Brace algebras and the cohomology comparison theorem

The Gerstenhaber and Schack cohomology comparison theorem asserts that there is a cochain equivalence between the Hochschild complex of a certain algebra and the usual singular cochain complex of a space. We show that this comparison theorem preserves the brace algebra structures. This result gives a structural reason for the recent results establishing fine topological structures on the Hochschild cohomology, and a simple way to derive them from the corresponding properties of cochain complexes.Comment: Revised version of "The bar construction as a Hopf algebra", Dec. 200

Convergence of U-statistics for interacting particle systems

The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework

Deciphering Clusters With a Deterministic Measure of Clustering Tendency

Clustering, a key aspect of exploratory data analysis, plays a crucial role in various fields such as information retrieval. Yet, the sheer volume and variety of available clustering algorithms hinder their application to specific tasks, especially given their propensity to enforce partitions, even when no clear clusters exist, often leading to fruitless efforts and erroneous conclusions. This issue highlights the importance of accurately assessing clustering tendencies prior to clustering. However, existing methods either rely on subjective visual assessment, which hinders automation of downstream tasks, or on correlations between subsets of target datasets and random distributions, limiting their practical use. Therefore, we introduce the Proximal Homogeneity Index (PHI) , a novel and deterministic statistic that reliably assesses the clustering tendencies of datasets by analyzing their internal structures via knowledge graphs. Leveraging PHI and the boundaries between clusters, we establish the Partitioning Sensitivity Index (PSI) , a new statistic designed for cluster quality assessment and optimal clustering identification. Comparative studies using twelve synthetic and real-world datasets demonstrate PHI and PSI's superiority over existing metrics for clustering tendency assessment and cluster validation. Furthermore, we demonstrate the scalability of PHI to large and high-dimensional datasets, and PSI's broad effectiveness across diverse cluster analysis tasks

Amoeba: Circumventing ML-supported Network Censorship via Adversarial Reinforcement Learning

Embedding covert streams into a cover channel is a common approach to circumventing Internet censorship, due to censors' inability to examine encrypted information in otherwise permitted protocols (Skype, HTTPS, etc.). However, recent advances in machine learning (ML) enable detecting a range of anti-censorship systems by learning distinct statistical patterns hidden in traffic flows. Therefore, designing obfuscation solutions able to generate traffic that is statistically similar to innocuous network activity, in order to deceive ML-based classifiers at line speed, is difficult. In this paper, we formulate a practical adversarial attack strategy against flow classifiers as a method for circumventing censorship. Specifically, we cast the problem of finding adversarial flows that will be misclassified as a sequence generation task, which we solve with Amoeba, a novel reinforcement learning algorithm that we design. Amoeba works by interacting with censoring classifiers without any knowledge of their model structure, but by crafting packets and observing the classifiers' decisions, in order to guide the sequence generation process. Our experiments using data collected from two popular anti-censorship systems demonstrate that Amoeba can effectively shape adversarial flows that have on average 94% attack success rate against a range of ML algorithms. In addition, we show that these adversarial flows are robust in different network environments and possess transferability across various ML models, meaning that once trained against one, our agent can subvert other censoring classifiers without retraining

On particle Gibbs Markov chain Monte Carlo models

This article analyses a new class of advanced particle Markov chain Monte Carlo algorithms recently introduced by Andrieu, Doucet, and Holenstein (2010). We present a natural interpretation of these methods in terms of well known unbiasedness properties of Feynman-Kac particle measures, and a new duality with many-body Feynman-Kac models. This perspective sheds a new light on the foundations and the mathematical analysis of this class of methods. A key consequence is the equivalence between the backward and ancestral particle Markov chain Monte Carlo methods, and Gibbs sampling of a many-body Feynman-Kac target distribution. Our approach also presents a new stochastic differential calculus based on geometric combinatorial techniques to derive explicit non-asymptotic Taylor type series of the semigroup of a class of particle Markov chain Monte Carlo models around their invariant measures with respect to the population size of the auxiliary particle sampler. These results provide sharp quan- titative estimates of the convergence properties of conditional particle Markov chain models with respect to the time horizon and the size of the systems. We illustrate the implication of these results with sharp estimates of the contraction coefficient and the Lyapunov exponent of conditional particle samplers, and explicit and non-asymptotic Lp-mean error decompositions of the law of the random states around the limiting invariant measure. The abstract framework developed in the article also allows the design of natural extensions to island (also called SMC2) type particle methodologies

Rota-Baxter algebras and new combinatorial identities

The word problem for an arbitrary associative Rota-Baxter algebra is solved. This leads to a noncommutative generalization of the classical Spitzer identities. Links to other combinatorial aspects, particularly of interest in physics, are indicated.Comment: 8 pages, improved versio