I study the geometry of statistical models arising in statistical learning theory and finance. Using tools from geometry, I develop novel models of random phenomena and novel algorithms for estimation and decision-making with these models. I also seek to use geometric intuition to better understand classical methods. The questions that arise in this process can also lead to new mathematical results.
Papers
Submitted
10. Holographic algorithms without matchgates, with J.M. Landsberg and S. Norine.
9. Relations among conditional probabilities. Submitted (revising).
Some citations and discussion: Petrović, Rinaldo, and Fienberg, Slavković, Zhu, and Petrović.
Published
8. Geometry of the restricted Boltzmann machine with M. A. Cueto and B. Sturmfels. To appear in Algebraic Methods in Statistics and Probability, (eds. M. Viana and H. Wynn), American Mathematical Society, Contemporary Mathematics (2010).
7. Graphical models for correlated default, with I.O. Feliz, X.Guo, and B. Sturmfels. To appear in Mathematical Finance. Slides from a talk.
Some citations and discussion: Bayraktar and Yang.
6. Comparing pattern detection methods in microarray time series of the segmentation clock, with M. Dequeant, S. Ahnert, H. Edelsbrunner, T. Fink, E. Glynn, G. Hattem, A. Kudlicki, Y. Mileyko, A. Mushegian, L. Pachter, M. Rowicka, A. Shiu, B. Sturmfels, and O. Pourquie, PLoS ONE 3(8): e2856 doi:10.1371/journal.pone.0002856.
5. Convex rank tests and semigraphoids with L. Pachter, A. Shiu, B. Sturmfels, and O. Wienand, SIAM J. Discrete Math 23(3) 1117-1134, or here.
Some citations and discussion: Derksen and Fink.
4. The Cyclohedron test for finding periodic genes with L. Pachter, A. Shiu, and B. Sturmfels, Statistical Applications in Genetics and Molecular Biology 6 (2007) 1:21.
Some citations and discussion: Buck, Forcey and Springfield.
3. Three counterexamples on semigraphoids with R. Hemmecke, A. Shiu, B. Sturmfels and O. Wienand, Combinatorics, Probability and Computing 17 (2008) 239-257.
Some citations and discussion: Kahle, Huber (forthcoming), Niepert.
2. Geometry of rank tests, with L. Pachter, A. Shiu, B. Sturmfels, and O. Wienand, in Proceedings of the Third European Workshop on Probabilistic Graphical Models, Prague, Czech Republic, September 2006.
Some citations and discussion: Postnikov, Reiner, and Williams, Billera, Jia, and Reiner, Devadoss, Agnarsson and Morris.
1. Equations defining hidden markov models, with N. Bray, Chapter 11 of Algebraic Statistics for Computational Biology, Lior Pachter and Bernd Sturmfels, ed.s. Cambridge University Press, 2005.
Our Conjecture 11.9 has reportedly been proved by Alexander Schoenhuth; see the paper.
Books and theses
Geometry of conditional independence, my dissertation.
Workshop reports
Future Directions in Tensor-Based Computation and Modeling