New version page

Proposal

Upgrade to remove ads

This preview shows page 1-2-3 out of 8 pages.

Save
View Full Document
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 8 pages.
Access to all documents
Download any document
Ad free experience

Upgrade to remove ads
Unformatted text preview:

Algebraic Geometry in Algebraic Statistics andGeometric ModelingTexas Advanced Research Project Grant15 May 2008 — 14 May 2010Luis Garc´ıa-Puente & Frank Sottile, co-PIsSummaryGeometric modeling builds computer models for industrial design and manufacture frombasic units, called patches. Many patches, including Bezier curves and surfaces, are piecesof toric varieties, which are objects from algebraic geometry. Statistical models are familiesof probability distributions used in statistical inference to study the distribution of observeddata. Many statistical models, including the log-linear or discrete exponential models used toanalyze discrete data, are also pieces of toric varieties. While these connections of geometricmodeling and algebraic statistics to algebraic geometry were known, direct connections be-tween these applied subjects were only recently discovered by th e PIs [arXiv:/0706.2116]. Forexample, iterative proportional fitting (IPF) from statistics can be used to compute patches,and linear precision in geometric modeling is related to maximum likelihood estimation instatistics.Not only are th ese basic objects the same, but an important tool, the algebraic momentmap of a toric variety, is also shared. In geometric modeling it offers a prefered parametriza-tion of a patch, and in algebraic statistics it is the expectation map of a log-linear model.These same objects arise in dynamical systems; in [arXiv:0708.3431] the authors introducetoric dynamical systems whose space of systems is a toric variety with the algebraic momentmap giving the steady state solution.We will exploit these connections to transfer ideas and techniques between these threefields. Krasauskas’s multi-sided toric patches [Adv. Comput. Math. 17 (2002)] generalizethe classical Bezier patches. Several important problems, including the tuning of a toricpatch to achieve linear precision, are amenable to techniques from algebraic statistics. Wewill also develop IPF into a tool to compute and manipulate toric patches.While these fields use objects from algebraic geometry, the application of methods fromalgebraic geometry to these fields is in its infancy. We will deepen these applications. Forexample, there is a dictionary between toric degenerations and control polyhedra of patchesthat we will elucidate and use. Another direction is to find applications in modeling andstatistics of theoretical work of Sottile on bounds (both lower and upper) on numbers ofreal solutions to equations. We forsee many avenues of research, including several which aresuitable for drawing undergraduates into research and others which could form the foundationfor a Masters or PhD thesis.Student InvolvementThis project will link students at SHSU with the rest of Sottile’s group at TAMU. Weplan both individual supervised research projects for students and vertically integrated re-search projects between undergraduates, graduate students, postdocs and faculty at bothinstitutions. This will train junior members of our group in the art of collaborative scientificresearch, in the relevant mathematics, and in the use of computers in mathematical research.It will provide SHSU students, many of whom are the first in their families to attend college12or come from underrepresented groups, with the opportunity of being involved in researchat a major research university. Senior members will get experience in mentoring.This project will fund our joint research, two undergraduate students and a Masters stu-dent at SHSU and a PhD student and postdoc at TAMU.A. Research objectivesRecent work in geometric modeling [4], dynamical systems [2], and algebraic statistics[9] has revealed a mathematical object common to these fields—the p ositive part of a toricvariety and its algebraic moment map, µ. In geometric modeling, the inverse of µ is a pre-ferred parametrization of a toric patch, in algebraic statistics it is the maximum likelihoodestimate of a log-linear model, and for toric dynamical systems it is the corresponding steadystate. We will exploit this connection to transfer ideas and techniques between these fields.This project will involve collaboration between teams of students and researchers at SamHouston State University and at Texas A&M University. Our team-based, experimentalapproach represents a new methodology for mathematical research. The outcomes will in-clude the training of junior researchers, published results, and the submission of proposalsfor federally-funded research.The problems we propose below are designed to advance our understanding of t hese fieldsacross a broad front. They will deepen the mathematical foundations of these areas, as wellas further the development of tools and introduce new objects which may eventually proveuseful in actual applications. Some will also impact pure mathematics as the specific needsof applications often highlight new mathematical structures to investigate.1. The algebraic moment map. An integer vector a = (α1, . . . , αd) corresponds to amonomialxa:= xα11xα22· · · xαddfor xi6= 0. (Typically, xi∈C×(:= C − {0}), but we may have xi> 0 so that xi∈ R>.)Lists c = (c0, c1, . . . , cn) ∈ Rn+1>of positive numbers (called a weight) and A = {a0, a1, . . . , an}of integer vectors together define a mapϕc,A: (C×)d→ Pn(n-dimensional projective space)via(1) ϕc,A(x) := [c0xa0, c1xa1, . . . , cnxan] .The closure of the image ϕc,A((C×)d) is thetoric variety cXA. The closure of ϕc,A(Rd>)is thepositive part cX+Aof cXA, which consists of those points of cXAwith nonnegativecoordinates.The algebraic moment map [10] µA: cXA→ Cd(or µA: Pn→ Cd) is defined bycXA∋ y = [y0, y1, . . . , yn] 7−→µA(y) :=Pni=0aiyiPni=0yi.The image of cX+Ais∆A, the convex hull of the vectors ai∈ A. The following result isfundamental.Theorem. The alge braic moment map µAis a homeomorphism between cX+Aand ∆A.32. Algebraic Statistics. In algebraic statistics, the positive part ∆nof Pnis identifiedwith the probability simplex {(p0, . . . , pn) | pi≥ 0 and p0+ · · · + pn= 1}. Algebraic subsetsX of ∆narestatistical models. These usually come with a parametrization from a set ofmeaningful parameters to the family of probability distributions X. A fundamental problemis statistical inference:Given a point q ∈ ∆nand a mo d el X, find the point p ∈ X which ‘best’ agrees with q.Typically, ‘best’ means the point corresponding to the maximum likelihood estimate (MLE)for


Download Proposal
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Proposal and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Proposal 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?