Jie Meng

Dario A. Bini, Bruno Iannazzo, Beatrice Meini et al.

Component-wise accurate algorithms for computing the principal square root of an M-matrix are designed in terms of triplet representations. A triplet representation of an M-matrix $A$ is the triple $(P, {\bf u},{\bf v})$, where the matrix $P$ is such that $p_{ij}=-a_{ij}$ for $i\ne j$, $p_{ii}=0$, and ${\bf u}>0$, ${\bf v}\ge 0$ are two vectors such that $A{\bf u}={\bf v}$. It is shown that if $A$ is an M-matrix representable by a triplet, then its principal square root exists and is an M-matrix represented by a triplet as well. New versions of the Cyclic Reduction and the Incremental Newton iterations are provided in terms of triplets, to compute the principal matrix square root of $A$. It is shown that these algorithms are component-wise numerically stable independently of the singularity of $A$ and of its condition number. Numerical experiments are shown to confirm the component-wise stability.

Yuezhi Wang, Gwi Soo Kim, Jie Meng

Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point iterations, including a relaxed fixed-point iteration and Newton iteration, are proposed and analyzed. A structure-preserving doubling algorithm is proved to be applicable in computing the required solution, the convergence is at least linear with rate 1/2. Numerical experiments are performed to demonstrate the effectiveness of the proposed algorithms.

Yi Dong, Ronghui Mu, Gaojie Jin et al.

As Large Language Models (LLMs) become more integrated into our daily lives, it is crucial to identify and mitigate their risks, especially when the risks can have profound impacts on human users and societies. Guardrails, which filter the inputs or outputs of LLMs, have emerged as a core safeguarding technology. This position paper takes a deep look at current open-source solutions (Llama Guard, Nvidia NeMo, Guardrails AI), and discusses the challenges and the road towards building more complete solutions. Drawing on robust evidence from previous research, we advocate for a systematic approach to construct guardrails for LLMs, based on comprehensive consideration of diverse contexts across various LLMs applications. We propose employing socio-technical methods through collaboration with a multi-disciplinary team to pinpoint precise technical requirements, exploring advanced neural-symbolic implementations to embrace the complexity of the requirements, and developing verification and testing to ensure the utmost quality of the final product.

Jie Meng, Jin Mao

Large language models (LLMs) are predominantly used as evaluators for natural language generation (NLG) tasks, but their application to broader evaluation scenarios remains limited. In this work, we explore the potential of LLMs as general evaluators across diverse tasks. Although LLM-based evaluators have made progress in different areas, existing methods struggle to generalize due to their reliance on costly, human-designed evaluation principles, which are often misaligned with both annotated data and LLMs' understanding.To address these challenges, we propose a rule-augmented evaluation paradigm. First, we introduce a rule distillation method that automatically extracts scoring rules from data using an LLM-assisted Monte Carlo Tree Search (MCTS), alleviating scalability issues and improving alignment with data. Second, to enable LLMs to effectively apply the learned rules, we propose two strategies: (1) Chain-of-Rule (CoR), which guides LLM to follow distilled rules, and (2) training a rule-augmented LLM evaluator (RuAE) via reinforcement learning, further bridging the gap between rules and LLMs' reasoning. Extensive experiments on diverse tasks demonstrate the effectiveness and generalizability of our approach across various evaluation scenarios.