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

• URL: http://arxiv.org/abs/2304.09242v2
• Date: Tue, 31 Oct 2023 21:00:01 GMT
• Title: A Framework for Analyzing Cross-correlators using Price's Theorem and Piecewise-Linear Decomposition
• Authors: Zhili Xiao and Shantanu Chakrabartty
• Abstract summary: We present a general mathematical framework that allows us to analyze cross-correlators constructed using a mixture of piece-wise linear functions. 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.
• Score: 5.094549132183797
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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