Chris White

Chris White

Half Moon Bay, California, United States
2K followers 500+ connections

About

Mathematician turned software engineer turned CTO, passionate about building…

Activity

Experience

  • Prefect Graphic
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    McLean, VA

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    McLean, VA

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    McLean, VA

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    Austin, Texas

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    Austin, TX

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    Greater Los Angeles Area

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    Malibu, CA

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    Greater Los Angeles Area

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    Baton Rouge, Louisiana Area

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    Lake Charles, Louisiana Area

Education

  • The University of Texas at Austin Graphic

    The University of Texas at Austin

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    Activities and Societies: Ran and Organized Undergraduate Math Club - every week I brought in a speaker from the math community to give a talk about his/her research to an enthusiastic group of undergrads.

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    Undergraduate Math Excellence Award 2009

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    Outstanding Undergraduate in Econometrics in 2008

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    Activities and Societies: Show Choir

Publications

  • The Local Convexity of Solving Systems of Quadratic Equations

    Springer Results in Mathematics

    This paper considers the recovery of a rank r positive semidefinite matrix XXT∈ℝn×n from m scalar measurements of the form yi:=aTiXXTai (i.e., quadratic measurements of X). Such problems arise in a variety of applications, including covariance sketching of high-dimensional data streams, quadratic regression, quantum state tomography, among others. A natural approach to this problem is to minimize the loss function f(U)=∑i(yi−aTiUUTai)^2 which has an entire manifold of solutions given by…

    This paper considers the recovery of a rank r positive semidefinite matrix XXT∈ℝn×n from m scalar measurements of the form yi:=aTiXXTai (i.e., quadratic measurements of X). Such problems arise in a variety of applications, including covariance sketching of high-dimensional data streams, quadratic regression, quantum state tomography, among others. A natural approach to this problem is to minimize the loss function f(U)=∑i(yi−aTiUUTai)^2 which has an entire manifold of solutions given by {XO}O∈r where r is the orthogonal group of r×r orthogonal matrices; this is non-convex in the n×r matrix U, but methods like gradient descent are simple and easy to implement (as compared to semidefinite relaxation approaches). In this paper we show that once we have m≥Cnrlog2(n) samples from isotropic gaussian ai, with high probability (a) this function admits a dimension-independent region of local strong convexity on lines perpendicular to the solution manifold, and (b) with an additional polynomial factor of r samples, a simple spectral initialization will land within the region of convexity with high probability. Together, this implies that gradient descent with initialization (but no re-sampling) will converge linearly to the correct X, up to an orthogonal transformation. We believe that this general technique (local convexity reachable by spectral initialization) should prove applicable to a broader class of nonconvex optimization problems.

    Other authors
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  • Minimal Dirichlet energy partitions for graphs

    SIAM Journal on Scientific Computing

    Motivated by a geometric problem, we introduce a new nonconvex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed…

    Motivated by a geometric problem, we introduce a new nonconvex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semisupervised extension and provides a natural representative for the clusters as well.

    Other authors
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  • Nonnegative Matrix Factorization of Transition Matrices via Eigenvalue Optimization

    Optimization@NIPS

    We consider the nonnegative matrix factorization (NMF) approach to clustering
    where the matrix to be factorized is a transition matrix for a Markov chain. We
    prove the equivalence of this problem to an eigenvalue optimization problem and
    based on this equivalence, interpret clustering NMF as finding a k-partition of the
    data for which the stationary states of random walkers associated to each component
    are optimally closed. One novel feature of this interpretation is that it…

    We consider the nonnegative matrix factorization (NMF) approach to clustering
    where the matrix to be factorized is a transition matrix for a Markov chain. We
    prove the equivalence of this problem to an eigenvalue optimization problem and
    based on this equivalence, interpret clustering NMF as finding a k-partition of the
    data for which the stationary states of random walkers associated to each component
    are optimally closed. One novel feature of this interpretation is that it simultaneously
    outputs clusters as well as a “local ranking” of the data within each
    cluster, in the sense of PageRank. The local ranking provides label confidences
    and naturally identifies cluster representatives. A relaxed formulation is identified
    and a novel algorithm is proposed, which we show is strictly decreasing and converges
    in a finite number of iterations to a local minimum of the relaxed objective
    function. A semi-supervised version of the algorithm yields excellent results for
    the MNIST handwritten digit dataset. We conclude with an intriguing relationship
    to a reaction-diffusion system for antagonistically-interacting random walkers.

    Other authors
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Projects

  • Sudoku Solver

    Wrote R code for solving Sudoku puzzles using Binary Integer Programming. Deployed using Shiny; link: http://moody-marlin.shinyapps.io/Sudoku_Solver

    See project

Languages

  • English

    Native or bilingual proficiency

  • French

    Limited working proficiency

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