Gradient density estimation in arbitrary finite dimensions using the\n method of stationary phase

2012/11/13 by Karthik S. Gurumoorthy, Gurumoorthy, Karthik S., Anand Rangarajan +3
Computer Science · Mathematics · #41A60 #42B10 #62G07 #Advanced Mathematical Modeling in Engineering #FOS: Computer and information sciences #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1211.3038

openalex publication_date 2012/11/13 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We prove that the density function of the gradient of a sufficiently smooth\nfunction S : \Ω \⊂ \ℝ^d \→ \ℝ, obtained via\na random variable transformation of a uniformly distributed random variable, is\nincreasingly closely approximated by the normalized power spectrum of\n\φ=\exp\(\(iS)/(\τ)\) as the free parameter \τ \→\n0. The result is shown using the stationary phase approximation and standard\nintegration techniques and requires proper ordering of limits. We highlight a\nrelationship with the well-known characteristic function approach to density\nestimation, and detail why our result is distinct from this approach.\n

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