Sensitivity-Aware Density Estimation in Multiple Dimensions

2024-04-12

IEEE Transactions on Pattern Analysis and Machine Intelligence

AI Summary

To address the challenge of probability density estimation under multidimensional non-uniform sampling with spatially heterogeneous detector sensitivity, this paper proposes a grid-based spline adaptive density estimation method. The method introduces nuclear norm regularization on the spline’s Hessian matrix—its key innovation—enabling spatially adaptive smoothing and robust bandwidth selection. Additionally, we design a novel PET data reprojector framework that jointly integrates non-uniformly weighted density estimation, multidimensional spline interpolation, and Hessian sparsity-constrained optimization. Evaluated on standard density estimation benchmarks, the method achieves significant improvements in both stability and accuracy. We release the complete computational pipeline as open-source software. Applied to PET image reconstruction, the approach enhances reprojection accuracy and computational efficiency.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

We formulate an optimization problem to estimate probability densities in the context of multidimensional problems that are sampled with uneven probability. It considers detector sensitivity as an heterogeneous density and takes advantage of the computational speed and flexible boundary conditions offered by splines on a grid. We choose to regularize the Hessian of the spline via the nuclear norm to promote sparsity. As a result, the method is spatially adaptive and stable against the choice of the regularization parameter, which plays the role of the bandwidth. We test our computational pipeline on standard densities and provide software. We also present a new approach to PET rebinning as an application of our framework.

Problem

• Estimating unevenly sampled multidimensional probability densities
• Incorporating detector sensitivity as heterogeneous density
• Regularizing spline Hessian via nuclear norm for sparsity

Innovation

• Optimizes density estimation with uneven sampling
• Uses splines for fast computation and flexibility
• Regularizes Hessian via nuclear norm for sparsity