# Step and Smooth Decompositions as Topological Clustering

arXiv: https://arxiv.org/abs/2311.05756 · HTML: https://mlsanity.com/paper/2311.05756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 9 Nov 2023.

## Builds on (most cited first, 6 of 6)

- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [On the Identifiability of Nonlinear ICA: Sparsity and Beyond](https://mlsanity.com/paper/2206.07751/) (2022)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Compressed sensing of low-rank plus sparse matrices](https://mlsanity.com/paper/2007.09457/) (2020)
- [Near-Optimal Estimation of Simultaneously Sparse and Low-Rank Matrices from Nested Linear Measurements](https://mlsanity.com/paper/1506.08159/) (2015)

## Abstract (arXiv)

We investigate a class of recovery problems for which observations are a noisy combination of continuous and step functions. These problems can be seen as non-injective instances of non-linear ICA with direct applications to image decontamination for magnetic resonance imaging. Alternately, the problem can be viewed as clustering in the presence of structured (smooth) contaminant. We show that a global topological property (graph connectivity) interacts with a local property (the degree of smoothness of the continuous component) to determine conditions under which the components are identifiable. Additionally, a practical estimation algorithm is provided for the case when the contaminant lies in a reproducing kernel Hilbert space of continuous functions. Algorithm effectiveness is demonstrated through a series of simulations and real-world studies.