Bona Fide Riesz Projections for Density Estimation

Abstract

The projection of sample measurements onto a reconstruction space represented by a basis on a regular grid is a powerful and simple approach to estimate a probability density function. In this paper, we focus on Riesz bases and propose a projection operator that, in contrast to previous works, guarantees the bona fide properties for the estimate, namely, nonnegativity and total probability mass 1. Our bona fide projection is defined as a convex problem. We propose solution techniques and evaluate them. Results suggest an improved performance, specifically in circumstances prone to rippling effects.

Spline Local Basis Methods for Nonparametric Density Estimation

“…6. Bona Fide Projection: a very recent approach to this problem, proposed by [87], is to solve a constrained convex optimization for the optimal coefficients. As discussed in Section 2.7, this has the advantage over alternative methods of ensuring that the density is everywhere positive (a bona fide density), although it no longer admits a fast closed-form solution.…”

“…For B-splines, a simple procedure of flooring the basis coefficients, and redistributing probability mass to ensure integration to one works well in many applications, see [17]. An alternative procedure is proposed in [87], which solves a convex optimization problem for the coefficients, subject to the constraints that the density integrates to one and is non-negative. Let p δ denote the empirical measure,…”

“…This estimator, coined the "Bona Fide Projection", is also promising for applications, and (2.34) is solvable using standard quadratic programming techniques, as demonstrated in Section 3 of [87].…”