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A Comparative Study of Invariance-Aware Loss Functions for Deep Learning-based Gridless Direction-of-Arrival Estimation

Kuan-Lin Chen, Bhaskar D. Rao

TL;DR

This work tackles gridless direction-of-arrival estimation for sparse linear arrays by moving beyond covariance-matrix reconstruction toward invariance-aware loss functions. It introduces a scale-invariant loss based on the scale-invariant SDR and analyzes its invariance properties, comparing it to affine- and geodesic-distance-based approaches and to a recently proposed subspace loss that operates on Grassmann manifolds. Experiments on minimum-redundancy and nested MRAs demonstrate that scale-invariant losses outperform non-invariant alternatives, while the subspace loss achieves the best performance due to its maximal invariance to basis changes. The findings indicate that increasing invariance in loss design expands the effective solution space and improves learning, providing practical guidance for loss design in gridless DoA estimation with DL.

Abstract

Covariance matrix reconstruction has been the most widely used guiding objective in gridless direction-of-arrival (DoA) estimation for sparse linear arrays. Many semidefinite programming (SDP)-based methods fall under this category. Although deep learning-based approaches enable the construction of more sophisticated objective functions, most methods still rely on covariance matrix reconstruction. In this paper, we propose new loss functions that are invariant to the scaling of the matrices and provide a comparative study of losses with varying degrees of invariance. The proposed loss functions are formulated based on the scale-invariant signal-to-distortion ratio between the target matrix and the Gram matrix of the prediction. Numerical results show that a scale-invariant loss outperforms its non-invariant counterpart but is inferior to the recently proposed subspace loss that is invariant to the change of basis. These results provide evidence that designing loss functions with greater degrees of invariance is advantageous in deep learning-based gridless DoA estimation.

A Comparative Study of Invariance-Aware Loss Functions for Deep Learning-based Gridless Direction-of-Arrival Estimation

TL;DR

This work tackles gridless direction-of-arrival estimation for sparse linear arrays by moving beyond covariance-matrix reconstruction toward invariance-aware loss functions. It introduces a scale-invariant loss based on the scale-invariant SDR and analyzes its invariance properties, comparing it to affine- and geodesic-distance-based approaches and to a recently proposed subspace loss that operates on Grassmann manifolds. Experiments on minimum-redundancy and nested MRAs demonstrate that scale-invariant losses outperform non-invariant alternatives, while the subspace loss achieves the best performance due to its maximal invariance to basis changes. The findings indicate that increasing invariance in loss design expands the effective solution space and improves learning, providing practical guidance for loss design in gridless DoA estimation with DL.

Abstract

Covariance matrix reconstruction has been the most widely used guiding objective in gridless direction-of-arrival (DoA) estimation for sparse linear arrays. Many semidefinite programming (SDP)-based methods fall under this category. Although deep learning-based approaches enable the construction of more sophisticated objective functions, most methods still rely on covariance matrix reconstruction. In this paper, we propose new loss functions that are invariant to the scaling of the matrices and provide a comparative study of losses with varying degrees of invariance. The proposed loss functions are formulated based on the scale-invariant signal-to-distortion ratio between the target matrix and the Gram matrix of the prediction. Numerical results show that a scale-invariant loss outperforms its non-invariant counterpart but is inferior to the recently proposed subspace loss that is invariant to the change of basis. These results provide evidence that designing loss functions with greater degrees of invariance is advantageous in deep learning-based gridless DoA estimation.

Paper Structure

This paper contains 13 sections, 1 theorem, 13 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Assumptions and the signal model
  3. Scale-invariant Loss Functions
  4. Geodesic distances and their invariances
  5. Numerical Results
  6. Settings
  7. Methods
  8. Models
  9. Training
  10. Benefits of scale invariances
  11. On greater degrees of invariance
  12. Other MRAs
  13. Conclusion

Key Result

Proposition 1

For every $m$-by-$m$ Hermitian matrix such that $\mathbf{R}\succ 0$ and for every $\alpha>0$,

Figures (3)

  • Figure 1: MSE vs. SNR. The proposed scale-invariant covariance matrix reconstruction approach (SI-Cov) outperforms DA, SPA, and the deep learning-based covariance matrix reconstruction approach (Cov) using the Frobenius norm when $k>2$.
  • Figure 2: MSE vs. number of snapshots. Despite these models are trained with their corresponding loss functions at $T=50$ snapshots, they are able to generalize to unseen numbers of snapshots. The proposed SI-Cov outperforms Cov, showing the advantage of using the scale-invariant strategy.
  • Figure 3: The subspace loss outperforms all the other loss functions designed for covariance matrix reconstruction. In general, increasing the degrees of invariance leads to a better optimization landscape and yields better performance. SI-Cov and Cov-Aff have mixed results.

Theorems & Definitions (2)

  • Proposition 1
  • proof