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This paper investigates the synchronization problems for general high-dimensional linear networks over finite fields. By using the technique of linear transformations and invariant subspaces for linear spaces over finite fields, several…

Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding…

Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the…

The nearest neighbor problem is defined as follows: Given a set $P$ of $n$ points in some metric space $(X,D)$, build a data structure that, given any point $q$, returns a point in $P$ that is closest to $q$ (its "nearest neighbor" in $P$).…

Binary optimization, a representative subclass of discrete optimization, plays an important role in mathematical optimization and has various applications in computer vision and machine learning. Usually, binary optimization problems are…

Tensor network contractions are widely used in statistical physics, quantum computing, and computer science. We introduce a method to efficiently approximate tensor network contractions using low-rank approximations, where each intermediate…

Community detection in graphs has many important and fundamental applications including in distributed systems, compression, image segmentation, divide-and-conquer graph algorithms such as nested dissection, document and word clustering,…

Proximity graph-based methods have emerged as a leading paradigm for approximate nearest neighbor (ANN) search in the system community. This paper presents fresh insights into the theoretical foundation of these methods. We describe an…

We propose efficient algorithms for enumerating maximal common subsequences (MCSs) of two strings. Efficiency of the algorithms are estimated by the preprocessing-time, space, and delay-time complexities. One algorithm prepares a…

In this paper we accomplish the development of the fast rank-adaptive solver for tensor-structured symmetric positive definite linear systems in higher dimensions. In [arXiv:1301.6068] this problem is approached by alternating minimization…

Metric embeddings are a widely used method in algorithm design, where generally a ``complex'' metric is embedded into a simpler, lower-dimensional one. Historically, the theoretical computer science community has focused on bi-Lipschitz…

Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and…

We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small…

In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain…

A linear time algorithm to find a set of nearest elements in a mesh.

Big data is becoming ever more ubiquitous, ranging over massive video repositories, document corpuses, image sets and Internet routing history. Proximity search and clustering are two algorithmic primitives fundamental to data analysis, but…

We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…

Differential network is an important tool to capture the changes of conditional correlations under two sample cases. In this paper, we introduce a fast iterative algorithm to recover the differential network for high-dimensional data. The…

The Lipschitz constant is an important quantity that arises in analysing the convergence of gradient-based optimization methods. It is generally unclear how to estimate the Lipschitz constant of a complex model. Thus, this paper studies an…

We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bounds on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite…