Shaowei Lin

Email: shaowei at math dot berkeley dot edu
Office: 737 Evans

I am a PhD student in the UC Berkeley Math Department. Broadly, my research interests are in combinatorics, computational algebraic geometry and non-traditional applications of mathematics such as algebraic statistics. Specifically, I am working on algebraic methods for evaluating marginal likelihood integrals, and on polynomial relations among principal minors of a matrix. I am fully funded by A*STAR (Agency for Science, Technology and Research), Singapore. My advisor is Bernd Sturmfels. I graduated from Stanford University in June 2005 with a B.S.(with honors) in mathematics and a minor in computer science. Here is my CV.

[Publications] [Talks] [Software] [Reports] [Awards] [Research] [Past Work] [Past Events]

Current Events

Publications

  1. S. Lin and B. Sturmfels, "Polynomial Relations among Principal Minors of a 4x4-Matrix," accepted by Journal of Algebra, Dec 2008. [Preprint] [Website]

  2. S. Lin, B. Sturmfels and Z. Xu, "Marginal Likelihood Integrals for Mixtures of Independence Models," Journal of Machine Learning Research, vol. 10, pp. 1611 - 1631, Jul 2009. [Preprint] [Website]

  3. S. Lin, W. W. L. Ho and Y. C. Liang, "Block Diagonal Geometric Mean Decomposition (BD-GMD) for MIMO Broadcast Channels," IEEE Trans. Wireless Communications, vol. 7, no. 7, pp. 2778 - 2789, Jul 2008. [Abstract] [Paper]

  4. S. Lin, W. W. L. Ho and Y. C. Liang, "Block-Diagonal Geometric Mean Decomposition (BD-GMD) for Multiuser MIMO Broadcast Channels," Proc. IEEE PIMRC, Helsinki, Finland, Sep. 2006. [Abstract] [Paper]

  5. S. Lin, W. W. L. Ho and Y. C. Liang, "MIMO Broadcast Communications Using Block-Diagonal Uniform Channel Decomposition (BD-UCD)," Proc. IEEE PIMRC, Helsinki, Finland, Sep. 2006. [Abstract] [Paper]

Talks

  1. 21 Apr 2009: [Slides] Asymptotic Approximation of Marginal Likelihood Integrals. UPenn, PA.

  2. 08 Apr 2009: [Slides] Log Canonical Thresholds and Statistical Models. MSRI, CA.

  3. 19 Mar 2009: [Slides] Polynomial Relations among Principal Minors of a Matrix. UCSD, CA.

  4. 19 Jan 2009: [Slides] Polynomial Relations among Principal Minors of a Matrix. NTU, Singapore.

  5. 15 Jan 2009: [Slides] Polynomial Relations among Principal Minors of a Matrix. NUS, Singapore.

  6. 17 Dec 2008: [Slides] Asymptotic Approximation of Marginal Likelihood Integrals. MSRI Workshop on Algebraic Statistics. Berkeley, CA.

  7. 13 Nov 2008: [Slides] Exact Evaluation of Marginal Likelihood Integrals. Math 239/Stat 260 Algebraic Statistics class lecture.

  8. 15 Sep 2008: [Poster] Exact Evaluation of Marginal Likelihood Integrals. SAMSI Program on Algebraic Methods in Systems Biology and Statistics. Research Triangle Park, NC.

  9. 27 Aug 2008: [Slides] Exact Evaluation of Marginal Likelihood Integrals. CACAO seminar. UC Davis, CA.

  10. 01 May 2008: [Slides] Exact Evaluation of Marginal Likelihood Integrals. Math 239 Discrete Mathematics for the Life Sciences project presentation.

  11. 29 Mar 2008: [Slides] Relations among Principal Minors of a Matrix. AMS Sectional Meeting on Algebraic Geometry of Matrices and Determinants. Baton Rouge, LA.

  12. 18 Dec 2007: [Slides] Relations among Principal Minors of a Matrix. Reine und Angewandte Algebra. MA 313.

  13. 28 Sep 2007: [Notes] Relations between the Principal Minors of a 4x4 Matrix. Algebraic Geometry Seminar. 939 Evans.

Software

  1. [Website] Polynomial Relations among Principal Minors of a 4x4 Matrix

  2. [Website] Exact Evaluation of Marginal Likelihood Integrals.

  3. [Website] Studying Legendrian Invariants with FrontLeg.

Reports

  1. Dec 2008: [Paper] Asymptotic Approximation of Marginal Likelihood Integrals. Math 239/Stat 260 Algebraic Statistics project report.

  2. May 2008: [Paper] Exact Evaluation of Marginal Likelihood Integrals. Math 239 Discrete Mathematics for the Life Sciences project report.

  3. May 2007: [Paper] Counting Gene Finding Functions. Math 127 Mathematics for Computational Biology project report.

  4. May 2005: [Paper] Studying Legendrian Invariants with FrontLeg. Stanford undergraduate research thesis.

Awards

  1. 2005: A*STAR Roll of Honour

  2. 2005: Stanford Mathematics Undergraduate Research Award

  3. 2005: Stanford Mathematics - Honors with Distinction

Research Interests

Evaluation of Marginal Likelihood Integrals

In statistics, two scoring systems are generally used for model selection: maximum likelihood and marginal likelihood. The first is very well studied and understood in algebraic statistics, while the second is an exciting new research topic. Interestingly, we can think of the maximum likelihood as the tropical version of the marginal likelihood.

My initial interest was in evaluating these integrals exactly for mixtures of independence models as rational numbers. The idea is simple: we expand the polynomial integrand and compute the integral of each monomial over a polytope using a well-known formula. We wrote Maple code for our algorithms which speed up the computation using certain tricks. Many interesting combinatorial problems arise from this study, such as the counting of lattice points in zonotopes. See [P4], [T6], [S2].

Recent work by Sumio Watanabe showed that we can approximate such integrals using the technique of resolution of singularities from algebraic geometry. I am currently studying the relationship between these integrals, Newton diagrams, toric modifications and real log canonical thresholds. See [R4], [T8].

Polynomial Relations among Principal Minors

Principal minors show up in many applied areas such as determinantal processes and graph theory. A key problem is to determine if a list of numbers are realizable as the principal minors of some matrix. We solve this problem by finding all polynomial relations satisfied by the principal minors.

A crucial result we proved is that the image of the principal minor map is closed. This was accomplished by rewriting the principal minors in terms of cycles and cycle-sums. We also found 65 generators of our affine ideal (as opposed to the relations for a projective version of this problem) using these cycle-sums. I believe these cycles and cycle-sums are important for studying other determinantal-type problems.

We also studied the projective version of this problem using representation theory. There is a Lie group action of the n-fold product GL2 x GL2 x ... x GL2 on the principal minors of the n x n matrix. The ideal of relations is invariant under this action. By passing to the Lie algebra, we can use raising and lowering operators to determine the highest weight vectors in the ideal. By exploiting the symmetry in this problem through representation theory, we were able to solve our problem efficiently. I believe that studying the interplay between group actions and Grobner bases is important to algebraic statistics and other applications. See [P5], [T3], [S3].

Past Work

Legendrian Knots

I did my undergraduate degree at Stanford University, 2002-2005. I majored in Mathematics (with honors) and minored in Computer Science. My advisor was Prof. Yakov Eliashberg, and I did my honors thesis on the invariants on Legendrian knots (Symplectic Geometry). See [R1], [S1].

Persistent Homology

In 2004, I did a summer research project with Prof. Gunnar Carlsson under the Topological Methods in Scientific Computing, Statistics and Computer Science (TMSCSCS) program.

Matrix Decompositions

I worked in the Institute for Infocomm Research (I2R) (Singapore) from Jul 2005 to Aug 2006, as part of the one-year attachment required under the A*STAR scholarship. I was in the Digital Wireless Department, and my advisor was Dr. Ying-Chang Liang. My work was mainly on Matrix Decompositions and their applications in MIMO Communications. We published and presented two conference papers in the PIMRC 2006 in Helsinki. We also have one journal paper in the IEEE Transactions on Wireless Communications. See [P1], [P2], [P3].

Past Events

08-09 Spring

08-09 Fall

07-08 Spring

07-08 Fall

06-07 Spring

06-07 Fall