# Multimeasurement Generative Models

arXiv: https://arxiv.org/abs/2112.09822 · HTML: https://mlsanity.com/paper/2112.09822/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 66 references that are in the map.
- First posted on arXiv 18 Dec 2021.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [U2-Net: Going Deeper with Nested U-Structure for Salient Object Detection](https://mlsanity.com/paper/2005.09007/) (2020)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Large image datasets: A pyrrhic win for computer vision?](https://mlsanity.com/paper/2006.16923/) (2020)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [Adversarial score matching and improved sampling for image generation](https://mlsanity.com/paper/2009.05475/) (2020)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)
- [Cooperative Training of Descriptor and Generator Networks](https://mlsanity.com/paper/1609.09408/) (2016)
- [Neural Empirical Bayes](https://mlsanity.com/paper/1903.02334/) (2019)
- [Learning Energy-Based Model with Variational Auto-Encoder as Amortized Sampler](https://mlsanity.com/paper/2012.14936/) (2020)
- [Deep Energy Estimator Networks](https://mlsanity.com/paper/1805.08306/) (2018)

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

- [3D molecule generation by denoising voxel grids](https://mlsanity.com/paper/2306.07473/) (2023)
- [Protein Discovery with Discrete Walk-Jump Sampling](https://mlsanity.com/paper/2306.12360/) (2023)
- [Stochastic Localization via Iterative Posterior Sampling](https://mlsanity.com/paper/2402.10758/) (2024)
- [Chain of Log-Concave Markov Chains](https://mlsanity.com/paper/2305.19473/) (2023)
- [Universal Smoothed Score Functions for Generative Modeling](https://mlsanity.com/paper/2303.11669/) (2023)
- [NEBULA: Neural Empirical Bayes Under Latent Representations for Efficient and Controllable Design of Molecular Libraries](https://mlsanity.com/paper/2407.03428/) (2024)
- [Unified all-atom molecule generation with neural fields](https://mlsanity.com/paper/2511.15906/) (2025)
- [Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped](https://mlsanity.com/paper/2510.03949/) (2025)

## Abstract (arXiv)

We formally map the problem of sampling from an unknown distribution with a density in ℝd to the problem of learning and sampling a smoother density in ℝMd obtained by convolution with a fixed factorial kernel: the new density is referred to as M-density and the kernel as multimeasurement noise model (MNM). The M-density in ℝMd is smoother than the original density in ℝd, easier to learn and sample from, yet for large M the two problems are mathematically equivalent since clean data can be estimated exactly given a multimeasurement noisy observation using the Bayes estimator. To formulate the problem, we derive the Bayes estimator for Poisson and Gaussian MNMs in closed form in terms of the unnormalized M-density. This leads to a simple least-squares objective for learning parametric energy and score functions. We present various parametrization schemes of interest including one in which studying Gaussian M-densities directly leads to multidenoising autoencoders--this is the first theoretical connection made between denoising autoencoders and empirical Bayes in the literature. Samples in ℝd are obtained by walk-jump sampling (Saremi & Hyvarinen, 2019) via underdamped Langevin MCMC (walk) to sample from M-density and the multimeasurement Bayes estimation (jump). We study permutation invariant Gaussian M-densities on MNIST, CIFAR-10, and FFHQ-256 datasets, and demonstrate the effectiveness of this framework for realizing fast-mixing stable Markov chains in high dimensions.