# The local low-dimensionality of natural images

arXiv: https://arxiv.org/abs/1412.6626 · HTML: https://mlsanity.com/paper/1412.6626/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2014.

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

- [Learning Efficient Coding of Natural Images with Maximum Manifold Capacity Representations](https://mlsanity.com/paper/2303.03307/) (2023)
- [Blind denoising diffusion models and the blessings of dimensionality](https://mlsanity.com/paper/2602.09639/) (2026)
- [Minimalistic Unsupervised Learning with the Sparse Manifold Transform](https://mlsanity.com/paper/2209.15261/) (2022)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)
- [Do Neural Networks Compress Manifolds Optimally?](https://mlsanity.com/paper/2205.08518/) (2022)
- [What Time Is It? How Data Geometry Makes Time Conditioning Optional for Flow Matching](https://mlsanity.com/paper/2605.08344/) (2026)
- [Spectral phase transitions and trainability in neural network learning dynamics](https://mlsanity.com/paper/2606.28486/) (2026)
- [The Effects of Image Distribution and Task on Adversarial Robustness](https://mlsanity.com/paper/2102.10534/) (2021)

## Abstract (arXiv)

We develop a new statistical model for photographic images, in which the local responses of a bank of linear filters are described as jointly Gaussian, with zero mean and a covariance that varies slowly over spatial position. We optimize sets of filters so as to minimize the nuclear norms of matrices of their local activations (i.e., the sum of the singular values), thus encouraging a flexible form of sparsity that is not tied to any particular dictionary or coordinate system. Filters optimized according to this objective are oriented and bandpass, and their responses exhibit substantial local correlation. We show that images can be reconstructed nearly perfectly from estimates of the local filter response covariances alone, and with minimal degradation (either visual or MSE) from low-rank approximations of these covariances. As such, this representation holds much promise for use in applications such as denoising, compression, and texture representation, and may form a useful substrate for hierarchical decompositions.