arXiv research

We present the first tree-based regressor whose convergence rate depends only on the intrinsic dimension of the data, namely its Assouad dimension. The regressor uses the RPtree partitioning procedure, a simple randomized variant of k-d trees.

A new k-means algorithm using cover trees accelerates clustering.

A new clustering algorithm reduces density peaks clustering's computational complexity.

Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees…

We introduce the class $Σ_k(d)$ of $k$-stellated (combinatorial) spheres of dimension $d$ ($0 \leq k \leq d + 1$) and compare and contrast it with the class ${\cal S}_k(d)$ ($0 \leq k \leq d$) of $k$-stacked homology $d$-spheres. We have $Σ_1(d) = {\cal S}_1(d)$, and $Σ_k(d) \subseteq {\cal S}_k(d)$ for $d \geq 2k - 1$…

Approximate nearest neighbor algorithms are used to speed up nearest neighbor search in a wide array of applications. However, current indexing methods feature several hyperparameters that need to be tuned to reach an acceptable accuracy--speed trade-off. A grid search in the parameter space is often impractically slow…

Predicts destinations and routes from partial trajectory data.

We propose a fast, model agnostic method for finding interpretable counterfactual explanations of classifier predictions by using class prototypes. We show that class prototypes, obtained using either an encoder or through class specific k-d trees, significantly speed up the the search for counterfactual instances and …

In this article we continue the study of the geometry of $k$-D'Atri spaces, $$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown that ${\…

A C_k-move is a local move that involves (k+1) strands of a link. A C_k-move is called a C_k^d-move if these (k+1) strands belong to mutually distinct components of a link. Since a C_k^d-move preserves all k-component sublinks of a link, we consider the converse implication: are two links with common k-component sublin…

Let $D$ be a diagram of an alternating knot with unknotting number one. The branched double cover of $S^3$ branched over $D$ is an L-space obtained by half integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by $\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a satellite k…

We consider a finite simplicial complex $K$ together with its successive barycentric subdivisions $Sd^d(K), d\geq0,$ and study the expected topology of a random subcomplex in $Sd^d(K), d\gg0$. We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse …

We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show th…

This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category. The geometric r…

The paper introduces explainable $k$-means with axis-parallel hyperplanes for $d$-dimensional data.

Study Bergman metric on Cartan-Hartogs domains and their duals.

We propose a novel nonparametric online predictor for discrete labels conditioned on multivariate continuous features. The predictor is based on a feature space discretization induced by a full-fledged k-d tree with randomly picked directions and a recursive Bayesian distribution, which allows to automatically learn th…

Consider a divisor D with simple normal crossings in a compact Kähler manifold X. We show in this article that a Kähler metric in an arbitrary class, with constant scalar curvature and cusp singularities along the divisor is unique in this class when K[D] is ample. This we do by generalizing Chen's construction of appr…

New algorithm learns halfspaces almost optimally with fewer queries.

Let EMBED(k,d) be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R^d? Known results easily imply polynomiality of EMBED(k,2) (k=1,2; the case k=1, d=2 is graph planarity) and of EMBED(k,2k) for all k>2 (even if k i…

Both supervised and unsupervised machine learning algorithms have been used to learn partition-based index structures for approximate nearest neighbor (ANN) search. Existing supervised algorithms formulate the learning task as finding a partition in which the nearest neighbors of a training set point belong to the same…

GraphGP: Scalable Gaussian Processes with Vecchia's Approximation

Let $M$ be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form $R^4_2(c)$ with constant sectional curvature $c$ and index two. In the first part of this article, we prove that the Gauss curvature $K$, the normal curvature $K^D$, and mean curvature vector $H$ of $M$ satisfy the general inequali…

We introduce the $k$-stellated spheres and compare and contrast them with $k$-stacked spheres. It is shown that for $d \geq 2k$, any $k$-stellated sphere of dimension $d$ bounds a unique and canonically defined $k$-stacked ball. In parallel, any $k$-stacked polytopal sphere of dimension $d\geq 2k$ bounds a unique and c…

Wintgen proved in [P. Wintgen, Sur l'inégalité de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature $K$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $K+|K^D|\leq H^2,$ where $H^2$ is the squared mean curvature. A surface $M$ in $\E4$ is called …

Study phase transitions in noisy transformer dynamics on spheres.

New bounds on learning from multiple distributions for VC classes.

Denoting by $C_n(X)$ the configuration space of $n$ distinct points in $X$, with $X$ being either Euclidean $3$-space $\mathbb{E}^3$ or hyperbolic $3$-space $\mathbb{H}^3$ or $\mathbb{C}P^1$ , by $\mathscr{P}_{k,d}$ the vector space of homogeneous complex polynomials in the variables $z_0, \ldots, z_k$ of degree $d$, a…

New method generates private synthetic data with optimal utility for smooth queries.

New proof shows incremental flow models are essential for universal generation.

We consider a stochastic continuum armed bandit problem where the arms are indexed by the $\ell_2$ ball $B_{d}(1+ν)$ of radius $1+ν$ in $\mathbb{R}^d$. The reward functions $r :B_{d}(1+ν) \rightarrow \mathbb{R}$ are considered to intrinsically depend on $k \ll d$ unknown linear parameters so that $r(\mathbf{x}) = g(\ma…

Improved algorithm for bandits with delayed feedback, combining adversarial and stochastic performance.

For $d \geq 2$, Walkup's class ${\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d \geq 4$, all connected members of ${\cal K}(d)$ are obtained from stacked $d$-spheres by finitely many elementary handle additions. According to …

K-means -- and the celebrated Lloyd algorithm -- is more than the clustering method it was originally designed to be. It has indeed proven pivotal to help increase the speed of many machine learning and data analysis techniques such as indexing, nearest-neighbor search and prediction, data compression; its beneficial u…

Entity resolution (ER; also known as record linkage or de-duplication) is the process of merging noisy databases, often in the absence of unique identifiers. A major advancement in ER methodology has been the application of Bayesian generative models, which provide a natural framework for inferring latent entities with…

Neural networks with ReLU^k approximate Sobolev functions efficiently via Radon transform.

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry. Given a closed spin manifold M with fundamental group G, Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to the surgery exact sequence in topology. It calculates a structure gr…

One of the most important problems of data processing in high energy and nuclear physics is the event reconstruction. Its main part is the track reconstruction procedure which consists in looking for all tracks that elementary particles leave when they pass through a detector among a huge number of points, so-called hi…

The paper proves a bound on eigenvalues for surfaces embedded in 3D space.

The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bun…

In this paper, we analyze the theory of meromorphic $(1,0)$-forms $ω\in\mathcal{M}Ω^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as poles, where each is counted acc…

Graph Laplacians converge under symmetric divergence conditions.

New analysis improves accuracy of Newton step and influence function data attributions.

Correlation matrices are omnipresent in multivariate data analysis. When the number d of variables is large, the sample estimates of correlation matrices are typically noisy and conceal underlying dependence patterns. We consider the case when the variables can be grouped into K clusters with exchangeable dependence; t…

Efficiently approximates Sparse PCA with significant speedups and minor error.

It was shown recently that the $K$ L1-norm principal components (L1-PCs) of a real-valued data matrix $\mathbf X \in \mathbb R^{D \times N}$ ($N$ data samples of $D$ dimensions) can be exactly calculated with cost $\mathcal{O}(2^{NK})$ or, when advantageous, $\mathcal{O}(N^{dK - K + 1})$ where $d=\mathrm{rank}(\mathbf …

One of the most important problems of data processing in high energy and nuclear physics is the event reconstruction. Its main part is the track reconstruction procedure which consists in looking for all tracks that elementary particles leave when they pass through a detector among a huge number of points, so-called hi…

We introduce canonical measures on a locally finite simplicial complex $K$ and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the $d^{th}$ barycentric subdivision $Sd^d(K)$ of $K$, $d\gg0$. It is a…