# Score Matching With Missing Data

arXiv: https://arxiv.org/abs/2506.00557 · HTML: https://mlsanity.com/paper/2506.00557/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 31 references that are in the map.
- First posted on arXiv 31 May 2025.

## Builds on (most cited first, 23 of 23)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [GAIN: Missing Data Imputation using Generative Adversarial Nets](https://mlsanity.com/paper/1806.02920/) (2018)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [MissDiff: Training Diffusion Models on Tabular Data with Missing Values](https://mlsanity.com/paper/2307.00467/) (2023)
- [MCFlow: Monte Carlo Flow Models for Data Imputation](https://mlsanity.com/paper/2003.12628/) (2020)
- [Learning Energy-Based Models in High-Dimensional Spaces with Multi-scale Denoising Score Matching](https://mlsanity.com/paper/1910.07762/) (2019)
- [Estimation of High-Dimensional Graphical Models Using Regularized Score Matching](https://mlsanity.com/paper/1507.00433/) (2015)
- [Estimating Density Models with Truncation Boundaries using Score Matching](https://mlsanity.com/paper/1910.03834/) (2019)
- [Generalized Score Matching for General Domains](https://mlsanity.com/paper/2009.11428/) (2020)
- [Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions](https://mlsanity.com/paper/1708.09479/) (2017)
- [Bi-level Score Matching for Learning Energy-based Latent Variable Models](https://mlsanity.com/paper/2010.07856/) (2020)
- [Variational (Gradient) Estimate of the Score Function in Energy-based Latent Variable Models](https://mlsanity.com/paper/2010.08258/) (2020)
- [Graphical Models for Non-Negative Data Using Generalized Score Matching](https://mlsanity.com/paper/1802.06340/) (2018)
- [Learning Deep Energy Models: Contrastive Divergence vs. Amortized MLE](https://mlsanity.com/paper/1707.00797/) (2017)
- [Clustering via Mode Seeking by Direct Estimation of the Gradient of a Log-Density](https://mlsanity.com/paper/1404.5028/) (2014)
- [Robust Gaussian Graphical Modeling with the Trimmed Graphical Lasso](https://mlsanity.com/paper/1510.08512/) (2015)
- [Imputation estimators for unnormalized models with missing data](https://mlsanity.com/paper/1903.03630/) (2019)

## Built on by (most cited first, 8 of 8)

- [AutoSOTA: An End-to-End Automated Research System for State-of-the-Art AI Model Discovery](https://mlsanity.com/paper/2604.05550/) (2026)
- [Incomplete Data, Complete Dynamics: A Diffusion Approach](https://mlsanity.com/paper/2509.20098/) (2025)
- [AugMask: Training Diffusion Models on Incomplete Tabular Data via Stochastic Augmentation and Masking](https://mlsanity.com/paper/2606.03347/) (2026)
- [Smooth Flow Matching for Synthesizing Functional Data](https://mlsanity.com/paper/2508.13831/) (2025)
- [Learning Robust Diffusion Models from Imprecise Supervision](https://mlsanity.com/paper/2510.03016/) (2025)
- [Latent Diffusion for Missing Data](https://mlsanity.com/paper/2605.28427/) (2026)
- [Flow Matching with Missing Data](https://mlsanity.com/paper/2607.28698/) (2026)
- [Learning Field Reconstruction from Incomplete Data by Globally Correcting Local Estimates](https://mlsanity.com/paper/2610.05375/) (2026)

## Abstract (arXiv)

Score matching is a vital tool for learning the distribution of data with applications across many areas including diffusion processes, energy based modelling, and graphical model estimation. Despite all these applications, little work explores its use when data is incomplete. We address this by adapting score matching (and its major extensions) to work with missing data in a flexible setting where data can be partially missing over any subset of the coordinates. We provide two separate score matching variations for general use, an importance weighting (IW) approach, and a variational approach. We provide finite sample bounds for our IW approach in finite domain settings and show it to have especially strong performance in small sample lower dimensional cases. Complementing this, we show our variational approach to be strongest in more complex high-dimensional settings which we demonstrate on graphical model estimation tasks on both real and simulated data.