# Subspace Representation Learning for Sparse Linear Arrays to Localize More Sources than Sensors: A Deep Learning Methodology

arXiv: https://arxiv.org/abs/2408.16605 · HTML: https://mlsanity.com/paper/2408.16605/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 56 references that are in the map.
- First posted on arXiv 29 Aug 2024.

## Builds on (most cited first, 12 of 12)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [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)
- [A Machine Learning Approach to DoA Estimation and Model Order Selection for Antenna Arrays with Subarray Sampling](https://mlsanity.com/paper/2009.12858/) (2020)
- [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)
- [Super-resolution with Sparse Arrays: A Non-Asymptotic Analysis of Spatio-temporal Trade-offs](https://mlsanity.com/paper/2301.01734/) (2023)

## Built on by (most cited first, 2 of 2)

- [A Comparative Study of Invariance-Aware Loss Functions for Deep Learning-based Gridless Direction-of-Arrival Estimation](https://mlsanity.com/paper/2503.12386/) (2025)
- [Reliable Near-Field Multi-User Positioning Informed by Two-Stage MUSIC](https://mlsanity.com/paper/2609.09409/) (2026)

## Abstract (arXiv)

Localizing more sources than sensors with a sparse linear array (SLA) has long relied on minimizing a distance between two covariance matrices and recent algorithms often utilize semidefinite programming (SDP). Although deep neural network (DNN)-based methods offer new alternatives, they still depend on covariance matrix fitting. In this paper, we develop a novel methodology that estimates the co-array subspaces from a sample covariance for SLAs. Our methodology trains a DNN to learn signal and noise subspace representations that are invariant to the selection of bases. To learn such representations, we propose loss functions that gauge the separation between the desired and the estimated subspace. In particular, we propose losses that measure the length of the shortest path between subspaces viewed on a union of Grassmannians, and prove that it is possible for a DNN to approximate signal subspaces. The computation of learning subspaces of different dimensions is accelerated by a new batch sampling strategy called consistent rank sampling. The methodology is robust to array imperfections due to its geometry-agnostic and data-driven nature. In addition, we propose a fully end-to-end gridless approach that directly learns angles to study the possibility of bypassing subspace methods. Numerical results show that learning such subspace representations is more beneficial than learning covariances or angles. It outperforms conventional SDP-based methods such as the sparse and parametric approach (SPA) and existing DNN-based covariance reconstruction methods for a wide range of signal-to-noise ratios (SNRs), snapshots, and source numbers for both perfect and imperfect arrays.