Our analysis of the generalization error is based on an extension of Gordon's Gaussian process inequality [ R is a continuous function, which is convex in the first argument and concave in the second argument. The main result of CGMT is to connect the above two random optimization problems. The CGMT framework has been used to infer statistical properties of estimators in certain high-dimensional asymptotic regime. Second, derive the point-wise limit of the AO objective in terms of a convex-concave optimization problem, over only few scalar variables.

Open-source LLMs have shown great potential as fine-tuned chatbots, and demonstrate robust abilities in reasoning and surpass many existing benchmarks. Retrieval-Augmented Generation (RAG) is a technique for improving the performance of LLMs on tasks that the models weren't explicitly trained on, by leveraging external knowledge databases. Numerous studies have demonstrated the effectiveness of RAG to more successfully accomplish downstream tasks when using vector datasets that consist of relevant background information. It has been implicitly assumed by those in the field that if adversarial background information is utilized in this context, that the success of using a RAG-based approach would be nonexistent or even negatively impact the results. To address this assumption, we tested several open-source LLMs on the ability of RAG to improve their success in answering multiple-choice questions (MCQ) in the medical subspecialty field of Nephrology. Unlike previous studies, we examined the effect of RAG in utilizing both relevant and adversarial background databases. We set up several open-source LLMs, including Llama 3, Phi-3, Mixtral 8x7b, Zephyr$\beta$, and Gemma 7B Instruct, in a zero-shot RAG pipeline. As adversarial sources of information, text from the Bible and a Random Words generated database were used for comparison. Our data show that most of the open-source LLMs improve their multiple-choice test-taking success as expected when incorporating relevant information vector databases. Surprisingly however, adversarial Bible text significantly improved the success of many LLMs and even random word text improved test taking ability of some of the models. In summary, our results demonstrate for the first time the countertintuitive ability of adversarial information datasets to improve the RAG-based LLM success.

Timeseries classification as stochastic (noise-like) or non-stochastic (structured), helps understand the underlying dynamics, in several domains. Here we propose a two-legged matrix decomposition-based algorithm utilizing two complementary techniques for classification. In Singular Value Decomposition (SVD) based analysis leg, we perform topological analysis (Betti numbers) on singular vectors containing temporal information, leading to SVD-label. Parallely, temporal-ordering agnostic Principal Component Analysis (PCA) is performed, and the proposed PCA-derived features are computed. These features, extracted from synthetic timeseries of the two labels, are observed to map the timeseries to a linearly separable feature space. Support Vector Machine (SVM) is used to produce PCA-label. The proposed methods have been applied to synthetic data, comprising 41 realisations of white-noise, pink-noise (stochastic), Logistic-map at growth-rate 4 and Lorentz-system (non-stochastic), as proof-of-concept. Proposed algorithm is applied on astronomical data: 12 temporal-classes of timeseries of black hole GRS 1915+105, obtained from RXTE satellite with average length 25000. For a given timeseries, if SVD-label and PCA-label concur, then the label is retained; else deemed "Uncertain". Comparison of obtained results with those in literature are presented. It's found that out of 12 temporal classes of GRS 1915+105, concurrence between SVD-label and PCA-label is obtained on 11 of them.