# ℓ1-norm constrained multi-block sparse canonical correlation analysis via proximal gradient descent

arXiv: https://arxiv.org/abs/2201.05289 · HTML: https://mlsanity.com/paper/2201.05289/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 14 Jan 2022.

## Builds on (most cited first, 8 of 8)

- [Sparse CCA: Adaptive Estimation and Computational Barriers](https://mlsanity.com/paper/1409.8565/) (2014)
- [Sparse Generalized Eigenvalue Problem: Optimal Statistical Rates via Truncated Rayleigh Flow](https://mlsanity.com/paper/1604.08697/) (2016)
- [Sparse CCA via Precision Adjusted Iterative Thresholding](https://mlsanity.com/paper/1311.6186/) (2013)
- [Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem](https://mlsanity.com/paper/1711.03170/) (2017)
- [Structured SUMCOR Multiview Canonical Correlation Analysis for Large-Scale Data](https://mlsanity.com/paper/1804.08806/) (2018)
- [Sparse GCA and Thresholded Gradient Descent](https://mlsanity.com/paper/2107.00371/) (2021)
- [Penalized versus constrained generalized eigenvalue problems](https://mlsanity.com/paper/1410.6131/) (2014)
- [An Inverse-free Truncated Rayleigh-Ritz Method for Sparse Generalized Eigenvalue Problem](https://mlsanity.com/paper/2003.10897/) (2020)

## Built on by (most cited first, 1 of 1)

- [Penalized Principal Component Analysis Using Smoothing](https://mlsanity.com/paper/2309.13838/) (2023)

## Abstract (arXiv)

Multi-block CCA constructs linear relationships explaining coherent variations across multiple blocks of data. We view the multi-block CCA problem as finding leading generalized eigenvectors and propose to solve it via a proximal gradient descent algorithm with ℓ1 constraint for high dimensional data. In particular, we use a decaying sequence of constraints over proximal iterations, and show that the resulting estimate is rate-optimal under suitable assumptions. Although several previous works have demonstrated such optimality for the ℓ0 constrained problem using iterative approaches, the same level of theoretical understanding for the ℓ1 constrained formulation is still lacking. We also describe an easy-to-implement deflation procedure to estimate multiple eigenvectors sequentially. We compare our proposals to several existing methods whose implementations are available on R CRAN, and the proposed methods show competitive performances in both simulations and a real data example.