# Closed-Form, Provable, and Robust PCA via Leverage Statistics and Innovation Search

arXiv: https://arxiv.org/abs/2106.12190 · HTML: https://mlsanity.com/paper/2106.12190/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 49 references that are in the map.
- First posted on arXiv 23 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [Dual Principal Component Pursuit](https://mlsanity.com/paper/1510.04390/) (2015)
- [Provable Self-Representation Based Outlier Detection in a Union of Subspaces](https://mlsanity.com/paper/1704.03925/) (2017)
- [Optimal Algorithms for L1-subspace Signal Processing](https://mlsanity.com/paper/1405.6785/) (2014)
- [Coherence Pursuit: Fast, Simple, and Robust Principal Component Analysis](https://mlsanity.com/paper/1609.04789/) (2016)
- [Greedy Subspace Clustering](https://mlsanity.com/paper/1410.8864/) (2014)
- [Innovation Pursuit: A New Approach to Subspace Clustering](https://mlsanity.com/paper/1512.00907/) (2015)
- [Subspace Clustering via Optimal Direction Search](https://mlsanity.com/paper/1706.03860/) (2017)
- [PCA leverage: outlier detection for high-dimensional functional magnetic resonance imaging data](https://mlsanity.com/paper/1509.00882/) (2015)

## Abstract (arXiv)

The idea of Innovation Search, which was initially proposed for data clustering, was recently used for outlier detection. In the application of Innovation Search for outlier detection, the directions of innovation were utilized to measure the innovation of the data points. We study the Innovation Values computed by the Innovation Search algorithm under a quadratic cost function and it is proved that Innovation Values with the new cost function are equivalent to Leverage Scores. This interesting connection is utilized to establish several theoretical guarantees for a Leverage Score based robust PCA method and to design a new robust PCA method. The theoretical results include performance guarantees with different models for the distribution of outliers and the distribution of inliers. In addition, we demonstrate the robustness of the algorithms against the presence of noise. The numerical and theoretical studies indicate that while the presented approach is fast and closed-form, it can outperform most of the existing algorithms.