Professor, Associate Director of the Center for Nonlinear Analysis
7123 Wean Hall
Department of Mathematical Sciences
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
Ph.D., University of Texas at Austin
My research lies primarily in applied analysis. I use partial differential equations, calculus of variations, optimal transportation, as well as tools of probability, statistics, functional analysis, and scientific computation to address applied problems. Over the last few years I have been working on applications to problems of data science. Modern data-acquisition techniques produce a wealth of data about the world we live in. Extracting the information from the data leads to machine learning/statistics tasks such as clustering, classification, dimensionality reduction, regression, and others. These tasks are regularly cast as optimization problems where one minimizes an objective functional that models the desired properties of the object sought. The objective functionals and the resulting minimization are often posed on the available data sample which leads to discrete variational and PDE problems on graphs and related structures representing the data.
I work on developing a mathematical framework to study variational and PDE based problems on random data samples. I use insights from the continuum based variational problems and PDE to improve existing approaches in the discrete setting and introduce new models and algorithms for problems of data science.
I am also interested in collective behavior of systems of interacting particles (or agents). Systems with long range interactions arise as models of collective behavior of a variety of living organisms, as well as in physical sciences. I have been working on mathematically describing the evolution of such systems, explaining why are they stable, and exploring how they interact with their environment.
I am further interested in energy-driven systems — broadly defined as the systems that dissipate an associated free energy. Many pattern-forming systems are governed by energy which is dissipated during the evolution. The dissipation mechanism endows the underlying configuration space with a geometric structure. Studying how the energy changes as the configuration space is traversed provides understanding of the energy landscape, which in turn provides information about the scaling laws present in the dynamics. I have worked on understanding the dynamics of patterns of thin liquid films, demixing of fluids, evolution of interfaces, and evolution of grain-boundary networks.
M.M. Dunlop, D. Slepčev, A.M. Stuart, M. Thorpe, Large data and zero noise limits of graph-based semi-supervised learning algorithms, arXiv:1805.09450
N. Garcia Trillos, M. Gerlach, M. Hein, D. Slepčev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace–Beltrami operator, arXiv:1801.10108
M. Thorpe, D. Slepčev, Analysis of p-Laplacian regularization in semi-supervised learning, arXiv:1707.06213
J-G. Liu, R.L. Pego, D. Slepčev, Least action principles for incompressible flows and optimal transport between shapes, preprint, arXiv:1604.03387.
N. Garcia Trillos, D. Slepčev, A variational approach to the consistency of spectral clustering, Appl. Comput. Harmon. Anal. 45, no. 2, (2018), pp. 239-281.
N. Garcia Trillos, D. Slepčev, J. von Brecht, T. Laurent, and X. Bresson, Consistency of Cheeger and ratio graph cuts, J. Mach. Learn. Res. 17, no. 181, (2016), pp. 1-46.
N. Garcia Trillos and D. Slepčev, Continuum limit of total variation on point clouds, Arch. Ration. Mech. Anal., 220 no. 1, (2016) 193-241.
X.Y. Lu and D. Slepčev, Properties of minimizers of average-distance problem via discrete approximation of measures, SIAM J. Math. Anal., 45, No. 5, (2013) pp. 3114-3131.
W. Wang, D. Slepčev, S. Basu, J. Ozolek, and G.K. Rohde, A linear optimal transportation framework for quantifying and visualizing variations in sets of images, International Journal of Computer Vision, 101, No. 2, (2013), pp. 254-269.
J.A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slepčev, Global-in- time weak measure solutions, and finite-time aggregation for nonlocal interaction equations, Duke Math J. 156 No 2, (2011) pp. 229–271.
F. Otto, T. Rump, and D. Slepčev, Coarsening rates for a droplet model: rigorous upper bounds, SIAM J. Math. Anal. 38 (2006) no. 2, pp. 503-529