We prove a Chernoff-type bound for sums of matrix-valued random variables sampled viaaregular (aperiodic and irreducible) finite Markovchain.

Inourlatestversion,wehaveallowedtheMarkov22 chain to start from an arbitrary initial distributionφ rather than the stationary distributionπ. To verify that is a meaning-34 ful range for tuningL, we enumerate trajectory lengthL from {104,,1010}, estimate the co-occurrence ma-35 trix with the single trajectory sampled from BlogCatalog, convert the co-occurrence matrix to the one required36 by NetMF, and factorize it with SVD.

This paper addresses the problem of designing efficient no-swap regret algorithms for combinatorial bandits, where the number of actions $N$ is exponentially large in the dimensionality of the problem. In this setting, designing efficient no-swap regret translates to sublinear -- in horizon $T$ -- swap regret with polylogarithmic dependence on $N$. In contrast to the weaker notion of external regret minimization - a problem which is fairly well understood in the literature - achieving no-swap regret with a polylogarithmic dependence on $N$ has remained elusive in combinatorial bandits. Our paper resolves this challenge, by introducing a no-swap-regret learning algorithm with regret that scales polylogarithmically in $N$ and is tight for the class of combinatorial bandits. To ground our results, we also demonstrate how to implement the proposed algorithm efficiently -- that is, with a per-iteration complexity that also scales polylogarithmically in $N$ -- across a wide range of well-studied applications.

We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a regular (aperiodic and irreducible) finite Markov chain. Specially, consider a random walk on a regular Markov chain and a Hermitian matrix-valued function on its state space. Our result gives exponentially decreasing bounds on the tail distributions of the extreme eigenvalues of the sample mean matrix. Our proof is based on the matrix expander (regular undirected graph) Chernoff bound [Garg et al.

Models such as Word2Vec and GloVe construct word embeddings based on the co-occurrence probability $P(i,j)$ of words $i$ and $j$ in text corpora. The resulting vectors $W_i$ not only group semantically similar words but also exhibit a striking linear analogy structure -- for example, $W_{\text{king}} - W_{\text{man}} + W_{\text{woman}} \approx W_{\text{queen}}$ -- whose theoretical origin remains unclear. Previous observations indicate that this analogy structure: (i) already emerges in the top eigenvectors of the matrix $M(i,j) = P(i,j)/P(i)P(j)$, (ii) strengthens and then saturates as more eigenvectors of $M (i, j)$, which controls the dimension of the embeddings, are included, (iii) is enhanced when using $\log M(i,j)$ rather than $M(i,j)$, and (iv) persists even when all word pairs involved in a specific analogy relation (e.g., king-queen, man-woman) are removed from the corpus. To explain these phenomena, we introduce a theoretical generative model in which words are defined by binary semantic attributes, and co-occurrence probabilities are derived from attribute-based interactions. This model analytically reproduces the emergence of linear analogy structure and naturally accounts for properties (i)-(iv). It can be viewed as giving fine-grained resolution into the role of each additional embedding dimension. It is robust to various forms of noise and agrees well with co-occurrence statistics measured on Wikipedia and the analogy benchmark introduced by Mikolov et al.

The widespread use of Large Language Models (LLMs) in many applications marks a significant advance in research and practice. However, their complexity and hard-to-understand nature make them vulnerable to attacks, especially jailbreaks designed to produce harmful responses. To counter these threats, developing strong detection methods is essential for the safe and reliable use of LLMs. This paper studies this detection problem using the Contextual Co-occurrence Matrix, a structure recognized for its efficacy in data-scarce environments. We propose a novel method leveraging the latent space characteristics of Contextual Co-occurrence Matrices and Tensors for the effective identification of adversarial and jailbreak prompts. Our evaluations show that this approach achieves a notable F1 score of 0.83 using only 0.5% of labeled prompts, which is a 96.6% improvement over baselines. This result highlights the strength of our learned patterns, especially when labeled data is scarce. Our method is also significantly faster, speedup ranging from 2.3 to 128.4 times compared to the baseline models.

We appreciate your positive feedback and will revise our presentation accordingly. Prior to this work, the walk length of DeepWalk has to be selected by cross-validation. Thank you for your comments. We appreciate your views and we would like to clarify a few points. We are open to reframing the work as "Matrix Thank you for your comments.