Theoretical Refinement of CLIP by Utilizing Linear Structure of Optimal Similarity

Contrastive objectives power state-of-the-art multimodal models, but their training remains slow, relying on long stochastic optimization. We propose a Unified Framework for Efficient Contrastive Alignment via Kernels (UniCon), which spans linear and nonlinear encoders as well as one-to-one and many-to-many alignments. At its core, UniCon introduces the contrastive similarity weight matrix $S(γ)$, which enables closed-form global solutions that provably replace minibatch back-propagation with exact updates. Through the lens of reproducing kernel Hilbert spaces (RKHS), UniCon provides a kernelized perspective that unifies contrastive alignment and reveals its connection to spectral methods. To validate the theory, we conduct experiments on synthetic, unimodal, multimodal, and zero-shot tasks, demonstrating that UniCon achieves substantial efficiency gains while preserving generality and strong empirical performance.

Vision-language foundation models such as CLIP and SigLIP provide widely used representations for multimodal learning systems. While these models are typically compared through downstream performance, such evaluations often do not explain how their representations differ structurally. In this work, we study this problem through the task of Contrastive Embedding Clustering: identifying sample subsets that are weakly clustered under one representation but strongly clustered under another. We propose \emph{Kernel Optimization for Discrepancy Analysis (KODA)}, a kernel-based framework for contrastive representation comparison and alignment. KODA constructs unified multimodal kernels through modality-wise kernel composition and formulates discrepancy discovery as a constrained optimization problem that searches for coherent structures in one representation while suppressing coherence in a reference representation. This yields interpretable discrepancy directions associated with specific sample subsets and modality interactions. To scale KODA to large vision-language datasets, we develop randomized low-dimensional approximations of joint kernels using random projections, including Random Fourier Features for shift-invariant kernels. Empirically, KODA identifies consistent and interpretable discrepancy structures across vision-language representations and provides sample subsets for representation alignment. The code is available at https://github.com/yokiwuuu/KODA.

Contrastive Language-Image Pre-training (CLIP) demonstrates strong zero-shot generalization, but due to substantial computational and memory costs, distillation into lightweight models is required. Existing relational objectives do not explicitly model multidirectional relationships between teacher and student embeddings, potentially leaving the geometric relationships insufficiently constrained. This may disrupt the modality-gap structure important for zero-shot transfer. To address these limitations, we propose a relational distillation framework, CLIP-RD, which introduces two relational methods, Vertical Relational Distillation (VRD) and Cross Relational Distillation (XRD). VRD aligns teacher-student intra-modal similarity distributions to enforce consistent distillation strength across image and text embeddings. Meanwhile, XRD aligns the teacher-image-student-text and teacher-text-student-image similarity distributions to impose bidirectional cross-modal symmetry. By jointly modeling these multidirectional relational structures, CLIP-RD aligns the student's embedding geometry more faithfully to the teacher's, outperforming CLIP-KD by 1.8%p. This performance improvement is maintained across diverse architectures, teacher scales, retrieval tasks, downstream tasks, and corruption settings, with negligible additional training-time overhead.