# The Sparse Manifold Transform

arXiv: https://arxiv.org/abs/1806.08887 · HTML: https://mlsanity.com/paper/1806.08887/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2018.

## Built on by (most cited first, 30 of 36)

- [White-Box Transformers via Sparse Rate Reduction](https://mlsanity.com/paper/2306.01129/) (2023)
- [Neural Manifold Clustering and Embedding](https://mlsanity.com/paper/2201.10000/) (2022)
- [Where Do We Stand with Implicit Neural Representations? A Technical and Performance Survey](https://mlsanity.com/paper/2411.03688/) (2024)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [Attention Approximates Sparse Distributed Memory](https://mlsanity.com/paper/2111.05498/) (2021)
- [Emergence of Segmentation with Minimalistic White-Box Transformers](https://mlsanity.com/paper/2308.16271/) (2023)
- [EMP-SSL: Towards Self-Supervised Learning in One Training Epoch](https://mlsanity.com/paper/2304.03977/) (2023)
- [Two Sparsities Are Better Than One: Unlocking the Performance Benefits of Sparse-Sparse Networks](https://mlsanity.com/paper/2112.13896/) (2021)
- [BeMap: Balanced Message Passing for Fair Graph Neural Network](https://mlsanity.com/paper/2306.04107/) (2023)
- [SynthSAEBench: Evaluating Sparse Autoencoders on Scalable Realistic Synthetic Data](https://mlsanity.com/paper/2602.14687/) (2026)
- [Bag of Image Patch Embedding Behind the Success of Self-Supervised Learning](https://mlsanity.com/paper/2206.08954/) (2022)
- [Identifying Interpretable Visual Features in Artificial and Biological Neural Systems](https://mlsanity.com/paper/2310.11431/) (2023)
- [Pose-Aware Self-Supervised Learning with Viewpoint Trajectory Regularization](https://mlsanity.com/paper/2403.14973/) (2024)
- [Minimalistic Unsupervised Learning with the Sparse Manifold Transform](https://mlsanity.com/paper/2209.15261/) (2022)
- [Simple Emergent Action Representations from Multi-Task Policy Training](https://mlsanity.com/paper/2210.09566/) (2022)
- [A polar prediction model for learning to represent visual transformations](https://mlsanity.com/paper/2303.03432/) (2023)
- [Context-sensitive neocortical neurons transform the effectiveness and efficiency of neural information processing](https://mlsanity.com/paper/2207.07338/) (2022)
- [Unlocking the potential of two-point cells for energy-efficient and resilient training of deep nets](https://mlsanity.com/paper/2211.01950/) (2022)
- [A Non-negative VAE:the Generalized Gamma Belief Network](https://mlsanity.com/paper/2408.03388/) (2024)
- [Sparse Coding with Multi-Layer Decoders using Variance Regularization](https://mlsanity.com/paper/2112.09214/) (2021)
- [Cloning Deterministic Worlds: The Critical Role of Latent Geometry in Long-Horizon World Models](https://mlsanity.com/paper/2510.26782/) (2025)
- [Improved Training of Sparse Coding Variational Autoencoder via Weight Normalization](https://mlsanity.com/paper/2101.09453/) (2021)
- [Block Sparse Bayesian Learning: A Diversified Scheme](https://mlsanity.com/paper/2402.04646/) (2024)
- [SL-CycleGAN: Blind Motion Deblurring in Cycles using Sparse Learning](https://mlsanity.com/paper/2111.04026/) (2021)
- [Manifold Contrastive Learning with Variational Lie Group Operators](https://mlsanity.com/paper/2306.13544/) (2023)
- [URLOST: Unsupervised Representation Learning without Stationarity or Topology](https://mlsanity.com/paper/2310.04496/) (2023)
- [Weaver: Kronecker Product Approximations of Spatiotemporal Attention for Traffic Network Forecasting](https://mlsanity.com/paper/2511.08888/) (2025)
- [Sensitivity of sparse codes to image distortions](https://mlsanity.com/paper/2204.07466/) (2022)
- [Commutativity and Disentanglement from the Manifold Perspective](https://mlsanity.com/paper/2210.07857/) (2022)
- [Predictive Sparse Manifold Transform](https://mlsanity.com/paper/2308.14207/) (2023)

## Abstract (arXiv)

We present a signal representation framework called the sparse manifold transform that combines key ideas from sparse coding, manifold learning, and slow feature analysis. It turns non-linear transformations in the primary sensory signal space into linear interpolations in a representational embedding space while maintaining approximate invertibility. The sparse manifold transform is an unsupervised and generative framework that explicitly and simultaneously models the sparse discreteness and low-dimensional manifold structure found in natural scenes. When stacked, it also models hierarchical composition. We provide a theoretical description of the transform and demonstrate properties of the learned representation on both synthetic data and natural videos.