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

   https://doi.org/10.1109/TSP.2025.3544170  

   Chen, Kuan-Lin; Rao, Bhaskar D (January 2025, IEEE Transactions on Signal Processing)

   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. 

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2. Robust semi-supervised non-negative matrix factorization for binary subspace learning

   https://doi.org/10.1007/s40747-021-00285-1  

   Dai, Xiangguang; Zhang, Keke; Li, Juntang; Xiong, Jiang; Zhang, Nian; Li, Huaqing (February 2021, Complex & Intelligent Systems)

   null (Ed.)

   Abstract Non-negative matrix factorization and its extensions were applied to various areas (i.e., dimensionality reduction, clustering, etc.). When the original data are corrupted by outliers and noise, most of non-negative matrix factorization methods cannot achieve robust factorization and learn a subspace with binary codes. This paper puts forward a robust semi-supervised non-negative matrix factorization method for binary subspace learning, called RSNMF, for image clustering. For better clustering performance on the dataset contaminated by outliers and noise, we propose a weighted constraint on the noise matrix and impose manifold learning into non-negative matrix factorization. Moreover, we utilize the discrete hashing learning method to constrain the learned subspace, which can achieve a binary subspace from the original data. Experimental results validate the robustness and effectiveness of RSNMF in binary subspace learning and image clustering on the face dataset corrupted by Salt and Pepper noise and Contiguous Occlusion. 

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

   https://doi.org/10.1109/ICASSP49660.2025.10889620  

   Chen, Kuan-Lin; Rao, Bhaskar D (April 2025, IEEE)

   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. 

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4. Radar Clutter Covariance Estimation: A Nonlinear Spectral Shrinkage Approach

   https://doi.org/10.1109/TAES.2023.3292069  

   Jain, Shashwat; Krishnamurthy, Vikram; Rangaswamy, Muralidhar; Kang, Bosung; Gogineni, Sandeep (December 2023, IEEE Transactions on Aerospace and Electronic Systems)

   In this article, we exploit the spiked covariance structure of the clutter plus noise covariance matrix for radar signal processing. Using state-of-the-art techniques high dimensional statistics, we propose a nonlinear shrinkage-based rotation invariant spiked covariance ma- trix estimator. We state the convergence of the estimated spiked eigen- values. We use a dataset generated from the high-fidelity, site-specific physics-based radar simulation software RFView to compare the proposed algorithm against the existing rank constrained maximum likelihood (RCML)-expected likelihood (EL) covariance estimation 

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5. Feature learning in deep classifiers through Intermediate Neural Collapse

   Rangamani, Akshay; Lindegaard, Marius; Galanti, Tomer; Poggio, Tomaso (February 2023, Center for Brains, Minds and Machines (CBMM))

   In this paper, we conduct an empirical study of the feature learning process in deep classifiers. Recent research has identified a training phenomenon called Neural Collapse (NC), in which the top-layer feature embeddings of samples from the same class tend to concentrate around their means, and the top layer’s weights align with those features. Our study aims to investigate if these properties extend to intermediate layers. We empirically study the evolution of the covariance and mean of representations across different layers and show that as we move deeper into a trained neural network, the within-class covariance decreases relative to the between-class covariance. Additionally, we find that in the top layers, where the between-class covariance is dominant, the subspace spanned by the class means aligns with the subspace spanned by the most significant singular vector components of the weight matrix in the corresponding layer. Finally, we discuss the relationship between NC and Associative Memories (Willshaw et al., 1969). 

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