# Generation Properties of Stochastic Interpolation under Finite Training Set

arXiv: https://arxiv.org/abs/2509.21925 · HTML: https://mlsanity.com/paper/2509.21925/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 32 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 30 of 31)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Improved Baselines with Momentum Contrastive Learning](https://mlsanity.com/paper/2003.04297/) (2020)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [MANIQA: Multi-dimension Attention Network for No-Reference Image Quality Assessment](https://mlsanity.com/paper/2204.08958/) (2022)
- [SeeSR: Towards Semantics-Aware Real-World Image Super-Resolution](https://mlsanity.com/paper/2311.16518/) (2023)
- [Scaling Diffusion Language Models via Adaptation from Autoregressive Models](https://mlsanity.com/paper/2410.17891/) (2024)
- [One-Step Effective Diffusion Network for Real-World Image Super-Resolution](https://mlsanity.com/paper/2406.08177/) (2024)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Understanding and Mitigating Copying in Diffusion Models](https://mlsanity.com/paper/2305.20086/) (2023)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [A Non-Parametric Test to Detect Data-Copying in Generative Models](https://mlsanity.com/paper/2004.05675/) (2020)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Diffusion Language Models Can Perform Many Tasks with Scaling and Instruction-Finetuning](https://mlsanity.com/paper/2308.12219/) (2023)
- [On the Generalization of Diffusion Model](https://mlsanity.com/paper/2305.14712/) (2023)
- [Do Generated Data Always Help Contrastive Learning?](https://mlsanity.com/paper/2403.12448/) (2024)
- [The Emergence of Reproducibility and Generalizability in Diffusion Models](https://mlsanity.com/paper/2310.05264/) (2023)
- [How Do Flow Matching Models Memorize and Generalize in Sample Data Subspaces?](https://mlsanity.com/paper/2410.23594/) (2024)
- [Data-Copying in Generative Models: A Formal Framework](https://mlsanity.com/paper/2302.13181/) (2023)

## Built on by (most cited first, 1 of 1)

- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)

## Abstract (arXiv)

This paper investigates the theoretical behavior of generative models under finite training populations. Within the stochastic interpolation generative framework, we derive closed-form expressions for the optimal velocity field and score function when only a finite number of training samples are available. We demonstrate that, under some regularity conditions, the deterministic generative process exactly recovers the training samples, while the stochastic generative process manifests as training samples with added Gaussian noise. Beyond the idealized setting, we consider model estimation errors and introduce formal definitions of underfitting and overfitting specific to generative models. Our theoretical analysis reveals that, in the presence of estimation errors, the stochastic generation process effectively produces convex combinations of training samples corrupted by a mixture of uniform and Gaussian noise. Experiments on generation tasks and downstream tasks such as classification support our theory.