Hierarchical Bayesian inversion using the Karhunen-Loève expansion with analytical eigenpairs of the squared exponential kernel

What the math gives to ML

The paper's transferable asset is a differentiable low-rank representation of a squared-exponential Gaussian field whose basis does not require repeatedly solving an integral eigenvalue problem as covariance hyperparameters change. In neural networks, the same construction can parameterize spatially varying weights, biases, or latent functions with Gaussian-Hermite modes, while learning correlation length and amplitude through ordinary backpropagation. The most promising first use is a compact neural-field or convolutional adapter in which a small set of analytic KL coefficients replaces a dense grid of trainable parameters and provides an explicit smoothness control.