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

Kuan-Lin Chen, Bhaskar D. Rao

TL;DR

The paper develops a subspace representation learning framework for localizing more sources than sensors with sparse linear arrays, using a DNN to learn signal and noise subspaces on Grassmannians in a basis-invariant manner. It introduces loss functions based on principal angles (notably the geodesic distance) and proves the DNN can approximate the signal subspaces, complemented by a consistent rank sampling strategy for efficient training. A gridless end-to-end variant is also proposed to directly map sample covariances to DoAs. Numerical results show the approach outperforms SDP-based methods (SPA, WDA) and previous DNN covariance reconstruction approaches across a wide range of SNRs, snapshots, and source counts, while remaining robust to array imperfections and scalable with different MRAs.

Abstract

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.

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

TL;DR

The paper develops a subspace representation learning framework for localizing more sources than sensors with sparse linear arrays, using a DNN to learn signal and noise subspaces on Grassmannians in a basis-invariant manner. It introduces loss functions based on principal angles (notably the geodesic distance) and proves the DNN can approximate the signal subspaces, complemented by a consistent rank sampling strategy for efficient training. A gridless end-to-end variant is also proposed to directly map sample covariances to DoAs. Numerical results show the approach outperforms SDP-based methods (SPA, WDA) and previous DNN covariance reconstruction approaches across a wide range of SNRs, snapshots, and source counts, while remaining robust to array imperfections and scalable with different MRAs.

Abstract

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.
Paper Structure (37 sections, 2 theorems, 42 equations, 11 figures, 1 table)

This paper contains 37 sections, 2 theorems, 42 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Assumptions
  4. SCMs and the DoA estimation problem
  5. Neural network models
  6. Prior Art
  7. The maximum likelihood problem
  8. Redundancy averaging and direct augmentation
  9. Direct SDP-based methods
  10. Majorization-Minimization
  11. Proxy covariance matrix estimation
  12. DNN-based covariance matrix reconstruction
  13. Subspace Representation Learning
  14. Subspace representations of different dimensions
  15. Distances between subspace representations
  16. ...and 22 more sections

Key Result

Theorem 1

For every $k\in[M-1]$ and every $\epsilon>0$, there exists a ReLU network $f:\mathbb{C}^{N\times N}\to\text{Gr}(k,M)$ such that

Figures (11)

  • Figure 1: An illustration of the gridless end-to-end model, which consists of an architecture and several output layers. The model simultaneously generates DoAs for every possible number of sources so there are $M-1$ heads (affine functions) at the output. The $k$-th head is picked when there are $k$ sources.
  • Figure 2: An illustration of a $3$-stage $L$-block ResNet model he2016deep. In the wide ResNet 16-8 (WRN-16-8) zagoruyko2016wide, there are $L=2$ blocks per stage, leading to $16$ layers in total. The widening factor is $8$, meaning that WRN-16-8 is $8$ times wider than the original ResNet. See Section \ref{['sec:DNN_models']} for more details.
  • Figure 3: MSE vs. SNR. Our approach is in general superior to all of the baselines. In most cases, it is significantly better than SPA, WDA, DCR-T, and DCR-G-Fro. DCR-G-Aff is the most competitive baseline. For $k>3$, our approach outperforms DCR-G-Aff. In comparison to DCR-G-Aff at $k=2$ or $k=3$, our approach is slightly better at low SNRs but worse at high SNRs.
  • Figure 4: MSE vs. number of snapshots. Although the DNN models are only trained on a single number of snapshots $T=50$, they are capable of performing well on a wide range of unseen scenarios from $T=10$ to $T=100$. Our approach is consistently better than SPA, WDA, and DCR-G-Aff.
  • Figure 5: MSE vs. SNR. $N=4$. $M=7$. Our approach is significantly better than all of the baselines when $k>2$. For $k=2$, it is better than all of the DNN-based baselines but slightly worse than the SPA at $20$ dB SNR. The main results obtained for the $5$-element MRA are similar to the $4$-element MRA.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • proof
  • proof