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Compositional data analysis has gained increased attention due to the widespread occurrence of simplex-valued data, including microbiome data and financial portfolios. Existing compositional two-sample tests often require $$\log$$-transformations and only detect mean differences, motivating the need for a more general framework without relying on $$\log$$-based methods. There is a close connection between compositional data and directional statistics, and we construct a unified non-parametric two-sample test framework. Our work is based on a studentized energy statistic constructed from spherical harmonics theory over a fixed dimensional underlying space, incorporating U-statistics theory and recent developments of studentization for both compositional and directional data. We establish asymptotic normality for our spherical harmonics based test statistics, thus avoiding the need for permutation tests or bootstrap procedures. Our proposed framework sheds new light on the connections between Non-Euclidean data analysis and classical asymptotic high-dimensional data techniques. 

Jailbreak attacks serve as essential red-teaming tools, proactively assessing whether LLMs can behave responsibly and safely in adversarial environments. Despite diverse strategies (eg, cipher, low-resource language, persuasions, and so on) that have been proposed and shown success, these strategies are still manually designed, limiting their scope and effectiveness as a red-teaming tool. In this paper, we propose AutoDAN-Turbo, a black-box jailbreak method that can automatically discover as many jailbreak strategies as possible from scratch, without any human intervention or predefined scopes (eg, specified candidate strategies), and use them for red-teaming. As a result, AutoDAN-Turbo can significantly outperform baseline methods, achieving a 74.3% higher average attack success rate on public benchmarks. Notably, AutoDAN-Turbo achieves an 88.5 attack success rate on GPT-4-1106-turbo. In addition, AutoDAN-Turbo is a unified framework that can incorporate existing human-designed jailbreak strategies in a plug-and-play manner. By integrating human-designed strategies, AutoDAN-Turbo can even achieve a higher attack success rate of 93.4 on GPT-4-1106-turbo.