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arXiv · 2304.05053

Density Estimation on the Binary Hypercube using Transformed Fourier-Walsh Diagonalizations

Arthur C. Campello
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Abstract

This article focuses on estimating distribution elements over a high-dimensional binary hypercube from multivariate binary data. A popular approach to this problem, optimizing Walsh basis coefficients, is made more interpretable by an alternative representation as a "Fourier-Walsh" diagonalization. Allowing monotonic transformations of the resulting matrix elements yields a versatile binary density estimator: the main contribution of this article. It is shown that the Aitchison and Aitken kernel emerges from a constrained exponential form of this estimator, and that relaxing these constraints yields a flexible variable-weighted version of the kernel that retains positive-definiteness. Estimators within this unifying framework mix together well and span over extremes of the speed-flexibility trade-off, allowing them to serve a wide range of statistical inference and learning problems.

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Arthur C. Campello. 2023-04-11. Density Estimation on the Binary Hypercube using Transformed Fourier-Walsh Diagonalizations. https://arxiv.org/abs/2304.05053

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