Chris White
Half Moon Bay, California, United States
2K followers
500+ connections
About
Mathematician turned software engineer turned CTO, passionate about building…
Activity
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Submitting high quality documentation to an open-source project is an S-tier form of contribution - thank you Ben Epstein!!
Submitting high quality documentation to an open-source project is an S-tier form of contribution - thank you Ben Epstein!!
Shared by Chris White
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Naming things is hard, and the hardest part of every week is working with the Prefect OSS team to come up with fun release titles. Today we…
Naming things is hard, and the hardest part of every week is working with the Prefect OSS team to come up with fun release titles. Today we…
Shared by Chris White
Experience
Education
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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
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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 authorsSee publication -
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 authorsSee publication -
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 authorsSee publication
Projects
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Sudoku Solver
Wrote R code for solving Sudoku puzzles using Binary Integer Programming. Deployed using Shiny; link: http://moody-marlin.shinyapps.io/Sudoku_Solver
Languages
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English
Native or bilingual proficiency
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French
Limited working proficiency
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