Learning Approximate Isometric Embeddings for Efficient Template Placement

The sphere covering problem has broad applications in science however, exact solutions are computationally infeasible on curved, high-dimensional manifolds because distances must be evaluated using a position-dependent metric. We present $\text{isobank}$, a normalizing flow that learns an approximately isometric coordinate transformation, such that Euclidean distances in the latent space approximate metric-induced proper distances in the physical parameter space. This enables template placement using Euclidean distance checks in the latent space, avoiding repeated metric evaluations during placement. We evaluate $\text{isobank}$ in the context of gravitational wave template bank construction, where the sphere covering problem corresponds to the template placement problem. On a five-dimensional eccentric binary parameter space, $\text{isobank}$ reaches 97% coverage at a maximum mismatch of $\mu =0.03$ with 23% fewer templates than the state-of-the-art placement code $\text{mbank}$, which reaches 94% coverage. Coverage is measured on independent injections drawn uniformly over the masses, spins and eccentricity, using the exact waveform match rather than the metric approximation. Training the flow takes about 2.5 h and placing the bank under 1 h on 32 CPU cores. $\text{isobank}$ will be made publicly available.