In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m<n$ samples drawn either uniformly or using approximate leverage scores from the initial observations. Our main result is an upper bound on the approximation error of this procedure for both sampling strategies. It yields sufficient conditions on the subsample size to recover the standard (optimal) $n^{-1/2}$ rate while reducing drastically the number of functions evaluations, and thus the overall computational cost. Moreover, we obtain rates with respect to the number $m$ of evaluations of the integrand which adapt to its smoothness, and match known optimal rates for instance for Sobolev spaces. We illustrate our theoretical findings with numerical experiments on real datasets, which highlight the attractive efficiency-accuracy tradeoff of our method compared to existing randomized and greedy quadrature methods. We note that, the problem of numerical integration in RKHS amounts to designing a discrete approximation of the kernel mean embedding of the target distribution. As a consequence, direct applications of our results also include the efficient computation of maximum mean discrepancies between distributions and the design of efficient kernel-based tests.
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Integration：Integration, the VLSI Journal。
Explanation：集(ji)成，VLSI雜志。
Publisher：Elsevier。
SIT：
We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD) applied to a Burer-Monteiro least-squares problem and show that it converges linearly to the incoherent ground-truth with respect to the condition number \k{appa} of ground-truth under a benign initialization and large samples. We next apply the scaled variant of PGD to deal with the ill-conditioned case when \k{appa} is large, and we show the algorithm converges at a linear rate independent of the condition number \k{appa} under similar conditions. Finally, we provide numerical experiments to corroborate our results.
In this paper, we propose a cooperative long-term task execution (LTTE) algorithm for protecting a moving target into the interior of an ordering-flexible convex hull by a team of robots resiliently in the changing environments. Particularly, by designing target-approaching and sensing-neighbor collision-free subtasks, and incorporating these subtasks into the constraints rather than the traditional cost function in an online constraint-based optimization framework, the proposed LTTE can systematically guarantee long-term target convoying under changing environments in the n-dimensional Euclidean space. Then, the introduction of slack variables allow for the constraint violation of different subtasks; i.e., the attraction from target-approaching constraints and the repulsion from time-varying collision-avoidance constraints, which results in the desired formation with arbitrary spatial ordering sequences. Rigorous analysis is provided to guarantee asymptotical convergence with challenging nonlinear couplings induced by time-varying collision-free constraints. Finally, 2D experiments using three autonomous mobile robots (AMRs) are conducted to validate the effectiveness of the proposed algorithm, and 3D simulations tackling changing environmental elements, such as different initial positions, some robots suddenly breakdown and static obstacles are presented to demonstrate the multi-dimensional adaptability, robustness and the ability of obstacle avoidance of the proposed method.
Inspired by the remarkable success of large neural networks, there has been significant interest in understanding the generalization performance of over-parameterized models. Substantial efforts have been invested in characterizing how optimization algorithms impact generalization through their "preferred" solutions, a phenomenon commonly referred to as implicit regularization. In particular, it has been argued that gradient descent (GD) induces an implicit $\ell_2$-norm regularization in regression and classification problems. However, the implicit regularization of different algorithms are confined to either a specific geometry or a particular class of learning problems, indicating a gap in a general approach for controlling the implicit regularization. To address this, we present a unified approach using mirror descent (MD), a notable generalization of GD, to control implicit regularization in both regression and classification settings. More specifically, we show that MD with the general class of homogeneous potential functions converges in direction to a generalized maximum-margin solution for linear classification problems, thereby answering a long-standing question in the classification setting. Further, we show that MD can be implemented efficiently and enjoys fast convergence under suitable conditions. Through comprehensive experiments, we demonstrate that MD is a versatile method to produce learned models with different regularizers, which in turn have different generalization performances.
The objective of this work is to train a chatbot capable of solving evolving problems through conversing with a user about a problem the chatbot cannot directly observe. The system consists of a virtual problem (in this case a simple game), a simulated user capable of answering natural language questions that can observe and perform actions on the problem, and a Deep Q-Network (DQN)-based chatbot architecture. The chatbot is trained with the goal of solving the problem through dialogue with the simulated user using reinforcement learning. The contributions of this paper are as follows: a proposed architecture to apply a conversational DQN-based agent to evolving problems, an exploration of training methods such as curriculum learning on model performance and the effect of modified reward functions in the case of increasing environment complexity.
Comparing spatial data sets is a ubiquitous task in data analysis, however the presence of spatial autocorrelation means that standard estimates of variance will be wrong and tend to over-estimate the statistical significance of correlations and other observations. While there are a number of existing approaches to this problem, none are ideal, requiring detailed analytical calculations, which are hard to generalise or detailed knowledge of the data generating process, which may not be available. In this work we propose a resampling approach based on Tobler's Law. By resampling the data with fixed spatial autocorrelation, measured by Moran's I, we generate a more realistic null model. Testing on real and synthetic data, we find that, as long as the spatial autocorrelation is not too strong, this approach works just as well as if we knew the data generating process.
In this work, simulation-based equations to calculate propagation constant in uniform or periodic structures (SES) are deduced and verified through simulations in various types of structures. The modeling of those structures are essentially based on field distributions from a driven-mode solver, and the field distributions are used as the input parameters of the FPPS. It allows the separation of forward and backward waves from a total wave inside such a uniform or periodic structure, and thus it can be used to calculate the propagation constants inside both uniform and periodic structures even with a strong reflection. In order to test the performance and function of the FPPS, it has been applied to a variety of typical structures, including uniform waveguides, lossfree closed structures, lossy closed structures, and open radiation structures, and compared with the results of eigenmode solvers, equivalent network methods, and spectral domain integral equation methods. The comparison shows the easy-to-use and adaptable nature of the FPPS. the FPPS. This FPPS could be also applied to open radiating structures, and even multi-dimensional periodic/uniform structures.
In this study, a versatile methodology for initiating polymerization from monomers in highly cross-linked materials is investigated. As polymerization progresses, force-field parameters undergo continuous modification due to the formation of new chemical bonds. This dynamic process not only impacts the atoms directly involved in bonding, but also influences the neighboring atomic environment. Monitoring these complex changes in highly cross-linked structures poses a challenge. To address this issue, we introduce a graph-network-based algorithm that offers both rapid and accurate predictions. The algorithm merges polymer construction protocols with LAMMPS, a large-scale molecular dynamics simulation software. The adaptability of this code has been demonstrated by its successful application to various amorphous polymers, including porous polymer networks (PPNs), and epoxy-resins, while the algorithm has been employed for additional tasks, such as implementing pore-piercing deformations and calculating material properties.
In pace with developments in the research field of artificial intelligence, knowledge graphs (KGs) have attracted a surge of interest from both academia and industry. As a representation of semantic relations between entities, KGs have proven to be particularly relevant for natural language processing (NLP), experiencing a rapid spread and wide adoption within recent years. Given the increasing amount of research work in this area, several KG-related approaches have been surveyed in the NLP research community. However, a comprehensive study that categorizes established topics and reviews the maturity of individual research streams remains absent to this day. Contributing to closing this gap, we systematically analyzed 507 papers from the literature on KGs in NLP. Our survey encompasses a multifaceted review of tasks, research types, and contributions. As a result, we present a structured overview of the research landscape, provide a taxonomy of tasks, summarize our findings, and highlight directions for future work.
How can we estimate the importance of nodes in a knowledge graph (KG)? A KG is a multi-relational graph that has proven valuable for many tasks including question answering and semantic search. In this paper, we present GENI, a method for tackling the problem of estimating node importance in KGs, which enables several downstream applications such as item recommendation and resource allocation. While a number of approaches have been developed to address this problem for general graphs, they do not fully utilize information available in KGs, or lack flexibility needed to model complex relationship between entities and their importance. To address these limitations, we explore supervised machine learning algorithms. In particular, building upon recent advancement of graph neural networks (GNNs), we develop GENI, a GNN-based method designed to deal with distinctive challenges involved with predicting node importance in KGs. Our method performs an aggregation of importance scores instead of aggregating node embeddings via predicate-aware attention mechanism and flexible centrality adjustment. In our evaluation of GENI and existing methods on predicting node importance in real-world KGs with different characteristics, GENI achieves 5-17% higher NDCG@100 than the state of the art.
In this paper, we propose a deep reinforcement learning framework called GCOMB to learn algorithms that can solve combinatorial problems over large graphs. GCOMB mimics the greedy algorithm in the original problem and incrementally constructs a solution. The proposed framework utilizes Graph Convolutional Network (GCN) to generate node embeddings that predicts the potential nodes in the solution set from the entire node set. These embeddings enable an efficient training process to learn the greedy policy via Q-learning. Through extensive evaluation on several real and synthetic datasets containing up to a million nodes, we establish that GCOMB is up to 41% better than the state of the art, up to seven times faster than the greedy algorithm, robust and scalable to large dynamic networks.
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