Log-concave Density Estimation with Independent Components

2024/01/03 by Kubal, Sharvaj, Campbell, Christian, Robeva, Elina · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2401.01500

Abstract

We propose a method for estimating a log-concave density on \mathbb R^d from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most O(ε^-4) samples (suppressing constants and log factors) suffice for our proposed estimator to be within ε of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use.

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