m e m o l i @ m a t h . s t a n f o r d . e d u

Research interests

• Shape and data analysis, shape comparison and matching.
• Computational topology and Topological data analysis
• Machine learning, clustering.

Research summary

Shape comparison and matching

I have been working on Shape comparison. I am interested in applying ideas borrowed from Metric Geometry to this problem. Gromov's ideas about comparison of metric spaces have proved useful for formalizing several preexisting practical ideas in the field. In my PhD thesis I proposed using the Gromov-Hausdorff distance for approaching this problem, see this paper.

Gromov-Wasserstein distances

In this paper I proposed to represent objects/shapes as metric measure spaces and introduced certain Lp Gromov-Hausdorff distances or Gromov-Wasserstein distances as an alternative to the standard notion. These distances are based on ideas from mass transport. By doing this, I obtained a much more computationally tractable framework and at the same time, many other seemingly unrelated approaches to the problem of Shape Matching were proved to be lower or upper bounds to this new distance. In a follow paper up I study the problem of partial shape matching using these Gromov-Wasserstein distances.

Spectral methods

Lately I have been studying spectral methods for matching shapes and data matching and in this paper I proposed a unifying framework for several preexisting ideas. Some applications of heat kerneel methods are in this paper.

Formalization of clustering algorithms

With Gunnar Carlsson we have become interested in questions regarding the theoretical properties of clustering algorithms. We have results characterizing single linkage and establishing metric stability and convergence of hierarchical clustering in this paper and this paper. We have started looking at multiparameter clustering methods see this talk and this paper. More recently, using the formalism of category theory we have been able to find expand the classification results contained in the papers above: see preprint on the arxiv.

Data analysis using tools from Algebraic Topology

In the TMSCSCS group we deal with high dimensional datasets and we try to apply tools inspired in Algebraic Topology to the understanding of these datasets. Here's a paper with our work on applying these ideas to neuroscience data. We have developed an algorithm called mapper that is helpful in analyzing data. Together with Yuri Dabaghian and Loren Frank from UCSF we are currently analyzing data from rat hippocampus and studying the extent to which that part of the brain is able to encode for "space". We are writing the paper-- here's a poster presented at SfN 2010.

Maps between spaces and estimation of geodesic distances

In the past, I've worked on applications of p-Harmonic Maps to Brain Warping. During my PhD thesis time I proposed looking at the plausible connection between \infty-Harmonic Maps and Lipschitz Extensions as a way of obtaining interpolating maps between two cortical surfaces. Here's a paper with these ideas.

During my masters I became aware of the fast marching method on triangulated surfaces as proposed by Sethian and Kimmel. I had been working with level set representation of surfaces so it seemed natural to look for a direct solution to the problem of estimating geodesic distances without having to triangulate the surface. This led to my master's thesis which is contained in this paper.

Later, my PhD advisor proposed me to try to extend my approach to deal with surfaces represented merely as point clouds. This was a very good idea and led me to having to generalize some ideas of the paper above. In particular, the conditions that guarantee convergence were probabilistic. You can find the paper right here.