# Weighed l1 on the simplex: Compressive sensing meets locality

arXiv: https://arxiv.org/abs/2104.13894 · HTML: https://mlsanity.com/paper/2104.13894/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 28 Apr 2021.

## Builds on (most cited first, 5 of 5)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Deep Unfolding: Model-Based Inspiration of Novel Deep Architectures](https://mlsanity.com/paper/1409.2574/) (2014)
- [Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application](https://mlsanity.com/paper/1309.1541/) (2013)
- [Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization](https://mlsanity.com/paper/1310.7991/) (2013)

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

- [Geometric Sparse Coding in Wasserstein Space](https://mlsanity.com/paper/2210.12135/) (2022)
- [Sparse, Geometric Autoencoder Models of V1](https://mlsanity.com/paper/2302.11162/) (2023)

## Abstract (arXiv)

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard ℓ1 minimization problem. We propose weighted ℓ0 and weighted ℓ1 metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted ℓ0 and weighted ℓ1. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.