# A Framework for Analyzing Cross-correlators using Price's Theorem and Piecewise-Linear Decomposition

arXiv: https://arxiv.org/abs/2304.09242 · HTML: https://mlsanity.com/paper/2304.09242/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 35 references that are in the map.
- First posted on arXiv 18 Apr 2023.

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

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [AdderNet: Do We Really Need Multiplications in Deep Learning?](https://mlsanity.com/paper/1912.13200/) (2019)
- [A Theoretical Perspective on Hyperdimensional Computing](https://mlsanity.com/paper/2010.07426/) (2020)
- [TransRPPG: Remote Photoplethysmography Transformer for 3D Mask Face Presentation Attack Detection](https://mlsanity.com/paper/2104.07419/) (2021)
- [Attention Approximates Sparse Distributed Memory](https://mlsanity.com/paper/2111.05498/) (2021)
- [Kernel Cross-Correlator](https://mlsanity.com/paper/1709.05936/) (2017)
- [A Universal Approximation Result for Difference of log-sum-exp Neural Networks](https://mlsanity.com/paper/1905.08503/) (2019)
- [Visual Memorability for Robotic Interestingness via Unsupervised Online Learning](https://mlsanity.com/paper/2005.08829/) (2020)
- [Multiplierless MP-Kernel Machine For Energy-efficient Edge Devices](https://mlsanity.com/paper/2106.01958/) (2021)

## Abstract (arXiv)

Precise estimation of cross-correlation or similarity between two random variables lies at the heart of signal detection, hyperdimensional computing, associative memories, and neural networks. Although a vast literature exists on different methods for estimating cross-correlations, the question what is the best and simplest method to estimate cross-correlations using finite samples ? is still unclear. In this paper, we first argue that the standard empirical approach might not be the optimal method even though the estimator exhibits uniform convergence to the true cross-correlation. Instead, we show that there exists a large class of simple non-linear functions that can be used to construct cross-correlators with a higher signal-to-noise ratio (SNR). To demonstrate this, we first present a general mathematical framework using Price's Theorem that allows us to analyze cross-correlators constructed using a mixture of piece-wise linear functions. Using this framework and high-dimensional embedding, we show that some of the most promising cross-correlators are based on Huber's loss functions, margin-propagation (MP) functions, and the log-sum-exp (LSE) functions.