GEOMETRIC DATA ANALYSIS group (old pages)
ELECTORAL GEOMETRY and GERRYMANDERING group
11/12/21 Hanyu Zhang is a co-awardee of the Departmental Z.W. Birnbaum Award for his General Exam! Congrats, Hanyu!
10/18/21 Our paper Manifold coordinates with physical meaning accepted in JMLR! Congrats Sam, Hanyu and Yu-Chia!
10/18/21 The AMS MRC 2022 Summer Conference on Data science at the crossroads of Analysis, Geometry and Topology, organized with Facundo Memoli, Jose Perea, Nicolas Garcia-Trillos, Soledad Villar is accepting applications!(click on link) This experience is open to PhD students in their 2 terminal years and to early carreer graduates from Marthmatics and related Data Science fields. Dates: May 29-June 4, 2022, in Buffalo, NY.
9/28/21 Our paper The decomposition of the higher-order homology embedding constructed from the k-Laplacian is accepted at NeurIPS 2021 for an ORAL presentation! Congrats, Yu-Chia!
8/6/21 Sheridan Grant defends his PhD Thesis on "Causality, fairness and information in peer review". Congrats, Sheridan!
7/26/21 Yu-Chia Chen defends his PhD Thesis "Learning topological structures and vector fields on manifolds with (higher order) discrete Laplacians". Congrats, Yu-Chia!
7/18/21 The ICML 2021 Conference starts today!
6/30/21 Our paper Tangent Space Least Adaptive Clustering is accepted at the ICML 2021 Workshop on Unsupervised Learning for RL! Congrats, James!Sam!
6/9/21 Alon Milchgrub defends his PhD Thesis "Principal curves in High Dimensions". Congrats, Alon!
Sheridan Grant wins a JSM student paper award from the Social Statistics, Government Statistics, and Survey Research Methods Sections and will be presenting his paper 'Refinement: Measuring Informativeness of Ratings in the Absence of a Gold Standard' at JSM 2021
11/2/20 Manifold coordinates with physical meaning via Riemannian geometry Applied Mathematics Seminar at Yale University
I work on machine learning by probabilistic methods and reasoning in
uncertainty. In this area, it is particularly important to develop
computationally aware methods and theories. In this sense, my research
is at the frontier between the sciences of computing and statistics. I
am particularly interested in combinatorics, algorithms and
optimization, on the computing side, and in solving data
analysis problems with many variables and combinatorial structure.
MANIFOLD LEARNING AND GEOMETRIC DATA ANALYSIS
Manifold learning algorithms find a non-linear representation of
high-dimensional data (like images) with a small number of
parameters. However, all such existing methods deform the data (except
in special simple cases). We construct low-dimensional representations
that are geometrically accurate under much more general conditions. As
a consequence of the kind of geometric faithfulness we aim for, one
should be able to do regressions, predictions, and other statistical
analyses directly on the low-dimensional representation of the
data. These analyses would not be correct in general, if one were not
preserving the original data geometry accurately.
FOUNDATIONS OF CLUSTERINGS
It was widely believed that little can be theoretically said about clustering and clustering algorithms, as most clustering problems are NP-hard. This part of my work aims to overcome these difficulties, and takes steps towards developing a rigurous and practically relevant theoretical understanding of the clustering algorithms in everyday use.
A fundamental concept is that of clusterability of the
data. Given that the dat contains clusters, one can show that: we can
devise initialization methods that lead w.h.p. to an almost correct
clustering, we can prove that a given clustering is almost optimal,
and that if the nubmer of clusters in the data is smaller than our
guess, we will obtain unstable results.
PERMUTATIONS, PARTIAL RANKINGS, INTRANSITIVITY AND CHOICE
It has been often noted that people's choices are not transitive: in
other words, their preferences between K objects are not consistent
with an ordering. Economic theory of choice has introduced various
theories explaining how the observed intransitivity may
arise. However, there is no work to date on how one may infer these
models from data. Among the things I want to do: to formulate
estimation problems for the hidden context and other models of
intransitivity that are relevant to practical domains; to define when
the model is identifiable (it may not be when the number of components
K is large) and to design rigorously founded algorithms to estimate
CLUSTERING BY EIGENVALUES AND EIGENVECTORS
...is a technique rooted in graph theory for finding groups (or other
structure) in data. It already has applications in image segmentation,
web and document clustering, social networks, bioinformatics and
linguistics. My recent work concentrates on the study of asymmetric links, or, in other words, of directed graphs. More
GRAVIMETRIC INVERSION WITH SPARSITY CONSTRAINTS
This works deals with recovering the shape of an unknown body from
gravity measurements. As a mathematical physics problem, this one is
old, well-studied, and one of the hardest type of inverse problems. My
team is interested in finding algorithmic solutions, under realistic
scenarios, that recover given features of the unknown underground
density in noise. We showed that this problem can be mapped to a
linear program with sparsity constraints, for which we formulated
various continuous and integer approaches. The methodological and
theoretical work on this problem continues, as we exploit the
connections with Compressed Sensing, QBPs and submodularity. The
practical results led to intriguing new research questions, since the
restricted isometry assumptions that usually underlie compressed
sensing algorithms can be proved not to hold for the gravimetry
problem. (Collaboration with Caren Marzban and Ulvi Yurtsever.)
Interpreting the very complex signature of an
amino-acid sequence that is subjected to collision induced dissociation
(CID). Probabilistic identification of the protein composition of a
complex mixture from high throughput mass spectrometry data.