The Geometric Mechanics of Contrastive Representation Learning: Alignment Potentials, Entropic Dispersion, and Cross-modal Divergence

We study the modality gap in CLIP-style dual-encoder contrastive learning, where image and text embeddings remain misaligned despite being trained in a shared space. We argue that the gap is induced by a failure of the InfoNCE formulation with independent encoders. We conduct a uni-modal experiment with two independent encoders and identical initialization conditions and find that InfoNCE actively generates a gap at low temperatures. We provide a theoretical analysis of this phenomenon and show that the modality gap is indeed a mode-failure of InfoNCE, but only at low temperatures. We propose a simple modification called xNCE, which uses intermodal as well as intra-modality negative contrastive pairs. xNCE matches retrieval performance on MS-COCO while consistently reducing the gap even at low temperatures. Notably, xNCE improves zero-shot classification over the InfoNCE baseline across all benchmarks, whereas high-temperature InfoNCE and regularized InfoNCE both fail to do so, demonstrating that xNCE reduces the modality gap without sacrificing the discriminative geometry needed for transfer.

Contrastive learning is increasingly moving toward settings with three or more modalities instead of image-text pairs. Yet, extending models from pairwise to higher-order multimodal alignment can introduce optimization and representation challenges. We identify encoder Jacobian conditioning as a key factor in trimodal contrastive learning: poorly conditioned encoders exhibit collapsing or amplified singular-value spectra, leading to exploding Jacobian condition numbers and degraded multimodal alignment. We introduce geometry-preserving encoders (GPEs) by directly conditioning the Jacobian through regularization and demonstrating that simple modifications like LeakyReLU activations and residual paths recover these geometric benefits. Across a synthetic benchmark and four real-world datasets including missing modalities, improving Jacobian conditioning boosts retrieval and linear probe performance across multiple contrastive objectives, whereas expressive objectives yield little benefit in linear probes. More broadly, our results show that multimodal contrastive learning depends not only on objective expressivity, but also on the geometric and optimization properties of the underlying encoders.

Contrastive learning has become a leading paradigm for self-supervised representation learning, yet the conditions under which it recovers meaningful latent geometry remain incompletely understood. We develop a measure-theoretic framework formalizing the diversity condition, a support requirement on positive-pair sampling that is necessary for isometric latent recovery. We show that the standard full-support von Mises-Fisher setting implies the satisfaction of the diversity condition and as a consequence global contrastive loss minimizers recover latent geometry up to orthogonal transformation, while restricted conditionals can make non-orthogonal maps attain strictly lower asymptotic contrastive loss. We introduce a support-corrected Information Noise Contrastive Estimation (InfoNCE) variant as a theoretical fix: this correction makes orthogonal latent space recovery achievable but does not uniquely select it. Experiments on synthetic benchmarks validate the identifiability predictions, and CIFAR-10 experiments are consistent with the qualitative prediction that architectural inductive bias becomes more important when sampling diversity is limited. Together, our results clarify how sampling mechanisms and encoder inductive bias interact in contrastive representation learning.