# How Do Flow Matching Models Memorize and Generalize in Sample Data Subspaces?

arXiv: https://arxiv.org/abs/2410.23594 · HTML: https://mlsanity.com/paper/2410.23594/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 30 of its 43 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [PacGAN: The power of two samples in generative adversarial networks](https://mlsanity.com/paper/1712.04086/) (2017)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [Multi-Agent Diverse Generative Adversarial Networks](https://mlsanity.com/paper/1704.02906/) (2017)
- [Dual Discriminator Generative Adversarial Nets](https://mlsanity.com/paper/1709.03831/) (2017)
- [Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later](https://mlsanity.com/paper/2303.02490/) (2023)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [Score-based generative models learn manifold-like structures with constrained mixing](https://mlsanity.com/paper/2311.09952/) (2023)

## Built on by (most cited first, 19 of 19)

- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [A solvable model of learning generative diffusion: theory and insights](https://mlsanity.com/paper/2501.03937/) (2025)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [Momentum Guidance: Plug-and-Play Guidance for Flow Models](https://mlsanity.com/paper/2602.20360/) (2026)
- [Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers](https://mlsanity.com/paper/2412.18730/) (2024)
- [UniFlowRestore: A General Video Restoration Framework via Flow Matching and Prompt Guidance](https://mlsanity.com/paper/2504.09069/) (2025)
- [Tree Reward-Aligned Search for TReASURe in Masked Diffusion Language Models](https://mlsanity.com/paper/2509.23146/) (2025)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [Diffusion Models Memorize in Training -- and Generalize in Inference](https://mlsanity.com/paper/2603.13419/) (2026)
- [The Amazing Stability of Flow Matching](https://mlsanity.com/paper/2604.16079/) (2026)
- [Generation Properties of Stochastic Interpolation under Finite Training Set](https://mlsanity.com/paper/2509.21925/) (2025)
- [A Quantitative Approximation Framework for Flow Distillation in Diffusion Models](https://mlsanity.com/paper/2606.03820/) (2026)
- [Follow the Mean: Reference-Guided Flow Matching](https://mlsanity.com/paper/2605.10302/) (2026)
- [Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine](https://mlsanity.com/paper/2605.20235/) (2026)
- [A Theoretical Analysis of Memory and Overfitting Phenomena in Stochastic Interpolation Models](https://mlsanity.com/paper/2606.08554/) (2026)
- [Ensemble Controlled-Flow Filtering for Implicit Data Assimilation](https://mlsanity.com/paper/2607.12975/) (2026)
- [Points as Tori: Fast Pointwise Signed Distance for Point Clouds](https://mlsanity.com/paper/2607.16946/) (2026)
- [Tracing Generated Samples to Training-Data Clusters in Flow-Matching Models](https://mlsanity.com/paper/2608.30081/) (2026)

## Abstract (arXiv)

Real-world data is often assumed to lie within a low-dimensional structure embedded in high-dimensional space. In practical settings, we observe only a finite set of samples, forming what we refer to as the sample data subspace. It serves an essential approximation supporting tasks such as dimensionality reduction and generation. A major challenge lies in whether generative models can reliably synthesize samples that stay within this subspace rather than drifting away from the underlying structure. In this work, we provide theoretical insights into this challenge by leveraging Flow Matching models, which transform a simple prior into a complex target distribution via a learned velocity field. By treating the real data distribution as discrete, we derive analytical expressions for the optimal velocity field under a Gaussian prior, showing that generated samples memorize real data points and represent the sample data subspace exactly. To generalize to suboptimal scenarios, we introduce the Orthogonal Subspace Decomposition Network (OSDNet), which systematically decomposes the velocity field into subspace and off-subspace components. Our analysis shows that the off-subspace component decays, while the subspace component generalizes within the sample data subspace, ensuring generated samples preserve both proximity and diversity.