# Transport of Algebraic Structure to Latent Embeddings

arXiv: https://arxiv.org/abs/2405.16763 · HTML: https://mlsanity.com/paper/2405.16763/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 27 May 2024.

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

- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Text Embeddings by Weakly-Supervised Contrastive Pre-training](https://mlsanity.com/paper/2212.03533/) (2022)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Expressive Power of Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/2006.15646/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Deep Learning on Implicit Neural Representations of Shapes](https://mlsanity.com/paper/2302.05438/) (2023)
- [Occupancy Flow Fields for Motion Forecasting in Autonomous Driving](https://mlsanity.com/paper/2203.03875/) (2022)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Learning Algebraic Structures: Preliminary Investigations](https://mlsanity.com/paper/1905.02263/) (2019)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [SDF-Pack: Towards Compact Bin Packing with Signed-Distance-Field Minimization](https://mlsanity.com/paper/2307.07356/) (2023)
- [Reachability-based Trajectory Design with Neural Implicit Safety Constraints](https://mlsanity.com/paper/2302.07352/) (2023)
- [Algebraic Machine Learning](https://mlsanity.com/paper/1803.05252/) (2018)
- [Computing with Categories in Machine Learning](https://mlsanity.com/paper/2303.04156/) (2023)

## Built on by (most cited first, 2 of 2)

- [Towards a Categorical Foundation of Deep Learning: A Survey](https://mlsanity.com/paper/2410.05353/) (2024)
- [A Pre-Specified Construction-Confirmation Test of Operation-Level Causal Transfer Across Finite Isomorphic Symbolic Domains](https://mlsanity.com/paper/2608.15809/) (2026)

## Abstract (arXiv)

Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. For example, in the field of 3D modeling, subsets of Euclidean space can be embedded as vectors using implicit neural representations. Such subsets also have a natural algebraic structure including operations (e.g., union) and corresponding laws (e.g., associativity). How can we learn to "union" two sets using only their latent embeddings while respecting associativity? We propose a general procedure for parameterizing latent space operations that are provably consistent with the laws on the input space. This is achieved by learning a bijection from the latent space to a carefully designed mirrored algebra which is constructed on Euclidean space in accordance with desired laws. We evaluate these structural transport nets for a range of mirrored algebras against baselines that operate directly on the latent space. Our experiments provide strong evidence that respecting the underlying algebraic structure of the input space is key for learning accurate and self-consistent operations.