Riemannian neural networks, which extend deep learning techniques to Riemannian spaces, have gained significant attention in machine learning. To better classify the manifold-valued features, researchers have started extending Euclidean multinomial logistic regression (MLR) into Riemannian manifolds. However, existing approaches suffer from limited applicability due to their strong reliance on specific geometric properties. This paper proposes a framework for designing Riemannian MLR over general geometries, referred to as RMLR. Our framework only requires minimal geometric properties, thus exhibiting broad applicability and enabling its use with a wide range of geometries.

Neural Information Processing Systems http://nips.cc/

In this case, one can hardly distinguish non-robust data with LPS being 10 and safe data that are insensitive to be attacked.

Developing provable learning algorithms is one of the central challenges in learning theory. The study of such algorithms has led to significant advances in both the theory and practice of passive and active learning.

This paper bridges internal and external analysis approaches to large language models (LLMs) by demonstrating that geometric properties of internal model representations serve as reliable proxies for evaluating generated text quality. We validate a set of metrics--including Maximum Explainable V ariance, Effective Rank, Intrinsic Dimensionality, MAUVE score, and Schatten Norms measured across different layers of LLMs, demonstrating that Intrinsic Dimensionality and Effective Rank can serve as universal assessments of text naturalness and quality. Our key finding reveals that different models consistently rank text from various sources in the same order based on these geometric properties, indicating that these metrics reflect inherent text characteristics rather than model-specific artifacts. This allows a reference-free text quality evaluation that does not require human-annotated datasets, offering practical advantages for automated evaluation pipelines. The rapid advancement of large language models (LLMs) has necessitated the development of methods for analyzing their internal mechanisms and the properties of generated text. Approaches to studying the geometric properties of representations in language models can be broadly categorized into two categories: internal or mechanistic methods, which investigate the model's intermediate representations, and external methods, which analyze the properties of text embeddings captured via some embedding model. Internal evaluation mainly considers model properties within which, these measures were made (Yin et al., 2024; Viswanathan et al., 2025; Roy & V etterli, 2007), while external measures mainly focus on evaluation of text properties given text embeddings (Zhao et al., 2019; Tulchinskii et al., 2023; Kuznetsov et al., 2024).

Neural Information Processing Systems http://nips.cc/

To this end, we pose two problems.