arXiv research

Introduces bounded scale measure and generalizes property A.

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gr…

Introduces resemblance structure for large scale geometry.

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

We introduce the notion of large scale inductive dimension for asymptotic resemblance spaces. We prove that the large scale inductive dimension and the asymptotic dimensiongrad are equal in the class of r-convex metric spaces. This class contains the class of all geodesic metric spaces and all finitely generated groups…

Decomposes ultrametric spaces into scaled simplices.

ParK efficiently solves kernel ridge regression for large datasets.

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the h…

SEMASIA provides a large dataset of latent representations for model comparison.

FibeRed reduces complex data dimensions while preserving topology.

This paper optimizes slate decision systems for large action spaces.

Proposes GBBHE for efficient large-scale regression.

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zi…

Chemical space is so large that brute force searches for new interesting molecules are infeasible. High-throughput virtual screening via computer cluster simulations can speed up the discovery process by collecting very large amounts of data in parallel, e.g., up to hundreds or thousands of parallel measurements. Bayes…

Efficient RL in large POMDPs with latent determinism and embeddings.

String kernels are attractive data analysis tools for analyzing string data. Among them, alignment kernels are known for their high prediction accuracies in string classifications when tested in combination with SVM in various applications. However, alignment kernels have a crucial drawback in that they scale poorly du…

Contractible Vietoris-Rips complexes for integer n proved using discrete Morse theory.

Uniformity and proximity are two different ways for defining small scale structures on a set. Coarse structures are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the concept of asymptotic resemblance as a relation between subsets of a set to define a…

CauScale efficiently discovers causal relationships in large graphs.

New method speeds up Gaussian process training and inference for large datasets.

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an…

Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and Čech complexes of $X/G$. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to z…

The paper introduces sections in metric spaces with properties related to Ahlfors-David regularity and convexity.

Bipartite graphs have been used to represent data relationships in many data-mining applications such as in E-commerce recommendation systems. Since learning in graph space is more complicated than in Euclidian space, recent studies have extensively utilized neural nets to effectively and efficiently embed a graph's no…

Large-scale Gaussian process inference has long faced practical challenges due to time and space complexity that is superlinear in dataset size. While sparse variational Gaussian process models are capable of learning from large-scale data, standard strategies for sparsifying the model can prevent the approximation of …

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

Deep Retrieval learns a retrievable structure for efficient large-scale recommendations.

Prove first-band large-diameter asymptotics for Dirichlet spectrum on horoconvex domains in real hyperbolic space.

We propose a new method for input variable selection in nonlinear regression. The method is embedded into a kernel regression machine that can model general nonlinear functions, not being a priori limited to additive models. This is the first kernel-based variable selection method applicable to large datasets. It sides…

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

Representations of probability measures in reproducing kernel Hilbert spaces provide a flexible framework for fully nonparametric hypothesis tests of independence, which can capture any type of departure from independence, including nonlinear associations and multivariate interactions. However, these approaches come wi…

Efficiently attacks large-scale graphs without using the whole graph.

This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona $ν(X)$ of $X$. The first …

New method improves Kalman filtering and smoothing for large state spaces.

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left an…

Characterizes fundamental groups of disjointly tree-graded spaces.

New method speeds up learning of complex dynamical systems.

Interactive machine learning improves learning efficiency with user input.

Develops a new framework for large-scale geometry.

New method discovers causal relationships in large-scale data.

New coarse LS-category introduced for groups and spaces.

Multi-label classification has received considerable interest in recent years. Multi-label classifiers have to address many problems including: handling large-scale datasets with many instances and a large set of labels, compensating missing label assignments in the training set, considering correlations between labels…

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …

A new framework scales active search for large datasets.

Study large-scale geometry of graph braid groups via cubical structures.

We derive scaling laws for optimizing neural networks in hardware.

Proposes SSL framework for improving item recommendations in large-scale systems.

We study the concept of coarse disjointness and large scale $n$-to-$1$ functions. As a byproduct, we obtain an Ostrand-type characterization of asymptotic dimension for coarse structures. It is shown that properties like finite asymptotic dimension, coarse finitism, large scale weak paracompactness, ect. are all invari…