Contrastive learning involves learning representations via a loss function that encourages each (unlabeled) sample to be far from other samples, but close to its own . In this paper, we aim to understand why this simple idea performs remarkably well, by theoretically analyzing it for a simple, natural problem setting: dimensionality reduction in Gaussian Mixture Models (GMMs). Note that the standard GMM setup lacks the concept of augmentations. We study an intuitive extension: we define the pair of data sample and its augmentation as a coupled random draw from the GMM such that the marginal over the noisy augmentation is towards the component of the data sample. For this setup, we show that vanilla contrastive loss, e.g., InfoNCE, is able to find the lower-dimensional subspace even when the Gaussian components are non-isotropic. In particular, we show that InfoNCE can match the performance of a fully supervised algorithm, e.g., LDA, (where each data point is labeled with the mixture component it comes from) even when the augmentations are noisy. We further extend our setup to the multi-modal case, and develop a GMM-like setting to study the contrastive CLIP loss. We corroborate our theoretical with real-data experiments on CIFAR100; representations learned by InfoNCE loss match the performance of LDA on clustering metrics.

We show that Contrastive Learning (CL) under a broad family of loss functions (including InfoNCE) has a unified formulation of coordinate-wise optimization on the network parameter $\vtheta$ and pairwise importance $\alpha$, where the \emph{max player} $\vtheta$ learns representation for contrastiveness, and the \emph{min player} $\alpha$ puts more weights on pairs of distinct samples that share similar representations. The resulting formulation, called \boldmethod{}, unifies not only various existing contrastive losses, which differ by how sample-pair importance $\alpha$ is constructed, but also is able to extrapolate to give novel contrastive losses beyond popular ones, opening a new avenue of contrastive loss design. These novel losses yield comparable (or better) performance on CIFAR10, STL-10 and CIFAR-100 than classic InfoNCE. Furthermore, we also analyze the max player in detail: we prove that with fixed $\alpha$, max player is equivalent to Principal Component Analysis (PCA) for deep linear network, and almost all local minima are global and rank-1, recovering optimal PCA solutions. Finally, we extend our analysis on max player to 2-layer ReLU networks, showing that its fixed points can have higher ranks.

Contrastive learning produces coherent semantic feature embeddings by encouraging positive samples to cluster closely while separating negative samples. However, existing contrastive learning methods lack principled guarantees on coverage within the semantic feature space. We extend conformal prediction to this setting by introducing minimum-volume covering sets equipped with learnable generalized multi-norm constraints. We propose a method that constructs conformal sets guaranteeing user-specified coverage of positive samples while maximizing negative sample exclusion. We establish theoretically that volume minimization serves as a proxy for negative exclusion, enabling our approach to operate effectively even when negative pairs are unavailable. The positive inclusion guarantee inherits the distribution-free coverage property of conformal prediction, while negative exclusion is maximized through learned set geometry optimized on a held-out training split. Experiments on simulated and real-world image datasets demonstrate improved inclusion-exclusion trade-offs compared to standard distance-based conformal baselines.

The code is available at Contrastive-Sampling .

Considering the complex graph structure, are some nodes consistently well-trained and following this principle even with different graph augmentations?

We propose information gating as a way to learn parsimonious representations that identify the minimal information required for a task.