I will present a forthcoming paper titled "Analysis of incomplete data and an intrinsic-dimension Helly Theorem" by Gao, Langberg, and Schulman. The authors extended the notion of point set clustering to a more general setting, where we are given a set of $k$-flats in $d$-dimensional space, and want to compute the smallest ball that intersects all flats. The authors proved that there exists a subset of the input flats of size $O(k^4/\eps^2)$ (a {\em coreset}), such that the optimal ball for this subset is an $\eps$-approximation to that for the original input. Note that the size of the subset does not depend on the dimension of the ambient space. Their proof is based on a nice application of the Helly theorem.

In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. In the past few years,researchers in Theoretical Computer Science, Databases, IP Networking and Computer Systems have developed new algorithms that work within these space and time constraints. The methods rely on metric embeddings, pseudo-random computations and sparse approximation theory. The applications include IP network traffic analysis, mining text message streams for Homeland Security and processing massive data sets in general.

I will present an overview of the principles, and discuss issues in building data stream systems that work at IP line speeds. I will also discuss open problems. This talk is based on an updated version of the survey at http://www.cs.rutgers.edu/~muthu/stream-1-1.ps.

A materialized view is a certain synopsis structure precomputed from one or more data sets (called \emph{base tables}) in order to facilitate various queries on the data. When the underlying base tables change, the materialized view also needs to be updated accordingly to reflect those changes. We consider the problem of batch-incrementally maintaining a materialized view under a response-time constraint. We propose techniques for selectively processing updates to some base tables while keeping others batched, with the goal of minimizing the total maintenance cost while meeting the response-time constraint.

Our main result is an online algorithm that achieves a constant competitive ratio for all concave maintenance cost functions while relaxing the response-time constraint by a constant factor. Our algorithms are based on emulating the behavior of an online paging algorithm on a page request sequence carefully designed from the maintenance cost function.

Joint work with Jun Yang and Hai Yu.
(Preliminary version appeared in ESA 2005)

Anomaly detection has important applications in biosurveilance and environmental monitoring. When comparing measured data to data drawn from a baseline distribution, merely, finding clusters in the measured data may not actually represent true anomalies. These clusters may likely be the clusters of the baseline distribution. Hence, a discrepancy function is often used to examine how different measured data is to baseline data within a region. An anomalous region is thus defined to be one with high discrepancy.

In this talk, I will present algorithms for maximizing statistical discrepancy functions over the space of axis-parallel rectangles. I will give provable approximation guarantees, both additive and relative, and these methods apply to any convex discrepancy function. The algorithms work by connecting statistical discrepancy to combinatorial discrepancy; roughly speaking, I will show that in order to maximize a convex discrepancy function over a class of shapes, one needs only maximize a linear discrepancy function over the same set of shapes.

I will derive general discrepancy functions for data generated from a one-parameter exponential family. This generalizes the widely-used Kulldorff scan statistic for data from a Poisson distribution. I will present an algorithm running in O(1/e n^2 log^2 n) time that computes the maximum discrepancy rectangle to within additive error e, for a size n data set, for the Kulldorff scan statistic. Similar results hold for relative error and for discrepancy functions for data coming from Gaussian, Bernoulli, and gamma distributions. Prior to our work, the best known algorithms were exact and ran in time O (n^4).

(Joint work with Deepak Agarwal and Suresh Venkatasubramanian)

Let f_t: X -> R be a homotopy of real-valued functions (t \in [0,1]). If we record every point (x,y) from the persistence diagram of every f_t in the point (x,y,t) in R^3, then from the stability theorem for persistence diagrams it follows that each off-diagonal point of a diagram D(f_t) moves in time tracing out a curve. We call these curves vines, and their collections vineyards. The basic problem that arises in computation of persistence vineyards is maintaining the persistence pairing of a filtration after two adjacent simplices transpose. We call this the transposition problem.

Suppose one is given a real-valued function f: M^2 -> R defined on a 2-manifold, and a parameter epsilon \in R. Let D(f) be the persistence diagram of f. Consider the problem of finding function g: M^2 -> R such that ||f-g|| <= epsilon, and the persistence diagram D(g) consist only of those points of D(f) that are more than epsilon away from the diagonal. We call this the simplification problem.

In this talk I will present solutions to both transposition and simplification problems, and will discuss research directions that stem from these problems that I intend to pursue for my thesis.

Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of CGAL, the Computational Geometry Algorithms Library, has been redesigned and re-implemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications, and is more efficient space- and time-wise. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. The new package is able to answer point-location queries very efficiently due to a new "landmark" strategy recently introduced. Arrangements are used, among the other, to implement (exact) Boolean-set operations on curved polygons, for Minkowski Sum constructions in 2D (arbitrary polygons) and 3D (convex polytopes), and for the computation of envelopes of surfaces in 3D.

This is joint work with Ron Wein, Baruch Zukerman, Idit Haran, Michal Meyerovitch, and Dan Halperin.

Many algorithms in geometric modeling are based on concepts from differential geometry. Classical notions such as the metric or the curvature of a surface were formulated in the continuous (smooth) setting and a rich mathematical apparatus exists to help us understand what is possible. When it comes to the computational realm the picture is far murkier. Simply discretizing continuous notions to transfer them to the setting of meshes, for example, often leads to the loss of important structures. In numerical practice this typically results in performance problems and difficulties "making things work." A possible way out of this state of affairs is to reinvent differential geometry in the discrete setting from the ground up.

In my talk I will consider an example from texture mapping, or, more generally, finding a "nice" parameterization of a surface given as a triangle mesh. For example, we may be interested in finding parameterizations which preserve angles (are conformal). What is a good way to capture this in the discrete setting? This is where circles enter! (This notion is not unfamiliar, for example, in the case of Delaunay triangulations where the empty circumcircle property becomes the defining tool in the construction of particular triangulations.) So called "circle patterns" lead to a mathematically clean (and deep) way to capture discrete conformality, result in well defined numerical problems with efficient algorithms and unique solutions, and, this is where it gets good, give us tools to control the pesky area distortion issue in conformal mappings.

Joint work with Liliya Kharevych and Boris Springborn.

Resource delivery networks are central to the functioning of many biological organisms. Recent work suggests that spatial network design should be "optimal" -- though controversy exists regarding the metric for and implications of "optimality" for organismal functioning. In this talk I present two examples of recent empirical and theoretical advances in our understanding of the scaling of vascular networks. The first example is taken from an extensive empirical study of the structure and design of hydraulic networks within trees. I will present evidence for the existence of an ontogenetically stable hydraulic design, i.e. a design that scales predictably as the tree grows and ages. The second example is a largely theoretical study of the optimal cardiovascular network within mammals as a function of body size. Throughout, I will show evidence for breakpoints in scaling, the importance of fluctuations, and data from real vascular networks. Slight deviations from optimality in the presentation of these examples will be considered further support of one of the central messages of the talk.

I will present a new algorithm for isotropic 3D meshing. The goal is to decompose a given 3D domain into tetrahedra having the best possible aspect ratio. The meshes we produce actually do not conform to the boundary of the domain, but rather approximate it. Our approach uses a variant of Lloyd's relaxation based on recent work from Long Chen. This method has the advantage to produce much less slivers, which are a certain type of badly-shaped tetrahedra, than standard Lloyd's relaxation or Delaunay refinement.

Joint work with Pierre Alliez (INRIA), Mathieu Desbrun (Caltech), Mariette Yvinec (INRIA).

We introduce an algorithm that learns gradients from samples in the supervised learning framework. An error analysis is given for the convergence of the gradient estimated by the algorithm to the true gradient. The utility of the algorithm for the problem of variable selection as well as determining variable covariance is illustrated on simulated data as well as two gene expression datasets. For square loss we provide a very efficient implementation with respect to both memory and time.

Joint work with Ding-Xuan Zhou.

An intuitive definition of the Reeb graph is as follows. Let $f$ be a continuous function over a closed manifold. We construct the Reeb graph of $f$ by contracting each component of the level set, $f^{-1}(s)$, to a point for every real value $s$.

Reeb graphs are a useful tool in capturing and visualizing the connectivity of level sets. This is helpful when analyzing data sets usually in the form of values assigned to points on a manifold. Reeb graphs are helpful in understanding a single function but it's not clear how we can use Reeb graphs to understand relationships between multiple functions on a manifold. For this project, we are interested in studying relationships between multiple smooth functions over a common closed manifold. We will define the Reeb set generalizing the idea of Reeb graphs in hopes of providing a useful tool in understanding topological differences between level sets of multiple smooth functions on a common closed manifold.

We present an algorithm for sampling and triangulating a smooth implicit surface. The only assumption we make is that the input surface representation is amenable to certain types of computations, namely computations of the intersection points of a line with the surface, computations of the critical points of some height functions defined on the surface and its restriction to a plane, and computations of some silhouette points. The algorithm ensures topological correctness, bounded aspect ratio, size optimality, and smoothness of the output triangulation. Unlike previous algorithms, this algorithm does not need to compute the local feature size for generating the sample points.

This is joint work with Tamal K. Dey, Edgar A. Ramos, and Tathagata Ray.

Computing topological invariants of the underlying topological space of a point cloud may provide qualitative information about the data set which is not readily available otherwise. Such computations are usually done by approximating the topological space with a simplicial complex, but using standard simplicial complexes for very large data sets may not be viable. Witness complexes allow us to handle large amount of data by restricting the vertex set of our complex. Such a restriction is realized by choosing 'landmark' points; the rest of the points are called 'witnesses', and define which simplices are actually added to the complex. It is also possible to produce a filtration for a witness complex and thus apply persistence homology to identify more stable features of the space. Unfortunately, existing variations of witness complexes produce filtrations with a big number of spurious simplices, generally created for large values of the persistence parameter, which makes it more difficult to distinguish important features, and also leads to a waste of resources. In this talk, I will introduce a new family of witness complexes, $(\alpha, \beta)$-witness complexes, where an additional parameter, $\beta$, is used to discard spurious simplices in the filtration induced by the parameter $\alpha$. Such a construction seems to have an advantage over the existing ones, and there is a nice geometric 'justification' for it which will be described along with some preliminary results. The question of finding the right value for $\beta$ brings us to the much more general question about the two-parameter persistence. This issue, and the the problem of choosing landmark points for witness complexes will also be discussed.