# Gradient flow encoding with distance optimization adaptive step size

arXiv: https://arxiv.org/abs/2105.05031 · HTML: https://mlsanity.com/paper/2105.05031/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 11 references that are in the map.
- First posted on arXiv 11 May 2021.

## Builds on (most cited first, 6 of 6)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Recent Advances in Autoencoder-Based Representation Learning](https://mlsanity.com/paper/1812.05069/) (2018)

## Abstract (arXiv)

The autoencoder model uses an encoder to map data samples to a lower dimensional latent space and then a decoder to map the latent space representations back to the data space. Implicitly, it relies on the encoder to approximate the inverse of the decoder network, so that samples can be mapped to and back from the latent space faithfully. This approximation may lead to sub-optimal latent space representations. In this work, we investigate a decoder-only method that uses gradient flow to encode data samples in the latent space. The gradient flow is defined based on a given decoder and aims to find the optimal latent space representation for any given sample through optimisation, eliminating the need of an approximate inversion through an encoder. Implementing gradient flow through ordinary differential equations (ODE), we leverage the adjoint method to train a given decoder. We further show empirically that the costly integrals in the adjoint method may not be entirely necessary. Additionally, we propose a 2nd order ODE variant to the method, which approximates Nesterov's accelerated gradient descent, with faster convergence per iteration. Commonly used ODE solvers can be quite sensitive to the integration step-size depending on the stiffness of the ODE. To overcome the sensitivity for gradient flow encoding, we use an adaptive solver that prioritises minimising loss at each integration step. We assess the proposed method in comparison to the autoencoding model. In our experiments, GFE showed a much higher data-efficiency than the autoencoding model, which can be crucial for data scarce applications.