle Approximation (GP-TBA), will also be discussed. This algorithm can deal with arbitrary manifold topology by decomposing the manifold into multiple local models, while also providing a probabilistic description of the data based on Gaussian process regression. Sparsity is enforced using l1 regularization.

Additionally, the model provided by GP-TBA can be used to simplify the dynamical identification and tracking problem. For this purpose, a multiple filter architecture, using e. g. Kalman or particle filtering, will be described. The GP-TBA algorithm and the filter bank framework will be illustrated with experimental results using real video sequences.

The large array of models of human disease that is now available in mice, has motivated us at the Center for In Vivo Microcopy (CIVM) to develop new tools and methods for in vivo small animal imaging in order to reveal both anatomical and functional aspects in rodents. Our imaging techniques include micro-CT, micro-digital subtraction angiography (DSA) and magnetic resonance microscopy (MRM). This talk presents an overview of our capabilities and concentrates on a few aspects that are potentially interesting for computer scientists such as geometric calibration for micro-CT, and image reconstruction approaches for micro-CT and MRM in dynamic imaging such as in cardiac and perfusion applications.

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

`Revenue management' has been used over the past twenty years, most notably by airlines to segment customers and manage capacity in face of uncertain demand. In recent years, with increased ability of data collection and availability of effective numerical solution methods for dynamic programming problems, several other industries are increasingly using revenue management and dynamic pricing. After an overview of problems in the area and approaches to their solution, I discuss a typology of problems that allows for the derivation of bounds on the value of different dynamic pricing strategies and to establish in which cases price exploration is potentially valuable. I also discuss approximate solutions to a dynamic programming problem that arises in a multi-period problem with demand learning.