Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase retrieval, and (c) finite reflection groups. Here, we prove that for all three of these cases, the smallest possible distortion is nearly achieved by a composition of a "max filter bank" with a linear transformation. Our proof amounts to a two-step process: first, we show it suffices to demonstrate a certain inclusion of Lipschitz function spaces, and second, we prove that inclusion, using fundamentally different approaches for the three cases. We also show that these cases interact differently with a few related function spaces, which suggests that a unified treatment would be nontrivial.

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 which the ambient vector space is acted on by either a unitary group or general linear group. If isomorphism also allows permutations of the subspaces, then the problem is at least as hard as graph isomorphism. Otherwise, we provide a variety of polynomial-time algorithms with Matlab implementations to test for isomorphism. Keywords: subspace isomorphism, Grassmannian, Bargmann invariants, $H^\ast$-algebras, quivers, graph isomorphism

We resolve a $1,000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his "favorite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we found that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe prove both Hall's and our counterexamples in Lean.

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 features is encoded through an overcomplete dictionary W, and feature recovery is performed by applying ReLU(W^⊤ W x+b) with an appropriate bias vector b. We prove several recovery theorems for this model. In the random-support setting, we establish high-probability support recovery for nearly tight, low-coherence dictionaries, with guarantees when the expected sparsity is up to order d/log n. In the worst-case support setting, we give a sharp and computable criterion for which sparsity levels permit support recovery. We apply this criterion to Gaussian random matrices and equiangular tight frames. For real equiangular tight frames with n>d+1, we determine the exact recovery threshold in terms of the coherence. The proof of this result for real equiangular tight frames relies on a novel characterization—which should be of independent interest to frame theorists—of the distribution of signs in the Gram matrix.

We show that there is no extension of the Naimark complement to arbitrary frames that satisfies three fundamental properties of the Naimark complement of Parseval frames.

We show that there does not exist a complex d× n equiangular tight frame with d^2-d+1<n<d^2. The proof, which originated from an internal model at OpenAI, mimics the relationship between real equiangular tight frames and strongly regular graphs.

When training a neural network for classification, the feature vectors of the training set are known to collapse to the vertices of a regular simplex, provided the dimension d of the feature space and the number n of classes satisfies n≤ d+1. This phenomenon is known as neural collapse. For other applications like language models, one instead takes n≫ d. Here, the neural collapse phenomenon still occurs, but with different emergent geometric figures. We characterize these geometric figures in the orthoplex regime where d+2≤ n≤ 2d. The techniques in our analysis primarily involve Radon's theorem and convexity.

We study the minimum arclength of spherical t-design curves, i.e., closed rectifiable curves on S^d whose normalized arclength measure exactly integrates every polynomial of degree at most t. We prove an explicit spectral lower bound that is sharp for t=1 in all spheres and for t=2 in every odd-dimensional sphere, yielding the first exact optimality results for spherical t-design curves with t>1. For even-dimensional spheres, we construct 2-design curves whose lengths asymptotically match the lower bound as d→∞, and in S^2, we use numerical optimization and the calculus of variations to derive a candidate for the shortest 2-design curve.

Given an inner product space V and a group G of linear isometries, max filtering offers a rich class of convex G-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on R(G), the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space V/G. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in R(G)/G and may be of independent interest. As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp. quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp. quaternionic) representations of the group of unit complex numbers S^1≅SO(2) (resp. unit quaternions S^3≅SU(2)). We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.

We build on the recent breakthrough of Faulhuber, Petersen, van Velthoven, and Voigtlaender that disproved the HRT conjecture with a Schwartz function and a 12-point configuration. We give a human-readable treatment of their mechanism and find HRT counterexample functions with exponential (or faster) decay. By a result of Bownik and Speegle, this is the fastest possible decay for an HRT counterexample, up to a logarithmic factor in the exponent.

For an unknown finite group G of automorphisms of a finite-dimensional Hilbert space, we find sharp bounds on the number of generic G-orbits needed to recover G up to group isomorphism, as well as the number needed to recover G as a concrete set of automorphisms.

Given a finite-dimensional inner product space V and a group G of isometries, we consider the problem of embedding the orbit space V/G into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.

Things can go spectacularly wrong when clustering timeseries data that has been preprocessed with a sliding window. We highlight three surprising failures that emerge depending on how the window size compares with the timeseries length. In addition to computational examples, we present theoretical explanations for each of these failure modes.

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.

We consider the fundamental problem of balanced k-means clustering. In particular, we introduce an optimal transport approach to alternating minimization called BalLOT, and we show that it delivers a fast and effective solution to this problem. We establish this with a variety of numerical experiments before proving several theoretical guarantees. First, we prove that for generic data, BalLOT produces integral couplings at each step. Next, we perform a landscape analysis to provide theoretical guarantees for both exact and partial recoveries of planted clusters under the stochastic ball model. Finally, we propose initialization schemes that achieve one-step recovery of planted clusters.

What fraction of points in a planar Poisson process are not the nearest neighbor of any other point? That is, what is the probability that a given particle is not the nearest neighbor of any other particle in a classical two-dimensional ideal gas model? In 1987, Tao and Wu presented an award-winning tour de force of integration to solve this problem, but in the time since, errors were discovered in their solution, and the technical difficulty precluded researchers from correcting them. In this paper, we rectify this situation by providing a complete solution to the original problem. First, we prove a correct version of Tao and Wu's intended change of variables, and in doing so, we identify an important constraint that was missing in one of the integration variables. Next, we use ideas from Fourier–Motzkin elimination to prove bounds of integration that correctly incorporate this missing constraint. The result is a weighted sum of two double integrals, four quadruple integrals, four sextuple integrals, and two octuple integrals, and we apply a variety of techniques to numerically compute each of these integrals. Considering the error-riddled history of this problem, we conclude by discussing the various ways we verified different portions of our solution.

Motivated by a popular code golf challenge, we review some key ideas from information theory and discuss how to efficiently compress a streaming file with an acceptable error rate.

Given a real inner product space V and a group G of linear isometries, max filtering offers a rich class of G-invariant maps. In this paper, we identify nearly sharp conditions under which these maps injectively embed the orbit space V/G into Euclidean space, and when G is finite, we estimate the map's distortion of the quotient metric. We also characterize when max filtering is a positive definite kernel.

We use dense Sidon sets to construct small weighted projective 2-designs. This represents quantitative progress on Zauner's conjecture.

In this paper, we study approximate Hadamard matrices, that is, well-conditioned n× n matrices with all entries in {±1}. We show that the smallest-possible condition number goes to 1 as n→∞, and we identify some explicit infinite families of approximate Hadamard matrices.