38 citations · 100 across the 18 of their papers we have counts for
28 papers
Transversal Pooling Neural Networks
Emily J. King, Dustin G. Mixon, Michael Perlmutter +1
Many learning tasks require stability to small transformations while retaining sensitivity to larger ones. We introduce \emph{transversal pooling neural networks}, which generalize…
Towards a mathematical theory of superposition
Michael I. Ivanitskiy, John Jasper, Emily J. King +1
We develop a mathematical theory of superposition in neural networks using tools from frame theory and compressed sensing. In our model, a sparse binary vector \(x\) of active feat…
Short spherical -design curves
Emily J. King, Dustin G. Mixon
We study the minimum arclength of spherical -design curves, i.e., closed rectifiable curves on whose normalized arclength measure exactly integrates every polynomial of de…
Dual Graphs of Polyhedral Decompositions for the Detection of Adversarial Attacks
Huma Jamil, Yajing Liu, Christina M. Cole +4
Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. These decompo…
A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science
Lander Ver Hoef, Henry Adams, Emily J. King +1
Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive…
The Flag Median and FlagIRLS
Nathan Mankovich, Emily King, Chris Peterson +1
Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust represent…
Formulating Beurling LASSO for Source Separation via Proximal Gradient Iteration
Sören Schulze, Emily J. King
Beurling LASSO generalizes the LASSO problem to finite Radon measures regularized via their total variation. Despite its theoretical appeal, this space is hard to parametrize, whic…
Blind Source Separation in Polyphonic Music Recordings Using Deep Neural Networks Trained via Policy Gradients
Sören Schulze, Johannes Leuschner, Emily J. King
We propose a method for the blind separation of sounds of musical instruments in audio signals. We describe the individual tones via a parametric model, training a dictionary to ca…
Testing isomorphism between tuples of subspaces
Emily J. King, Dustin G. Mixon, Shayne Waldron
Given two tuples of subspaces, can you tell whether the tuples are isomorphic? We develop theory and algorithms to address this fundamental question. We focus on isomorphisms in wh…
A note on tight projective 2-designs
Joseph W. Iverson, Emily J. King, Dustin G. Mixon
We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimensio…
Grassmannian codes from paired difference sets
Matthew Fickus, Joseph W. Iverson, John Jasper +1
An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal…
Uniquely optimal codes of low complexity are symmetric
Emily J. King, Dustin G. Mixon, Hans Parshall +1
We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predict…
Nonclosedness of Sets of Neural Networks in Sobolev Spaces
Scott Mahan, Emily King, Alex Cloninger
We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly -times differentiable activation function , we constru…
Game of Sloanes: Best known packings in complex projective space
John Jasper, Emily J. King, Dustin G. Mixon
It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations…
Constructing Subspace Packings from Other Packings
Emily J. King
The desirable properties when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identit…
2- and 3-Covariant Equiangular Tight Frames
Emily J. King
Equiangular tight frames (ETFs) are configurations of vectors which are optimally geometrically spread apart and provide resolutions of the identity. Many known constructions of ET…
Edge, Ridge, and Blob Detection with Symmetric Molecules
Rafael Reisenhofer, Emily J. King
We present a novel approach to the detection and characterization of edges, ridges, and blobs in two-dimensional images which exploits the symmetry properties of directionally sens…
Singular Values for ReLU Layers
Sören Dittmer, Emily J. King, Peter Maass
Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by stu…
Sparse Pursuit and Dictionary Learning for Blind Source Separation in Polyphonic Music Recordings
Sören Schulze, Emily J. King
We propose an algorithm for the blind separation of single-channel audio signals. It is based on a parametric model that describes the spectral properties of the sounds of musical…
Equiangular tight frames that contain regular simplices
Matthew Fickus, John Jasper, Emily J. King +1
An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more tha…
A theoretical guarantee for data completion via geometric separation
Emily J. King, James M. Murphy
Scientific and commercial data is often incomplete. Recovery of the missing information is an important pre-processing step in data analysis. Real-world data can in many cases be r…
New Constructions and Characterizations of Flat and Almost Flat Grassmannian Fusion Frames
Emily J. King
Configurations of subspaces like equichordal and equiisoclinic tight fusion frames, which are in some sense optimally spread apart and which also have reconstruction properties emu…
New Upper Bounds for Equiangular Lines by Pillar Decomposition
Emily J. King, Xiaoxian Tang
We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition develope…
Shearlet-Based Detection of Flame Fronts
Rafael Reisenhofer, Johannes Kiefer, Emily J. King
Identifying and characterizing flame fronts is the most common task in the computer-assisted analysis of data obtained from imaging techniques such as planar laser-induced fluoresc…