Learning a Flow to Self-Supervised Representations

Most self-supervised learning objectives defend against collapse but leave the target representation law unspecified. We formulate representation learning as Distribution Matching (DM), learning an augmentation-invariant encoder whose induced law matches an explicit geometric reference. The reference law specifies what the learned representation distribution should look like, whereas a separately chosen discrepancy determines how deviations from this target are measured; here we use Mallows distance. The DM framework reveals a directional inverse: generative learning maps a tractable reference to data, whereas representation learning maps data to a designed reference law. We connect the population objective to class-centre separation and classification error and prove a non-asymptotic neural-sieve guarantee. Simulations and image benchmarks show manifold rectification, fine-grained structure and transfer across label spaces.

Flow matching (FM) trains a time-dependent vector field that transports samples from a simple prior to a complex data distribution. However, for high-dimensional images, each training sample supervises only a single trajectory and intermediate point, yielding an extremely sparse and high-variance training signal. This under-constrained supervision can cause flow collapse, where the learned dynamics memorize specific source-target pairings, mapping diverse inputs to overly similar outputs, failing to generalize. We introduce Posterior-Augmented Flow Matching (PAFM), a theoretically grounded generalization of FM that replaces single-target supervision with an expectation over an approximate posterior of valid target completions for a given intermediate state and condition. PAFM factorizes this intractable posterior into (i) the likelihood of the intermediate under a hypothesized endpoint and (ii) the prior probability of that endpoint under the condition, and uses an importance sampling scheme to construct a mixture over multiple candidate targets. We prove that PAFM yields an unbiased estimator of the original FM objective while substantially reducing gradient variance during training by aggregating information from many plausible continuation trajectories per intermediate. Finally, we show that PAFM improves over FM by up to 3.4 FID50K across different model scales (SiT-B/2 and SiT-XL/2), different architectures (SiT and MMDiT), and in both class and text conditioned benchmarks (ImageNet and CC12M), with a negligible increase in the compute overhead. Code: https://github.com/gstoica27/PAFM.git.

Dataset distillation seeks to synthesize a compact surrogate dataset that enables performance comparable to training on the original dataset for downstream tasks. For the scenario where pre-trained self-supervised models serve as priors, traditional Linear Gradient Matching optimizes synthetic images by encouraging them to mimic the gradient updates induced by real images on the linear probe. However, this batch-level formulation requires loading thousands of real images and applying multiple differentiable augmentations to synthetic images at each distillation step, leading to substantial computational and memory overheads. In this paper, we revisit the linear gradient and theoretically derive that it is essentially a local relative distribution directed from target class centers toward non-target class centers, which we term flow. This property causes suboptimality and instability, often necessitating expensive multiple augmentations to compensate. To address this, we introduce Statistical Flow Matching, an optimal, stable, and efficient supervised learning framework that optimizes synthetic images by aligning global statistical flows in the original data. Our approach loads raw statistics only once and performs a single augmentation pass on the synthetic data, achieving performance comparable to or better than the state-of-the-art method with 10x less GPU memory usage and 4x faster distillation time. Moreover, increasing the number of augmentations for our method yields further performance gains while incurring lower additional cost. Our code is publicly available at https://github.com/einsteinxia/SFM.