# Energy-Based Models for Functional Data using Path Measure Tilting

arXiv: https://arxiv.org/abs/2202.01929 · HTML: https://mlsanity.com/paper/2202.01929/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 57 references that are in the map.
- First posted on arXiv 4 Feb 2022.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Image Generators with Conditionally-Independent Pixel Synthesis](https://mlsanity.com/paper/2011.13775/) (2020)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [Stochastic Variational Deep Kernel Learning](https://mlsanity.com/paper/1611.00336/) (2016)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Your GAN is Secretly an Energy-based Model and You Should use Discriminator Driven Latent Sampling](https://mlsanity.com/paper/2003.06060/) (2020)
- [A Contrastive Learning Approach for Training Variational Autoencoder Priors](https://mlsanity.com/paper/2010.02917/) (2020)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Variational Implicit Processes](https://mlsanity.com/paper/1806.02390/) (2018)
- [The continuous Bernoulli: fixing a pervasive error in variational autoencoders](https://mlsanity.com/paper/1907.06845/) (2019)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [Visualizing Data using GTSNE](https://mlsanity.com/paper/2108.01301/) (2021)
- [Learning Scalable Deep Kernels with Recurrent Structure](https://mlsanity.com/paper/1610.08936/) (2016)
- [Energy-based models for atomic-resolution protein conformations](https://mlsanity.com/paper/2004.13167/) (2020)
- [Energy-Based Processes for Exchangeable Data](https://mlsanity.com/paper/2003.07521/) (2020)
- [Exponential Tilting of Generative Models: Improving Sample Quality by Training and Sampling from Latent Energy](https://mlsanity.com/paper/2006.08100/) (2020)

## Built on by (most cited first, 5 of 5)

- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Functional Mean Flow in Hilbert Space](https://mlsanity.com/paper/2511.12898/) (2025)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Deep Optimal Sensor Placement for Black Box Stochastic Simulations](https://mlsanity.com/paper/2410.12036/) (2024)
- [FuncGenFoil: Airfoil Generation and Editing Model in Function Space](https://mlsanity.com/paper/2502.10712/) (2025)

## Abstract (arXiv)

Energy-Based Models (EBMs) have proven to be a highly effective approach for modelling densities on finite-dimensional spaces. Their ability to incorporate domain-specific choices and constraints into the structure of the model through composition make EBMs an appealing candidate for applications in physics, biology and computer vision and various other fields. Recently, Energy-Based Processes (EBP) for modelling stochastic processes was proposed for textitunconditional exchangeable data (e.g., point clouds). In this work, we present a novel subclass of EBPs, called F-EBM for textitconditional exchangeable data, which is able to learn distributions of functions (such as curves or surfaces) from functional samples evaluated at finitely many points. Two unique challenges arise in the functional context. Firstly, training data is often not evaluated along a fixed set of points. Secondly, steps must be taken to control the behaviour of the model between evaluation points, to mitigate overfitting. The proposed model is an energy based model on function space that is decomposed spectrally, where a Gaussian Process path measure is used to reweight the distribution to capture smoothness properties of the underlying process being modelled. The resulting model has the ability to utilize irregularly sampled training data and can output predictions at any resolution, providing an effective approach to up-scaling functional data. We demonstrate the efficacy of our proposed approach for modelling a range of datasets, including data collected from Standard and Poor's 500 (S&P) and UK National grid.