# Robust Principal Component Analysis Using a Novel Kernel Related with the L1-Norm

arXiv: https://arxiv.org/abs/2105.11634 · HTML: https://mlsanity.com/paper/2105.11634/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 30 references that are in the map.
- First posted on arXiv 25 May 2021.

## Builds on (most cited first, 5 of 5)

- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [Turning Big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering](https://mlsanity.com/paper/1807.04518/) (2018)
- [Optimal Algorithms for L1-subspace Signal Processing](https://mlsanity.com/paper/1405.6785/) (2014)
- [Robust and Computationally-Efficient Anomaly Detection using Powers-of-Two Networks](https://mlsanity.com/paper/1910.14096/) (2019)
- [A Generalization of Principal Component Analysis](https://mlsanity.com/paper/1910.13511/) (2019)

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

- [Multiplication-Avoiding Variant of Power Iteration with Applications](https://mlsanity.com/paper/2110.12065/) (2021)

## Abstract (arXiv)

We consider a family of vector dot products that can be implemented using sign changes and addition operations only. The dot products are energy-efficient as they avoid the multiplication operation entirely. Moreover, the dot products induce the ℓ1-norm, thus providing robustness to impulsive noise. First, we analytically prove that the dot products yield symmetric, positive semi-definite generalized covariance matrices, thus enabling principal component analysis (PCA). Moreover, the generalized covariance matrices can be constructed in an Energy Efficient (EEF) manner due to the multiplication-free property of the underlying vector products. We present image reconstruction examples in which our EEF PCA method result in the highest peak signal-to-noise ratios compared to the ordinary ℓ2-PCA and the recursive ℓ1-PCA.