Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces

What the math gives to ML

The paper identifies an unavoidable geometric barrier for recovering a diffused signal from observations restricted to a subset: information originating at essential maximal distance \(\mathcal{L}(\omega)\) is exponentially suppressed at time \(t\) on the scale \(e^{-\mathcal{L}(\omega)^2/(2t)}\). This can be transferred to graph and manifold neural operators that supervise or decode a global latent state from sparse sensor nodes. Rather than treating all early diffusion states as equally observable, use a geometry-calibrated time weight or attention bias whose exponent is determined by sensor coverage. The resulting module is falsifiable on sparse-observation graph tasks, where it should reduce unstable early-time gradients and improve reconstruction at fixed sensor count.