# EqNIO: Subequivariant Neural Inertial Odometry

arXiv: https://arxiv.org/abs/2408.06321 · HTML: https://mlsanity.com/paper/2408.06321/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 54 references that are in the map.
- First posted on arXiv 12 Aug 2024.

## Builds on (most cited first, 30 of 36)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [RoNIN: Robust Neural Inertial Navigation in the Wild: Benchmark, Evaluations, and New Methods](https://mlsanity.com/paper/1905.12853/) (2019)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [IONet: Learning to Cure the Curse of Drift in Inertial Odometry](https://mlsanity.com/paper/1802.02209/) (2018)
- [TLIO: Tight Learned Inertial Odometry](https://mlsanity.com/paper/2007.01867/) (2020)
- [Spin-Weighted Spherical CNNs](https://mlsanity.com/paper/2006.10731/) (2020)
- [Equivariant Multi-View Networks](https://mlsanity.com/paper/1904.00993/) (2019)
- [RIDI: Robust IMU Double Integration](https://mlsanity.com/paper/1712.09004/) (2017)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Aria Everyday Activities Dataset](https://mlsanity.com/paper/2402.13349/) (2024)
- [The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry](https://mlsanity.com/paper/2211.09231/) (2022)
- [AI-IMU Dead-Reckoning](https://mlsanity.com/paper/1904.06064/) (2019)
- [Multivariate Uncertainty in Deep Learning](https://mlsanity.com/paper/1910.14215/) (2019)
- [Denoising IMU Gyroscopes with Deep Learning for Open-Loop Attitude Estimation](https://mlsanity.com/paper/2002.10718/) (2020)
- [IDOL: Inertial Deep Orientation-Estimation and Localization](https://mlsanity.com/paper/2102.04024/) (2021)
- [OxIOD: The Dataset for Deep Inertial Odometry](https://mlsanity.com/paper/1809.07491/) (2018)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)

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

- [Learning IMU Bias with Diffusion Model](https://mlsanity.com/paper/2505.11763/) (2025)
- [IONext: Unlocking the Next Era of Inertial Odometry](https://mlsanity.com/paper/2507.17089/) (2025)
- [X-IONet: Cross-Platform Inertial Odometry Network for Pedestrian and Legged Robot](https://mlsanity.com/paper/2511.08277/) (2025)
- [FDIO: Frequency Decomposed Inertial Odometry](https://mlsanity.com/paper/2511.15645/) (2025)
- [Tunable Soft Equivariance with Guarantees](https://mlsanity.com/paper/2603.26657/) (2026)
- [Tracking Large-scale Shared Bikes with Inertial Motion Learning in GNSS Blocked Environments](https://mlsanity.com/paper/2605.07412/) (2026)
- [PedestrianDiffusion: Multimodal Generative Denoising and Dense State Estimation for Inertial Navigation](https://mlsanity.com/paper/2607.03349/) (2026)
- [GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry](https://mlsanity.com/paper/2609.25338/) (2026)

## Abstract (arXiv)

Neural networks are seeing rapid adoption in purely inertial odometry, where accelerometer and gyroscope measurements from commodity inertial measurement units (IMU) are used to regress displacements and associated uncertainties. They can learn informative displacement priors, which can be directly fused with the raw data with off-the-shelf non-linear filters. Nevertheless, these networks do not consider the physical roto-reflective symmetries inherent in IMU data, leading to the need to memorize the same priors for every possible motion direction, which hinders generalization. In this work, we characterize these symmetries and show that the IMU data and the resulting displacement and covariance transform equivariantly, when rotated around the gravity vector and reflected with respect to arbitrary planes parallel to gravity. We design a neural network that respects these symmetries by design through equivariant processing in three steps: First, it estimates an equivariant gravity-aligned frame from equivariant vectors and invariant scalars derived from IMU data, leveraging expressive linear and non-linear layers tailored to commute with the underlying symmetry transformation. We then map the IMU data into this frame, thereby achieving an invariant canonicalization that can be directly used with off-the-shelf inertial odometry networks. Finally, we map these network outputs back into the original frame, thereby obtaining equivariant covariances and displacements. We demonstrate the generality of our framework by applying it to the filter-based approach based on TLIO, and the end-to-end RONIN architecture, and show better performance on the TLIO, Aria, RIDI and OxIOD datasets than existing methods.