Exploring Complex Time-series Representations for Riemannian Machine Learning of Radar Data

Abstract

Classification of radar observations with machine learning tools is of primary importance for the identification of noncooperative radar targets such as drones. These observations are made of complex-valued time series which possess a strong underlying structure. These signals can be processed through a time-frequency analysis, through their selfcorrelation (or covariance) matrices or directly as the raw signal. All representations are linked but distinct and it is known that the input representation is critical for the success of any machine learning method. In this article, we explore these three possible input representation spaces with the help of two kinds of neural networks: a temporal fully convolutional network and a Riemannian network working direcly on the manifold of covariances matrices. We show that all the considered input representations are a particular case of a generic machine learning pipeline which goes from the raw complex data to the final classification stage through convolutional layers and Riemannian layers. This pipeline can be learnt end-to-end and is shown experimentally to give the best classification accuracy together with the best robustness to lack of data.

Modeling Tree-like Heterophily on Symmetric Matrix Manifolds

“…Recent research endeavors have shifted towards the development of foundational components of neural networks within the covariance matrix space. This includes techniques for feature transformation, such as mapping Euclidean features to covariance matrices using geodesic Gaussian kernels [40], non-linear operations applied to the eigenvalues of covariance matrices [41], convolutional operations employing SPD filters [42], and the Frechét mean [43]. Furthermore, proposals for Riemannian recurrent networks [44] and Riemannian batch normalization [45] have been put forth.…”

Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices

“…Previous work has proposed alternatives to the basic neural building blocks respecting the geometry of the space. For example, transformation layers [29,33,40], alternate convolutional layers based on SPDs [94] and Riemannian means [23], or appended after the convolution [21], recurrent models [24], projections onto Euclidean spaces [50,57] and batch normalization [20]. Our work follows this line, providing explicit formulas for translating Euclidean arithmetic notions into SPDs.…”

Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network

“…The main difficulties in training SPDnet lie both in backpropagation through the structured Riemannian functions and in manifold-constrained optimization. For more details, please refer to [ 12 , 21 ].…”

“…In recent years, the manifold of SPD (symmetric positive definite) matrices has attracted much attention due to its powerful statistical representations. In medical imaging, such manifolds are used in diffusion tensor magnetic resonance imaging [ 5 ], and in the computer vision community, they are widely used in face recognition [ 6 , 7 ], object classification [ 8 ], transfer learning [ 9 ], action recognition in videos [ 10 , 11 ], radar signal processing [ 12 , 13 , 14 ] and sonar signal processing [ 15 ]. The powerful statistical representations of SPD matrices lie in the fact that they inherently belong to the curved Riemannian manifold.…”