# Simultaneous Best Subset Selection and Dimension Reduction via Primal-Dual Iterations

arXiv: https://arxiv.org/abs/2211.15889 · HTML: https://mlsanity.com/paper/2211.15889/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 29 Nov 2022.

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

- [High-Dimensional Linear Regression via Implicit Regularization](https://mlsanity.com/paper/1903.09367/) (2019)
- [Tuning parameter selection in high dimensional penalized likelihood](https://mlsanity.com/paper/1605.03321/) (2016)
- [Boosted Sparse and Low-Rank Tensor Regression](https://mlsanity.com/paper/1811.01158/) (2018)
- [SOFAR: large-scale association network learning](https://mlsanity.com/paper/1704.08349/) (2017)
- [Gradient-based Sparse Principal Component Analysis with Extensions to Online Learning](https://mlsanity.com/paper/1911.08048/) (2019)
- [Scalable Interpretable Multi-Response Regression via SEED](https://mlsanity.com/paper/1608.03686/) (2016)
- [Asymptotic properties for combined L1 and concave regularization](https://mlsanity.com/paper/1605.03335/) (2016)
- [Adaptive Estimation in Two-way Sparse Reduced-rank Regression](https://mlsanity.com/paper/1403.1922/) (2014)

## Abstract (arXiv)

Sparse reduced rank regression is an essential statistical learning method. In the contemporary literature, estimation is typically formulated as a nonconvex optimization that often yields to a local optimum in numerical computation. Yet, their theoretical analysis is always centered on the global optimum, resulting in a discrepancy between the statistical guarantee and the numerical computation. In this research, we offer a new algorithm to address the problem and establish an almost optimal rate for the algorithmic solution. We also demonstrate that the algorithm achieves the estimation with a polynomial number of iterations. In addition, we present a generalized information criterion to simultaneously ensure the consistency of support set recovery and rank estimation. Under the proposed criterion, we show that our algorithm can achieve the oracle reduced rank estimation with a significant probability. The numerical studies and an application in the ovarian cancer genetic data demonstrate the effectiveness and scalability of our approach.