# Structured Sparse Principal Components Analysis with the TV-Elastic Net penalty

arXiv: https://arxiv.org/abs/1609.01423 · HTML: https://mlsanity.com/paper/1609.01423/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2016.

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

- [Analysis of Truncated Orthogonal Iteration for Sparse Eigenvector Problems](https://mlsanity.com/paper/2103.13523/) (2021)
- [Calendar-Structured Sparse Principal Component Analysis for Interpretable Multi-Periodic Electricity Consumption Profiles](https://mlsanity.com/paper/2609.06060/) (2026)

## Abstract (arXiv)

Principal component analysis (PCA) is an exploratory tool widely used in data analysis to uncover dominant patterns of variability within a population. Despite its ability to represent a data set in a low-dimensional space, the interpretability of PCA remains limited. However, in neuroimaging, it is essential to uncover clinically interpretable phenotypic markers that would account for the main variability in the brain images of a population. Recently, some alternatives to the standard PCA approach, such as Sparse PCA, have been proposed, their aim being to limit the density of the components. Nonetheless, sparsity alone does not entirely solve the interpretability problem, since it may yield scattered and unstable components. We hypothesized that the incorporation of prior information regarding the structure of the data may lead to improved relevance and interpretability of brain patterns. We therefore present a simple extension of the popular PCA framework that adds structured sparsity penalties on the loading vectors in order to identify the few stable regions in the brain images accounting for most of the variability. Such structured sparsity can be obtained by combining l1 and total variation (TV) penalties, where the TV regularization encodes higher order information about the structure of the data. This paper presents the structured sparse PCA (denoted SPCA-TV) optimization framework and its resolution. We demonstrate the efficiency and versatility of SPCA-TV on three different data sets. The gains of SPCA-TV over unstructured approaches are significant,since SPCA-TV reveals the variability within a data set in the form of intelligible brain patterns that are easy to interpret, and are more stable across different samples.