Gridless Evolutionary Approach for Line Spectral Estimation with Unknown Model Order

TL;DR

A novel idea of simultaneously estimating the frequencies and model order using the atomic <inline-formula> <tex-math notation="LaTeX">$l_{0}$ </tex-math></inline-formula> norm is proposed, which builds a multiobjective optimization model and designs a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations.

Abstract

Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic $l_0$ norm (i.e., the continuous analog of $l_0$ norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of atomic $l_0$ norm, such as nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However, the relaxations give rise to a resolution limit, subsequently leading to biased model order and convergence error. To overcome the above shortcomings of relaxation, we propose a novel idea of simultaneously estimating the frequencies and model order by means of the atomic $l_0$ norm. To accomplish this idea, we build a multiobjective optimization model. The measurment error and the atomic $l_0$ norm are taken as the two optimization objectives. The proposed model directly exploits the model order via the atomic $l_0$ norm, thus breaking the resolution limit. We further design a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations. One is a variable-length coding and search strategy. It flexibly codes and interactively searches diverse solutions with different model orders. These solutions act as steppingstones that help fully exploring the variable and open-ended frequency search space and provide extensive potentials towards the optima. Another innovation is a model order pruning mechanism, which heuristically prunes less contributive frequencies within the solutions, thus significantly enhancing convergence and diversity. Simulation results confirm the superiority of our approach in both frequency estimation and model order selection.

Figures (9)

• Figure 1: Variable-length coding of frequency combinations. Each row represents a frequency combination.
• Figure 2: Variable-length crossover. (a) Link the most similar counterparts between the two parents by black oblique lines. (b) Align the two parents, and segment the two parents into $\bar{n}+1$ paired subsets by $\bar{n}$ red dotted lines, where $\bar{n}=3$ is the number of crossover points. (c) Crossover within each even paired subsets, respectively. (d) Glue subsets to produce two offsprings.
• Figure 3: Archiving and model order pruning mechanism. Vertical dotted lines refer to different model orders. Points labeled with cross symbols indicate being discarded.
• Figure 4: Pareto front (a) and slope variance (b) of final archive versus SNR for $K=4$, $M=15$, and $T=20$. The points inside the dotted line denote identified knee solutions.
• Figure 5: RMSE($\hat{\boldsymbol{\theta}}$) (a) and success rate (b) results of three versions of MVESA versus model order $K$ for $M=15$, $T=10$, and SNR$=10$dB.