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Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge 2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb R^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is $\mathcal O\!\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nystr\"om-based one) on $\mathbb R^d$.

Foundation models, such as Large language Models (LLMs), have attracted significant amount of interest due to their large number of applications. Existing works show that appropriate prompt design, such as Chain-of-Thoughts, can unlock LLM's powerful capacity in diverse areas. However, when handling tasks involving repetitive sub-tasks and/or deceptive contents, such as arithmetic calculation and article-level fake news detection, existing prompting strategies either suffers from insufficient expressive power or intermediate errors triggered by hallucination. To make LLM more discerning to such intermediate errors, we propose to guide LLM with a Divide-and-Conquer program that simultaneously ensures superior expressive power and disentangles task decomposition, sub-task resolution, and resolution assembly process. Theoretic analysis reveals that our strategy can guide LLM to extend the expressive power of fixed-depth Transformer. Experiments indicate that our proposed method can achieve better performance than typical prompting strategies in tasks bothered by intermediate errors and deceptive contents, such as large integer multiplication, hallucination detection and misinformation detection.

The techniques used to generate pseudo-random numbers for Monte Carlo (MC) applications bear many implications on the quality and speed of that programs work. As a random number generator (RNG) slows, the production of random numbers begins to dominate runtime. As RNG output grows in correlation, the final product becomes less reliable. These difficulties are further compounded by the need for reproducibility and parallelism. For reproducibility, the numbers generated to determine any outcome must be the same each time a simulation is run. However, the concurrency that comes with most parallelism introduces race conditions. To have both reproducibility and concurrency, separate RNG states must be tracked for each independently schedulable unit of simulation, forming independent random number streams. We propose an alternative to the stride-based parallel LCG seeding approach that scales more practically with increased concurrency and workload by generating seeds through hashing and allowing for repeated outputs. Data gathered from normality tests of tally results from simple MC transport benchmark calculations indicates that the proposed hash-based RNG does not significantly affect the tally result normality property as compared to the conventional stride-based RNG.

In the field of Learning from Demonstration (LfD), Dynamical Systems (DSs) have gained significant attention due to their ability to generate real-time motions and reach predefined targets. However, the conventional convergence-centric behavior exhibited by DSs may fall short in safety-critical tasks, specifically, those requiring precise replication of demonstrated trajectories or strict adherence to constrained regions even in the presence of perturbations or human intervention. Moreover, existing DS research often assumes demonstrations solely in Euclidean space, overlooking the crucial aspect of orientation in various applications. To alleviate these shortcomings, we present an innovative approach geared toward ensuring the safe execution of learned orientation skills within constrained regions surrounding a reference trajectory. This involves learning a stable DS on SO(3), extracting time-varying conic constraints from the variability observed in expert demonstrations, and bounding the evolution of the DS with Conic Control Barrier Function (CCBF) to fulfill the constraints. We validated our approach through extensive evaluation in simulation and showcased its effectiveness for a cutting skill in the context of assisted teleoperation.

In the Network Revenue Management (NRM) problem, products composed of up to L resources are sold to stochastically arriving customers. We take a randomized rounding approach to NRM, motivated by developments in Online Contention Resolution Schemes (OCRS). The goal is to take a fractional solution to NRM that satisfies the resource constraints in expectation, and implement it in an online policy that satisfies the resource constraints in any state, while (approximately) preserving all of the sales that were prescribed by the fractional solution. OCRS cannot be naively applied to NRM or revenue management problems in general, because customer substitution induces a negative correlation in products being demanded. We start by deriving an OCRS that achieves a guarantee of 1/(1+L) for NRM with customer substitution, matching a common benchmark in the literature. We then show how to beat this benchmark for all integers L>1 assuming no substitution, i.e., in the standard OCRS setting. By contrast, we show that this benchmark is unbeatable using OCRS or any fractional relaxation if there is customer substitution, for all integers L that are the power of a prime number. Finally, we show how to beat 1/(1+L) even with customer substitution, if the products comprise one item from each of up to L groups. Our results have corresponding implications for Online Combinatorial Auctions, in which buyers bid for bundles of up to L items, and buyers being single-minded is akin to no substitution. Our final result also beats 1/(1+L) for Prophet Inequality on the intersection of L partition matroids. All in all, our paper provides a unifying framework for applying OCRS to these problems, delineating the impact of substitution, and establishing a separation between the guarantees achievable with vs. without substitution under general resource constraints parametrized by L.

Contextualized embeddings are the preferred tool for modeling Lexical Semantic Change (LSC). Current evaluations typically focus on a specific task known as Graded Change Detection (GCD). However, performance comparison across work are often misleading due to their reliance on diverse settings. In this paper, we evaluate state-of-the-art models and approaches for GCD under equal conditions. We further break the LSC problem into Word-in-Context (WiC) and Word Sense Induction (WSI) tasks, and compare models across these different levels. Our evaluation is performed across different languages on eight available benchmarks for LSC, and shows that (i) APD outperforms other approaches for GCD; (ii) XL-LEXEME outperforms other contextualized models for WiC, WSI, and GCD, while being comparable to GPT-4; (iii) there is a clear need for improving the modeling of word meanings, as well as focus on how, when, and why these meanings change, rather than solely focusing on the extent of semantic change.

The arithmetic-geometric index is a newly proposed degree-based graph invariant in mathematical chemistry. We give a sharp upper bound on the value of this invariant for connected chemical graphs of given order and size and characterize the connected chemical graphs that reach the bound. We also prove that the removal of the constraint that extremal chemical graphs must be connected does not allow to increase the upper bound.

League of Legends (LoL) has been a dominant esport for a decade, yet the inherent complexity of the game has stymied the creation of analytical measures of player skill and performance. Current industry standards are limited to easy-to-procure individual player statistics that are incomplete and lacking context as they do not take into account teamplay or game state. We present a unified performance model for League of Legends which blends together measures of a player's contribution within the context of their team, insights from traditional sports metrics such as the Plus-Minus model, and the intricacies of LoL as a complex team invasion sport. Using hierarchical Bayesian models, we outline the use of gold and damage dealt as a measure of skill, detailing players' impact on their own-, their allies'- and their enemies' statistics throughout the course of the game. Our results showcase the model's increased efficacy in separating professional players when compared to a Plus-Minus model and to current esports industry standards, while metric quality is rigorously assessed for discrimination, independence, and stability. Readers might also find additional qualitative analytics which explore champion proficiency and the impact of collaborative team-play. Future work is proposed to refine and expand the SIDO performance model, offering a comprehensive framework for esports analytics in team performance management, scouting and research realms.

Pre-trained Language Models (PLMs) which are trained on large text corpus via self-supervised learning method, have yielded promising performance on various tasks in Natural Language Processing (NLP). However, though PLMs with huge parameters can effectively possess rich knowledge learned from massive training text and benefit downstream tasks at the fine-tuning stage, they still have some limitations such as poor reasoning ability due to the lack of external knowledge. Research has been dedicated to incorporating knowledge into PLMs to tackle these issues. In this paper, we present a comprehensive review of Knowledge-Enhanced Pre-trained Language Models (KE-PLMs) to provide a clear insight into this thriving field. We introduce appropriate taxonomies respectively for Natural Language Understanding (NLU) and Natural Language Generation (NLG) to highlight these two main tasks of NLP. For NLU, we divide the types of knowledge into four categories: linguistic knowledge, text knowledge, knowledge graph (KG), and rule knowledge. The KE-PLMs for NLG are categorized into KG-based and retrieval-based methods. Finally, we point out some promising future directions of KE-PLMs.

The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN) based models generate richer and more expressive feature embeddings and hence also perform well on relation prediction. However, we observe that these KG embeddings treat triples independently and thus fail to cover the complex and hidden information that is inherently implicit in the local neighborhood surrounding a triple. To this effect, our paper proposes a novel attention based feature embedding that captures both entity and relation features in any given entity's neighborhood. Additionally, we also encapsulate relation clusters and multihop relations in our model. Our empirical study offers insights into the efficacy of our attention based model and we show marked performance gains in comparison to state of the art methods on all datasets.