arXiv subjects

Hengjie Zhang

Publications and source records attributed to Hengjie Zhang.

10 recordsLinked to original sources

arXiv2024

Faster Algorithms for Average-Case Orthogonal Vectors and Closest Pair Problems

We study the average-case version of the Orthogonal Vectors problem, in which one is given as input $n$ vectors from $\{0,1\}^d$ which are chosen randomly so that each coordinate is $1$ independently with probability $p$. Kane and Williams [ITCS 2019] showed how to solve this problem in time $O(n^{2 - δ_p})$ for a constant $δ_p > 0$ that depends only on $p$. However, it was previously unclear how to solve the problem faster in the hardest parameter regime where $p$ may depend on $d$. The best prior algorithm was the best worst-case algorithm by Abboud, Williams and Yu [SODA 2014], which in dimension $d = c \cdot \log n$, solves the problem in time $n^{2 - Ω(1/\log c)}$. In this paper, we give a new algorithm which improves this to $n^{2 - Ω(\log\log c /\log c)}$ in the average case for any parameter $p$. As in the prior work, our algorithm uses the polynomial method. We make use of a very simple polynomial over the reals, and use a new method to analyze its performance based on computing how its value degrades as the input vectors get farther from orthogonal. To demonstrate the generality of our approach, we also solve the average-case version of the closest pair problem in the same running time.

cs.DS↗

arXiv2024

A Dynamic Low-Rank Fast Gaussian Transform

The \emph{Fast Gaussian Transform} (FGT) enables subquadratic-time multiplication of an $n\times n$ Gaussian kernel matrix $\mathsf{K}_{i,j}= \exp ( - \| x_i - x_j \|_2^2 ) $ with an arbitrary vector $h \in \mathbb{R}^n$, where $x_1,\dots, x_n \in \mathbb{R}^d$ are a set of \emph{fixed} source points. This kernel plays a central role in machine learning and random feature maps. Nevertheless, in most modern data analysis applications, datasets are dynamically changing (yet often have low rank), and recomputing the FGT from scratch in (kernel-based) algorithms incurs a major computational overhead ($\gtrsim n$ time for a single source update $\in \mathbb{R}^d$). These applications motivate a \emph{dynamic FGT} algorithm, which maintains a dynamic set of sources under \emph{kernel-density estimation} (KDE) queries in \emph{sublinear time} while retaining Mat-Vec multiplication accuracy and speed. Assuming the dynamic data-points $x_i$ lie in a (possibly changing) $k$-dimensional subspace ($k\leq d$), our main result is an efficient dynamic FGT algorithm, supporting the following operations in $\log^{O(k)}(n/\varepsilon)$ time: (1) Adding or deleting a source point, and (2) Estimating the ``kernel-density'' of a query point with respect to sources with $\varepsilon$ additive accuracy. The core of the algorithm is a dynamic data structure for maintaining the \emph{projected} ``interaction rank'' between source and target boxes, decoupled into finite truncation of Taylor and Hermite expansions.

cs.DS↗

arXiv2023

Generalizations of Matrix Multiplication can solve the Light Bulb Problem

In the light bulb problem, one is given uniformly random vectors $x_1, \ldots, x_n, y_1, \ldots, y_n \in \{-1,1\}^d$. They are all chosen independently except a planted pair $(x_{i^*}, y_{j^*})$ is chosen with correlation $ρ>0$. The goal is to find the planted pair. This problem was introduced over 30 years ago by L.~Valiant, and is known to have many applications in data analysis, statistics, and learning theory. The naive algorithm runs in $Ω(n^2)$ time, and algorithms based on Locality-Sensitive Hashing approach quadratic time as $ρ\to 0$. In 2012, G.~Valiant gave a breakthrough algorithm using fast matrix multiplication that runs in time $O(n^{(5-ω)/(4-ω)}) < O(n^{1.615})$, no matter how small $ρ>0$ is. This was subsequently refined by Karppa, Kaski, and Kohonen in 2016 to $O(n^{2 ω/ 3}) < O(n^{1.582})$. In this paper, we propose a new approach which can replace matrix multiplication tensor with other tensors. Those tensors can omit some terms one is supposed to compute, and include additional error terms. Our new approach can make use of any tensors which previously had no known algorithmic applications, including tensors which arise naturally as intermediate steps in border rank methods and in the Laser method. We further show that our approach can be combined with locality-sensitive hashing to design an algorithm whose running time improves as $ρ$ gets larger. To our knowledge, this is the first algorithm which combines fast matrix multiplication with hashing for the light bulb problem or any closest pair problem, and it leads to faster algorithms for small $ρ>0$. We also introduce a new tensor $T_{2112}$, which has the same size of $2 \times 2$ matrix multiplication tensor, but runs faster than the Strassen's algorithm for light bulb problem.

cs.DS↗

arXiv2023

Sub-quadratic (1+\eps)-approximate Euclidean Spanners, with Applications

We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch <\sqrt{2} and subquadratic size are not possible, even if we add Steiner points. On the other hand, if we add extra nodes to the graph (non-metric Steiner points), then we can obtain (1+\eps)-approximate spanners of subquadratic size. We show how to construct a spanner of size n^{2-Ω(\eps^3)}, as well as a directed version of the spanner of size n^{2-Ω(\eps^2)}. We use our directed spanner to obtain an algorithm for computing (1+\eps)-approximation to Earth-Mover Distance (optimal transport) between two sets of size n in time n^{2-Ω(\eps^2)}.

cs.CG↗

arXiv2023

Space-Efficient Interior Point Method, with applications to Linear Programming and Maximum Weight Bipartite Matching

We study the problem of solving linear program in the streaming model. Given a constraint matrix $A\in \mathbb{R}^{m\times n}$ and vectors $b\in \mathbb{R}^m, c\in \mathbb{R}^n$, we develop a space-efficient interior point method that optimizes solely on the dual program. To this end, we obtain efficient algorithms for various different problems: * For general linear programs, we can solve them in $\widetilde O(\sqrt n\log(1/ε))$ passes and $\widetilde O(n^2)$ space for an $ε$-approximate solution. To the best of our knowledge, this is the most efficient LP solver in streaming with no polynomial dependence on $m$ for both space and passes. * For bipartite graphs, we can solve the minimum vertex cover and maximum weight matching problem in $\widetilde O(\sqrt{m})$ passes and $\widetilde O(n)$ space. In addition to our space-efficient IPM, we also give algorithms for solving SDD systems and isolation lemma in $\widetilde O(n)$ spaces, which are the cornerstones for our graph results.

cs.DS↗

arXiv2022

Tight Revenue Gaps among Multi-Unit Mechanisms

This paper considers Bayesian revenue maximization in the $k$-unit setting, where a monopolist seller has $k$ copies of an indivisible item and faces $n$ unit-demand buyers (whose value distributions can be non-identical). Four basic mechanisms among others have been widely employed in practice and widely studied in the literature: {\sf Myerson Auction}, {\sf Sequential Posted-Pricing}, {\sf $(k + 1)$-th Price Auction with Anonymous Reserve}, and {\sf Anonymous Pricing}. Regarding a pair of mechanisms, we investigate the largest possible ratio between the two revenues (a.k.a.\ the revenue gap), over all possible value distributions of the buyers. Divide these four mechanisms into two groups: (i)~the discriminating mechanism group, {\sf Myerson Auction} and {\sf Sequential Posted-Pricing}, and (ii)~the anonymous mechanism group, {\sf Anonymous Reserve} and {\sf Anonymous Pricing}. Within one group, the involved two mechanisms have an asymptotically tight revenue gap of $1 + Θ(1 / \sqrt{k})$. In contrast, any two mechanisms from the different groups have an asymptotically tight revenue gap of $Θ(\log k)$.

cs.GT↗

arXiv2020

Distributed Linguistic Representations in Decision Making: Taxonomy, Key Elements and Applications, and Challenges in Data Science and Explainable Artificial Intelligence

Distributed linguistic representations are powerful tools for modelling the uncertainty and complexity of preference information in linguistic decision making. To provide a comprehensive perspective on the development of distributed linguistic representations in decision making, we present the taxonomy of existing distributed linguistic representations. Then, we review the key elements of distributed linguistic information processing in decision making, including the distance measurement, aggregation methods, distributed linguistic preference relations, and distributed linguistic multiple attribute decision making models. Next, we provide a discussion on ongoing challenges and future research directions from the perspective of data science and explainable artificial intelligence.

cs.AI↗

arXiv2020

Faster Dynamic Matrix Inverse for Faster LPs

Motivated by recent Linear Programming solvers, we design dynamic data structures for maintaining the inverse of an $n\times n$ real matrix under $\textit{low-rank}$ updates, with polynomially faster amortized running time. Our data structure is based on a recursive application of the Woodbury-Morrison identity for implementing $\textit{cascading}$ low-rank updates, combined with recent sketching technology. Our techniques and amortized analysis of multi-level partial updates, may be of broader interest to dynamic matrix problems. This data structure leads to the fastest known LP solver for general (dense) linear programs, improving the running time of the recent algorithms of (Cohen et al.'19, Lee et al.'19, Brand'20) from $O^*(n^{2+ \max\{\frac{1}{6}, ω-2, \frac{1-α}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, ω-2, \frac{1-α}{2}\}})$, where $ω$ and $α$ are the fast matrix multiplication exponent and its dual. Hence, under the common belief that $ω\approx 2$ and $α\approx 1$, our LP solver runs in $O^*(n^{2.055})$ time instead of $O^*(n^{2.16})$.

cs.DS↗

arXiv2018

Distributed Triangle Detection via Expander Decomposition

We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $\tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $\tilde{O}(n^{2/3})$ and $\tilde{O}(n^{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $\tilde{O}(n^{1-δ})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_m\cup E_s\cup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $Ω(n^δ)$ and conductance $Ω(1/\text{poly} \log(n))$. As a consequence the mixing time of a random walk within the component is $O(\text{poly} \log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n^δ$. (c) $|E_r| \leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.

cs.DS↗

arXiv2018

Improved Algorithms for Fully Dynamic Maximal Independent Set

Maintaining maximal independent set in dynamic graph is a fundamental open problem in graph theory and the first sublinear time deterministic algorithm was came up by Assadi, Onak, Schieber and Solomon(STOC'18), which achieves $O(m^{3/4})$ amortized update time. We have two main contributions in this paper. We present a new simple deterministic algorithm with $O(m^{2/3}\sqrt{\log m})$ amortized update time, which improves the previous best result. And we also present the first randomized algorithm with expected $O(\sqrt{m}\log^{1.5}m)$ amortized time against an oblivious adversary.

cs.DS↗