We find a succinct expression for computing the sequence $x_t = a_t x_{t-1} + b_t$ in parallel with two prefix sums, given $t = (1, 2, \dots, n)$, $a_t \in \mathbb{R}^n$, $b_t \in \mathbb{R}^n$, and initial value $x_0 \in \mathbb{R}$. On $n$ parallel processors, the computation of $n$ elements incurs $\mathcal{O}(\log n)$ time and $\mathcal{O}(n)$ space. Sequences of this form are ubiquitous in science and engineering, making efficient parallelization useful for a vast number of applications. We implement our expression in software, test it on parallel hardware, and verify that it executes faster than sequential computation by a factor of $\frac{n}{\log n}$.
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We propose a new approach for approximating functions in $C([0,1]^d)$ via Kolmogorov superposition theorem (KST) based on the linear spline approximation of the K-outer function in Kolmogorov superposition representation. We improve the results in \cite{LaiShenKST21} by showing that the optimal approximation rate based on our proposed approach is $\mathcal{O}(\frac{1}{n^2})$, with $n$ being the number of knots over $[0,1]$, and the approximation constant increases linearly in $d$. We show that there is a dense subclass in $C([0,1]^d)$ whose approximation can achieve such optimal rate, and the number of parameters needed in such approximation is at most $\mathcal{O}(nd)$. Moreover, for $d\geq 4$, we apply the tensor product spline denoising technique to smooth the KB-splines and get the corresponding LKB-splines. We use those LKB-splines as the basis to approximate functions for the cases when $d=4$ and $d=6$, which extends the results in \cite{LaiShenKST21} for $d=2$ and $d=3$. Based on the idea of pivotal data locations introduced in \cite{LaiShenKST21}, we validate via numerical experiments that fewer than $\mathcal{O}(nd)$ function values are enough to achieve the approximation rates such as $\mathcal{O}(\frac{1}{n})$ or $\mathcal{O}(\frac{1}{n^2})$ based on the smoothness of the K-outer function. Finally, we demonstrate that our approach can be applied to numerically solving partial differential equation such as the Poisson equation with accurate approximation results.
We show that $n$-bit integers can be factorized by independently running a quantum circuit with $\tilde{O}(n^{3/2})$ gates for $\sqrt{n}+4$ times, and then using polynomial-time classical post-processing. The correctness of the algorithm relies on a number-theoretic heuristic assumption reminiscent of those used in subexponential classical factorization algorithms. It is currently not clear if the algorithm can lead to improved physical implementations in practice.
This paper investigates the spectral norm version of the column subset selection problem. Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a positive integer $k\leq\text{rank}(\mathbf{A})$, the objective is to select exactly $k$ columns of $\mathbf{A}$ that minimize the spectral norm of the residual matrix after projecting $\mathbf{A}$ onto the space spanned by the selected columns. We use the method of interlacing polynomials introduced by Marcus-Spielman-Srivastava to derive a new upper bound on the minimal approximation error. This new bound is asymptotically sharp when the matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ obeys a spectral power-law decay. The relevant expected characteristic polynomials can be written as an extension of the expected polynomial for the restricted invertibility problem, incorporating two extra variable substitution operators. Finally, we propose a deterministic polynomial-time algorithm that achieves this error bound up to a computational error.
An infinite sequence of sets $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ is said to be a heterochromatic sequence from an infinite sequence of families $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$, if there exists a strictly increasing sequence of natural numbers $\left\{ i_{n}\right\}_{n \in \mathbb{N}}$ such that for all $n \in \mathbb{N}$ we have $B_{n} \in \mathcal{F}_{i_{n}}$. In this paper, we have proved that if for each $n\in\mathbb{N}$, $\mathcal{F}_n$ is a family of {\em nicely shaped} convex sets in $\mathbb{R}^d$ such that each heterochromatic sequence $\left\{B_{n}\right\}_{n\in\mathbb{N}}$ from $\left\{ \mathcal{F}_{n} \right\}_{n \in \mathbb{N}}$ contains at least $k+2$ sets that can be pierced by a single $k$-flat ($k$-dimensional affine space) then all but finitely many families in $\left\{\mathcal{F}_{n}\right\}_{n\in \mathbb{N}}$ can be pierced by finitely many $k$-flats. This result can be considered as a {\em countably colorful} generalization of the $(\aleph_0, k+2)$-theorem proved by Keller and Perles (Symposium on Computational Geometry 2022). We have also established the tightness of our result by proving a number of no-go theorems.
In an instance of the weighted Nash Social Welfare problem, we are given a set of $m$ indivisible items, $\mathscr{G}$, and $n$ agents, $\mathscr{A}$, where each agent $i \in \mathscr{A}$ has a valuation $v_{ij}\geq 0$ for each item $j\in \mathscr{G}$. In addition, every agent $i$ has a non-negative weight $w_i$ such that the weights collectively sum up to $1$. The goal is to find an assignment $\sigma:\mathscr{G}\rightarrow \mathscr{A}$ that maximizes $\prod_{i\in \mathscr{A}} \left(\sum_{j\in \sigma^{-1}(i)} v_{ij}\right)^{w_i}$, the product of the weighted valuations of the players. When all the weights equal $\frac1n$, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a $5\cdot\exp\left(2\cdot D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})\right) = 5\cdot\exp\left(2\log{n} + 2\sum_{i=1}^n w_i \log{w_i}\right)$-approximation algorithm for the weighted Nash Social Welfare problem, where $D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})$ denotes the KL-divergence between the distribution induced by $\mathbf{w}$ and the uniform distribution on $[n]$. We show a novel connection between the convex programming relaxations for the unweighted variant of Nash Social Welfare presented in \cite{cole2017convex, anari2017nash}, and generalize the programs to two different mathematical programs for the weighted case. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation.
We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available.
Neural marked temporal point processes have been a valuable addition to the existing toolbox of statistical parametric models for continuous-time event data. These models are useful for sequences where each event is associated with a single item (a single type of event or a "mark") -- but such models are not suited for the practical situation where each event is associated with a set of items. In this work, we develop a general framework for modeling set-valued data in continuous-time, compatible with any intensity-based recurrent neural point process model. In addition, we develop inference methods that can use such models to answer probabilistic queries such as "the probability of item $A$ being observed before item $B$," conditioned on sequence history. Computing exact answers for such queries is generally intractable for neural models due to both the continuous-time nature of the problem setting and the combinatorially-large space of potential outcomes for each event. To address this, we develop a class of importance sampling methods for querying with set-based sequences and demonstrate orders-of-magnitude improvements in efficiency over direct sampling via systematic experiments with four real-world datasets. We also illustrate how to use this framework to perform model selection using likelihoods that do not involve one-step-ahead prediction.
Given a complex high-dimensional distribution over $\{\pm 1\}^n$, what is the best way to increase the expected number of $+1$'s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem cannot be solved in polynomial time under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.
For a permutation $\pi: [K]\rightarrow [K]$, a sequence $f: \{1,2,\cdots, n\}\rightarrow \mathbb R$ contains a $\pi$-pattern of size $K$, if there is a sequence of indices $(i_1, i_2, \cdots, i_K)$ ($i_1<i_2<\cdots<i_K$), satisfying that $f(i_a)<f(i_b)$ if $\pi(a)<\pi(b)$, for $a,b\in [K]$. Otherwise, $f$ is referred to as $\pi$-free. For the special case where $\pi = (1,2,\cdots, K)$, it is referred to as the monotone pattern. \cite{newman2017testing} initiated the study of testing $\pi$-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the $(1,3,2)$ permutation. For the problem of testing monotone permutation $(1,2,\cdots,K)$, \cite{ben2019finding} improved the $(\log n)^{O(K^2)}$ non-adaptive query complexity of \cite{newman2017testing} to $O((\log n)^{\lfloor \log_{2} K\rfloor})$. Further, \cite{ben2019optimal} proposed an adaptive algorithm with $O(\log n)$ query complexity. However, no progress has yet been made on the problem of testing $(1,3,2)$-freeness. In this work, we present an adaptive algorithm for testing $(1,3,2)$-freeness. The query complexity of our algorithm is $O(\epsilon^{-2}\log^4 n)$, which significantly improves over the $O(\epsilon^{-7}\log^{26}n)$-query adaptive algorithm of \cite{newman2017testing}. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.
We preprocess the input subdivision with $n$ points on the plane in $O(n\sqrt{\log n})$ time to facilitate point location in constant time. Previously the preprocessing time is $O(n\log n)$ and point location takes $O(\log n)$ time.
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