# Variational inference of fractional Brownian motion with linear computational complexity

arXiv: https://arxiv.org/abs/2203.07961 · HTML: https://mlsanity.com/paper/2203.07961/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 82 references that are in the map.
- First posted on arXiv 15 Mar 2022.

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

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Instance Normalization: The Missing Ingredient for Fast Stylization](https://mlsanity.com/paper/1607.08022/) (2016)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Amortized Bayesian model comparison with evidential deep learning](https://mlsanity.com/paper/2004.10629/) (2020)
- [Learning physical properties of anomalous random walks using graph neural networks](https://mlsanity.com/paper/2103.11738/) (2021)
- [Efficient recurrent neural network methods for anomalously diffusing single particle short and noisy trajectories](https://mlsanity.com/paper/2108.02834/) (2021)
- [Bayesian inference of scaled versus fractional Brownian motion](https://mlsanity.com/paper/2201.00654/) (2021)
- [AGSTN: Learning Attention-adjusted Graph Spatio-Temporal Networks for Short-term Urban Sensor Value Forecasting](https://mlsanity.com/paper/2101.12465/) (2021)
- [Fully differentiable model discovery](https://mlsanity.com/paper/2106.04886/) (2021)

## Built on by (most cited first, 4 of 4)

- [BayesFlow: Amortized Bayesian Workflows With Neural Networks](https://mlsanity.com/paper/2306.16015/) (2023)
- [Machine Learning Analysis of Anomalous Diffusion](https://mlsanity.com/paper/2412.01393/) (2024)
- [Machine-Learning Solutions for the Analysis of Single-Particle Diffusion Trajectories](https://mlsanity.com/paper/2308.09414/) (2023)
- [Data-Efficient Learning of Anomalous Diffusion with Wavelet Representations: Enabling Direct Learning from Experimental Trajectories](https://mlsanity.com/paper/2512.08510/) (2025)

## Abstract (arXiv)

We introduce a simulation-based, amortised Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. In the first step a graph neural network is trained on simulated data to learn optimized low-dimensional summary statistics of the random walk. In the second step an invertible neural network generates the posterior distribution of the parameters from the learnt summary statistics using variational inference. We apply our method to infer the parameters of the fractional Brownian motion model from single trajectories. The computational complexity of the amortized inference procedure scales linearly with trajectory length, and its precision scales similarly to the Cram'er-Rao bound over a wide range of lengths. The approach is robust to positional noise, and generalizes well to trajectories longer than those seen during training. Finally, we adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.