Related papers: Quantitative Bounds for Sorting-Based Permutation-…
We consider {\em monotone} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean…
Random Projections have been widely used to generate embeddings for various graph learning tasks due to their computational efficiency. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. In this paper,…
Distance measures between graphs are important primitives for a variety of learning tasks. In this work, we describe an unsupervised, optimal transport based approach to define a distance between graphs. Our idea is to derive…
We develop sharp bounds on the statistical distance between high-dimensional permutation mixtures and their i.i.d. counterparts. Our approach establishes a new geometric link between the spectrum of a complex channel overlap matrix and the…
Let $\mathcal{P}$ be an $n$-point subset of Euclidean space and $d\geq 3$ be an integer. In this paper we study the following question: What is the smallest (normalized) relative change of the volume of subsets of $\mathcal{P}$ when it is…
We consider the problem of embedding one i.i.d.\ collection of Bernoulli random variables indexed by $\mathbb{Z}^d$ into an independent copy in an injective $M$-Lipschitz manner. For the case $d=1$, it was shown by Basu and Sly (PTRF, 2014)…
Deep learning-based models have demonstrated remarkable success in solving illposed inverse problems; however, many fail to strictly adhere to the physical constraints imposed by the measurement process. In this work, we introduce a…
Understanding the metric structure of permutation families is fundamental to combinatorics and has applications in social choice theory, bioinformatics, and coding theory. We study permutation families defined by restriction…
Score-based diffusion models have demonstrated remarkable empirical success in learning high-dimensional distributions, particularly those exhibiting low-dimensional and multi-modal structures. However, theoretical understanding of their…
Nondegenerate quadratic forms over $p$-adic fields are classified by their dimension, discriminant, and Hasse invariant. This paper uses these three invariants, elementary facts about $p$-adic fields and the theory of quadratic forms to…
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation,…
Given a cloud of $n$ data points in $\mathbb{R}^d$, consider all projections onto $m$-dimensional subspaces of $\mathbb{R}^d$ and, for each such projection, the empirical distribution of the projected points. What does this collection of…
Let V be a rank N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that S^2 V^* \otimes L is ample. Suppose that the maximum rank of the…
The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces…
This work studies embedding of arbitrary VC classes in well-behaved VC classes, focusing particularly on extremal classes. Our main result expresses an impossibility: such embeddings necessarily require a significant increase in dimension.…
This paper combines two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names…
We study regression of $1$-Lipschitz functions under a log-concave measure $\mu$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are…
We introduce the Linearized Diffusion Map (LDM), a novel linear dimensionality reduction method constructed via a linear approximation of the diffusion-map kernel. LDM integrates the geometric intuition of diffusion-based nonlinear methods…
We consider non-parametric estimation and inference of conditional moment models in high dimensions. We show that even when the dimension $D$ of the conditioning variable is larger than the sample size $n$, estimation and inference is…
We propose a new method to obtain landmark-matching transformations between n-dimensional Euclidean spaces with large deformations. Given a set of feature correspondences, our algorithm searches for an optimal folding-free mapping that…