Sobolev-Regularized Filtering on Weighted Riemannian State Spaces for Kinematic Signal Recovery at Very Low Signal-to-Noise Ratio

Recovering a kinematically coherent signal from observations whose signal-to-noise ratio (SNR) lies far below 0 dB is a central difficulty in radar tracking under electronic countermeasures. Classical linear filters treat the observation as an unstructured scalar sequence and, at very low SNR, either discard the signal or require model assumptions that an adversary can violate. This paper develops a variational framework in which the observation is represented as a scalar field on a five-dimensional weighted Riemannian state space (planar position, planar velocity, acceleration magnitude) and is estimated by minimizing a variable-exponent Sobolev energy that couples a second-order smoothness penalty with a spatially adaptive data-fidelity term. We (i) establish existence and uniqueness of the minimizer, (ii) derive the optimality condition, including a curvature correction missing from the flat-space form, (iii) give a spectral bias-variance analysis showing that the estimator behaves as an adaptive low-pass filter whose achievable noise reduction is governed by the ratio of the observation dimension to the effective signal dimension, and (iv) obtain a mean-square error rate that matches the classical minimax exponent for Sobolev classes. Interactive Benchmark & Live Demo: https://5dvr.ai