Con set isa (scaled)`p ball, 1 p< 2, then, considering(2), the thebestmethod p d/logdtimeslar weprovidene . Formally, if 2 then s 2 { 1}d implies (sj j)j d 2 .

In practical situations, the clustering result must be stable against points missing in the input data so that we can make trustworthy andconsistentdecisions.

Multiple versions of this dataset exist in the literature; we use the version by Ravi and Larochelle [43]. The original version of the dataset contains43images that are also present in ImageNet. We remove these duplicates to avoid overestimating the transfer capability during evaluation. VGGFlowers: Originally introduced by Nilsback and Zisserman[38], VGGFlowers consists of 102 flower categories with each category containing between40 and 258 images. A.3 Trainingalgorithms For the metric-based family, we use ProtoNet with Euclidean [53] and scaled negative cosine similarity measures [20].

High dimensional representation drift is commonly quantified using Euclidean or cosine distances, which presuppose fixed coordinates when comparing representations across time, training or preprocessing stages. While effective in many settings, these measures entangle intrinsic changes in the data with variations induced by arbitrary parametrizations. We introduce a projective geometric view of representation drift grounded in the Fubini Study metric, which identifies representations that differ only by gauge transformations such as global rescalings or sign flips. Applying this framework to empirical high dimensional datasets, we explicitly construct representation trajectories and track their evolution through cumulative geometric drift. Comparing Euclidean, cosine and Fubini Study distances along these trajectories reveals that conventional metrics systematically overestimate change whenever representations carry genuine projective ambiguity. By contrast, the Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations. We further show that the difference between cosine and Fubini Study drift defines a computable, monotone quantity that directly captures representation churn attributable to gauge freedom. This separation provides a diagnostic for distinguishing meaningful structural evolution from parametrization artifacts, without introducing model-specific assumptions. Overall, we establish a geometric criterion for assessing representation stability in high-dimensional systems and clarify the limits of angular distances. Embedding representation dynamics in projective space connects data analysis with established geometric programs and yields observables that are directly testable in empirical workflows.

Deep learning models have achieved remarkable success across various domains, yet their learned representations and decision-making processes remain largely opaque and hard to interpret. This work introduces HOLE (Homological Observation of Latent Embeddings), a method for analyzing and interpreting deep neural networks through persistent homology. HOLE extracts topological features from neural activations and presents them using a suite of visualization techniques, including Sankey diagrams, heatmaps, dendrograms, and blob graphs. These tools facilitate the examination of representation structure and quality across layers. We evaluate HOLE on standard datasets using a range of discriminative models, focusing on representation quality, interpretability across layers, and robustness to input perturbations and model compression. The results indicate that topological analysis reveals patterns associated with class separation, feature disentanglement, and model robustness, providing a complementary perspective for understanding and improving deep learning systems.

Dataset distillation aims to synthesize a compact subset of the original data, enabling models trained on it to achieve performance comparable to those trained on the original large dataset. Existing distribution-matching methods are confined to Euclidean spaces, making them only capture linear structures and overlook the intrinsic geometry of real data, e.g., curvature. However, high-dimensional data often lie on low-dimensional manifolds, suggesting that dataset distillation should have the distilled data manifold aligned with the original data manifold. In this work, we propose a geometry-aware distribution-matching framework, called \textbf{GeoDM}, which operates in the Cartesian product of Euclidean, hyperbolic, and spherical manifolds, with flat, hierarchical, and cyclical structures all captured by a unified representation. To adapt to the underlying data geometry, we introduce learnable curvature and weight parameters for three kinds of geometries. At the same time, we design an optimal transport loss to enhance the distribution fidelity. Our theoretical analysis shows that the geometry-aware distribution matching in a product space yields a smaller generalization error bound than the Euclidean counterparts. Extensive experiments conducted on standard benchmarks demonstrate that our algorithm outperforms state-of-the-art data distillation methods and remains effective across various distribution-matching strategies for the single geometries.

In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental limitations. At a large scale, real-world data often exhibits inherently non-Euclidean structures, such as multi-way relationships, hierarchies, symmetries, and non-isotropic scaling, in a variety of domains, such as languages, vision, and the natural sciences. It is challenging to effectively capture these structures within the constraints of Euclidean spaces. This position paper argues that moving beyond Euclidean geometry is not merely an optional enhancement but a necessity to maintain the scaling law for the next-generation of foundation models. By adopting these geometries, foundation models could more efficiently leverage the aforementioned structures. Task-aware adaptability that dynamically reconfigures embeddings to match the geometry of downstream applications could further enhance efficiency and expressivity. Our position is supported by a series of theoretical and empirical investigations of prevalent foundation models. Finally, we outline a roadmap for integrating non-Euclidean geometries into foundation models, including strategies for building geometric foundation models via fine-tuning, training from scratch, and hybrid approaches.

Retrieval-augmented generation (RAG) enables large language models (LLMs) to access external knowledge, helping mitigate hallucinations and enhance domain-specific expertise. Graph-based RAG enhances structural reasoning by introducing explicit relational organization that enables information propagation across semantically connected text units. However, these methods typically rely on Euclidean embeddings that capture semantic similarity but lack a geometric notion of hierarchical depth, limiting their ability to represent abstraction relationships inherent in complex knowledge graphs. To capture both fine-grained semantics and global hierarchy, we propose HyperbolicRAG, a retrieval framework that integrates hyperbolic geometry into graph-based RAG. HyperbolicRAG introduces three key designs: (1) a depth-aware representation learner that embeds nodes within a shared Poincare manifold to align semantic similarity with hierarchical containment, (2) an unsupervised contrastive regularization that enforces geometric consistency across abstraction levels, and (3) a mutual-ranking fusion mechanism that jointly exploits retrieval signals from Euclidean and hyperbolic spaces, emphasizing cross-space agreement during inference. Extensive experiments across multiple QA benchmarks demonstrate that HyperbolicRAG outperforms competitive baselines, including both standard RAG and graph-augmented baselines.

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