Related papers: Polytopes, lattices, and spherical codes for the n…
In this paper, we consider the problem of classification of $M$ high dimensional queries $y^1,\cdots,y^M\in B^S$ to $N$ high dimensional classes $x^1,\cdots,x^N\in A^S$ where $A$ and $B$ are discrete alphabets and the probabilistic model…
We introduce our new binary tree code for neighbour search and gravitational force calculations in an N-particle system. The tree is built in a "top-down" fashion by "recursive coordinate bisection" where on each tree level we split the…
Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is…
Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces $H_+(\mathbb{S})$ and $H_-(\mathbb{S})$, respectively, play a role in various inverse potential field problems since…
Modern neural network technologies, including large language models, have achieved remarkable success in various applied artificial intelligence applications, however, they face a range of fundamental limitations. Among them are…
We report our experiences for the development of a neighborhood algorithm implemented via tree-codes to optimize the performance of a discrete element method (DEM) for convex polytopes. Our implementation of the two-dimensional tree code…
Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with…
We propose an extension of tree-based space-partitioning indexing structures for data with low intrinsic dimensionality embedded in a high dimensional space. We call this extension an Angle Tree. Our extension can be applied to both…
Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid…
This is an algorithm for finding neighbors when the objects can freely move and have no predefined position. The query consists in finding neighbors for a center location and a given radius. Space is discretized in cubic cells. This…
We address the problem of converting large-scale high-dimensional image data into binary codes so that approximate nearest-neighbor search over them can be efficiently performed. Different from most of the existing unsupervised approaches…
We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also…
The era of Big Data has spawned unprecedented interests in developing hashing algorithms for efficient storage and fast nearest neighbor search. Most existing work learn hash functions that are numeric quantizations of feature values in…
Emerging location-based systems and data analysis frameworks requires efficient management of spatial data for approximate and exact search. Exact similarity search can be done using space partitioning data structures, such as Kd-tree,…
In this work, we exhibit a hierarchy of polynomial time algorithms solving approximate variants of the Closest Vector Problem (CVP). Our first contribution is a heuristic algorithm achieving the same distance tradeoff as HSVP algorithms,…
Algorithms often carry out equally many computations for "easy" and "hard" problem instances. In particular, algorithms for finding nearest neighbors typically have the same running time regardless of the particular problem instance. In…
We present a new algorithm for the approximate near neighbor problem that combines classical ideas from group testing with locality-sensitive hashing (LSH). We reduce the near neighbor search problem to a group testing problem by…
Lattice-based cryptography has recently emerged as a prime candidate for efficient and secure post-quantum cryptography. The two main hard problems underlying its security are the shortest vector problem (SVP) and the closest vector problem…
We introduce a new variant of the nearest neighbor search problem, which allows for some coordinates of the dataset to be arbitrarily corrupted or unknown. Formally, given a dataset of $n$ points $P=\{ x_1,\ldots, x_n\}$ in high-dimensions,…
Point location problems for $n$ points in $d$-dimensional Euclidean space (and $\ell_p$ spaces more generally) have typically had two kinds of running-time solutions: * (Nearly-Linear) less than $d^{poly(d)} \cdot n \log^{O(d)} n$ time, or…