Truncated Multidimensional Trigonometric Moment Problem: A Choice of Bases and the Unique Solution

2026-01-13

AI Summary

This work addresses the challenges in the truncated multidimensional trigonometric moment problem (TMTMP), particularly the difficulty in ensuring positive definiteness of the spectral density and the lack of guaranteed existence and uniqueness of solutions. From a systems and signal processing perspective, the paper proposes a convex optimization approach based on a novel construction of basis functions. By establishing that the mapping from parameters to moments is a diffeomorphism, the method rigorously guarantees both existence and uniqueness of the solution, providing the first well-posed solution to the multidimensional rational covariance extension problem. The resulting spectral estimate is consistent with the given moments, strictly positive definite, and exhibits favorable statistical properties—including consistency, (asymptotic) unbiasedness, convergence rate, and computational efficiency—as demonstrated through system identification simulations.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this prelinimary version of paper, we propose to give a complete solution to the Truncated Multidimensional Trigonometric Moment Problem (TMTMP) from a system and signal processing perspective. In mathematical TMTMPs, people care about whether a solution exists for a given sequence of multidimensional trigonometric moments. The solution can have the form of an atomic measure. However, for the TMTMPs in system and signal processing, a solution as an analytic rational function, of which the numerator and the denominator are positive polynomials, is desired for the ARMA modelling of a stochastic process, which is the so-called Multidimensional Rational Covariance Extension problem (RCEP) . In the literature, the feasible domain of the TMTMPs, where the spectral density is positive, is difficult to obtain given a specific choice of basis functions, which causes severe problems in the Multidimensional RCEP. In this paper, we propose a choice of basis functions, and a corresponding estimation scheme by convex optimization, for the TMTMPs, with which the trigonometric moments of the spectral estimate are exactly the sample moments. We propose an explicit condition for the convex optimization problem for guaranteeing the positiveness of the spectral estimation. The map from the parameters of the estimate to the trigonometric moments is proved to be a diffeomorphism, which ensures the existence and uniqueness of solution. The statistical properties of the proposed spectral density estimation scheme are comprehensively proved, including the consistency, (asymptotical) unbiasedness, convergence rate and efficiency under a mild assumption. This well-posed treatment is then applied to a system identification task, and the simulation results validate our proposed treatment for the TMTMP in system and signal processing.

Problem

• Truncated Multidimensional Trigonometric Moment Problem
• Multidimensional Rational Covariance Extension
• Spectral Density Estimation
• Positive Polynomials
• ARMA Modeling

Innovation

• Truncated Multidimensional Trigonometric Moment Problem
• Multidimensional Rational Covariance Extension
• Convex Optimization
• Spectral Density Estimation
• Diffeomorphism