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

arXiv: https://arxiv.org/abs/2503.12386 · HTML: https://mlsanity.com/paper/2503.12386/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 16 Mar 2025.

## Builds on (most cited first, 10 of 10)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [SDR - half-baked or well done?](https://mlsanity.com/paper/1811.02508/) (2018)
- [Improved Bounds on Neural Complexity for Representing Piecewise Linear Functions](https://mlsanity.com/paper/2210.07236/) (2022)
- [Deep Networks for Direction-of-Arrival Estimation in Low SNR](https://mlsanity.com/paper/2011.08848/) (2020)
- [ResNEsts and DenseNEsts: Block-based DNN Models with Improved Representation Guarantees](https://mlsanity.com/paper/2111.05496/) (2021)
- [Maximum Likelihood-based Gridless DoA Estimation Using Structured Covariance Matrix Recovery and SBL with Grid Refinement](https://mlsanity.com/paper/2210.03266/) (2022)
- [A DNN based Normalized Time-frequency Weighted Criterion for Robust Wideband DoA Estimation](https://mlsanity.com/paper/2302.10147/) (2023)
- [Subspace Representation Learning for Sparse Linear Arrays to Localize More Sources than Sensors: A Deep Learning Methodology](https://mlsanity.com/paper/2408.16605/) (2024)

## Abstract (arXiv)

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.