Geo-NN: An End-to-End Framework for Geodesic Mean Estimation on the Manifold of Symmetric Positive Definite Matrices

Niharika Shimona D'Souza, Archana Venkataraman

Keywords: Symmetric Postive Definite Manifolds, Geodesic Mean, Matrix Autoencoder

TL;DR: We propose an end-to-end deep learning framework, the Geo-NN, to efficiently compute the geodesic mean of a collection of matrices lying on the SPD manifold

Abstract: The manifold of symmetric positive definite (SPD) matrices plays a key role in many domains, from network science to differential geometry to signal and image processing. However, leveraging the SPD manifold geometry during inference is challenging, as simple operations, such as mean estimation, do not have a closed-form or easily computable solution. In this paper, we propose an end-to-end deep learning framework, which we call a Geometric Neural Network (Geo-NN), to efficiently compute the geodesic mean of a collection of matrices lying on the SPD manifold. Geo-NN utilizes a Matrix-Autoencoder (MAE) architecture with intersecting fully connected layers as its backbone. We illustrate that the matrix-normal equation arising from Fr\'echet mean estimation can be converted into a loss function for optimizing the Geo-NN, which in turn approximates the geodesic mean of a collection of SPD matrices. We demonstrate the efficacy of our framework in both synthetic and real-world scenarios, as compared to commonly used alternative methods. Our simulated experiments demonstrate that Geo-NN is robust to various noise conditions and is scalable to increasing dataset size and dimensionality. Our real-world application of Geo-NN to functional connectomics data allows us to extract network patterns associated with patient/control differences.

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