Geometric Analysis of Self-Supervised Vision Representations for Semantic Image Retrieval

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

Recent work has shown that "Vision-Free'' approaches (representing images as text) can be effective for standard image retrieval tasks. However, it remains unclear whether this paradigm can effectively handle a more complex, multimodal task, Composed Image Retrieval (CIR), due to the inherent information loss in textual descriptions. In this paper, we introduce a Vision-Free CIR framework that addresses this challenge through two key techniques: (1) Attribute-Augmented Hybrid Scoring, which compensates for lost visual details via explicit attribute matching, and (2) LLM-Based Reranking, which verifies semantic consistency of top candidates. Experiments on the open-domain CIRR dataset show that our approach outperforms existing Zero-shot CIR methods (44.04% R@1, +8.79%). On FashionIQ, our results highlight the trade-off between semantic reasoning and fine-grained visual matching. Ablation studies reveal that both attribute-augmented scoring and LLM-Based Reranking consistently improve performance.

Embedding-based retrieval systems rely on the assumption that geometric proximity in highdimensional representation spaces reflects semantic relevance. However, high-dimensional geometry induces concentration phenomena that can reduce the discriminative power of similarity measures and can destabilize nearest-neighbor retrieval. This work studies distance concentration, cosine concentration, contrast collapse, hubness, and retrieval instability through controlled numerical experiments across multiple synthetic distributions. The results show that similarity signals progressively lose contrast as dimension increases, leading to unstable retrieval behavior and structural bias in nearest-neighbor selection. A simplified Retrieval-Augmented Generation experiment further suggests that these effects can degrade grounding reliability upstream of generation. These findings motivate geometry-aware diagnostics and robustness-oriented retrieval strategies for embedding-based AI systems. The experiments are intentionally synthetic in order to isolate intrinsic geometric effects. High-dimensional embedding space Distance and cosine concentration Score-gap collapse and hubness Retrieval instability under perturbations Weak or incomplete retrieved context Potential degradation of grounding 1.