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06/09/2023
Spectrahedral Geometry of Graph Sparsifiers
We propose an approach to graph sparsification based on the idea of pres...
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10/06/2022
Magnetic Schrödinger operators and landscape functions
We study localization properties of low-lying eigenfunctions of magnetic...
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08/13/2022
May the force be with you
Modern methods in dimensionality reduction are dominated by nonlinear at...
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07/07/2022
Approximate Solutions of Linear Systems at a Universal Rate
Let A ∈ℝ^n × n be invertible, x ∈ℝ^n unknown and b =Ax given. We are int...
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06/10/2022
Random Walks, Equidistribution and Graphical Designs
Let G=(V,E) be a d-regular graph on n vertices and let μ_0 be a probabil...
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04/28/2022
Sums of Distances on Graphs and Embeddings into Euclidean Space
Let G=(V,E) be a finite, connected graph. We consider a greedy selection...
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12/03/2021
Fundamental component enhancement via adaptive nonlinear activation functions
In many real world oscillatory signals, the fundamental component of a s...
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07/30/2021
A common variable minimax theorem for graphs
Let 𝒢 = {G_1 = (V, E_1), …, G_m = (V, E_m)} be a collection of m graphs ...
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07/12/2021
Quantile-Based Random Kaczmarz for corrupted linear systems of equations
We consider linear systems Ax = b where A ∈ℝ^m × n consists of normalize...
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05/17/2021
How well-conditioned can the eigenvalue problem be?
The condition number for eigenvalue computations is a well–studied qua...
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04/29/2021
A 0.502·MaxCut Approximation using Quadratic Programming
We study the MaxCut problem for graphs G=(V,E). The problem is NP-hard, ...
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03/19/2021
Refined Least Squares for Support Recovery
We study the problem of exact support recovery based on noisy observatio...
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02/25/2021
t-SNE, Forceful Colorings and Mean Field Limits
t-SNE is one of the most commonly used force-based nonlinear dimensional...
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02/09/2021
Max-Cut via Kuramoto-type Oscillators
We consider the Max-Cut problem. Let G = (V,E) be a graph with adjacency...
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12/15/2020
Neural Collapse with Cross-Entropy Loss
We consider the variational problem of cross-entropy loss with n feature...
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10/28/2020
Fast localization of eigenfunctions via smoothed potentials
We study the problem of predicting highly localized low-lying eigenfunct...
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09/03/2020
Surrounding the solution of a Linear System of Equations from all sides
Suppose A ∈ℝ^n × n is invertible and we are looking for the solution of ...
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08/03/2020
Using Expander Graphs to test whether samples are i.i.d
The purpose of this note is to point out that the theory of expander gra...
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07/27/2020
Stochastic Gradient Descent applied to Least Squares regularizes in Sobolev spaces
We study the behavior of stochastic gradient descent applied to Ax -b _2...
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07/06/2020
A Weighted Randomized Kaczmarz Method for Solving Linear Systems
The Kaczmarz method for solving a linear system Ax = b interprets such a...
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06/30/2020
Randomized Kaczmarz converges along small singular vectors
Randomized Kaczmarz is a simple iterative method for finding solutions o...
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04/02/2020
A Spectral Approach to the Shortest Path Problem
Let G=(V,E) be a simple, connected graph. One is often interested in a s...
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03/22/2020
Spectral Clustering Revisited: Information Hidden in the Fiedler Vector
We are interested in the clustering problem on graphs: it is known that ...
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03/16/2020
Randomly Aggregated Least Squares for Support Recovery
We study the problem of exact support recovery: given an (unknown) vecto...
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03/02/2020
Regularized Potentials of Schrödinger Operators and a Local Landscape Function
We study localization properties of low-lying eigenfunctions (-Δ +V...
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02/27/2020
The Spectral Underpinning of word2vec
word2vec due to Mikolov et al. (2013) is a word embedding method that is...
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01/05/2020
Non-Convex Planar Harmonic Maps
We formulate a novel characterization of a family of invertible maps bet...
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09/19/2019
On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure
We discuss the classical problem of measuring the regularity of distribu...
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02/15/2019
Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations
T-distributed stochastic neighbour embedding (t-SNE) is a widely used da...
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01/15/2019
A forgotten Theorem of Schoenberg on one-sided integral averages
Let f:R→R be a function for which we want to take local averages. Assumi...
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06/28/2018
Recovering Trees with Convex Clustering
Convex clustering refers, for given {x_1, ..., x_n}⊂^p, to the minimizat...
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04/25/2018
On the Dual Geometry of Laplacian Eigenfunctions
We discuss the geometry of Laplacian eigenfunctions -Δϕ = λϕ on compact ...
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03/19/2018
Numerical Integration on Graphs: where to sample and how to weigh
Let G=(V,E,w) be a finite, connected graph with weighted edges. We are i...
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12/25/2017
Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding
t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for ...
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11/13/2017
Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science
If we pick n random points uniformly in [0,1]^d and connect each point t...
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06/08/2017
Clustering with t-SNE, provably
t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering an...
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06/05/2017
The Geometry of Nodal Sets and Outlier Detection
Let (M,g) be a compact manifold and let -Δϕ_k = λ_k ϕ_k be the sequence ...
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02/09/2017
Stochastic Neighbor Embedding separates well-separated clusters
Stochastic Neighbor Embedding and its variants are widely used dimension...
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11/09/2016
On the Diffusion Geometry of Graph Laplacians and Applications
We study directed, weighted graphs G=(V,E) and consider the (not necessa...
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07/15/2016