Geometry-Aware Robust Denoising via Hierarchical Manifold Traversal

Existing manifold traversal algorithms denoise by navigating a graph via gradient-guided steps and zero-order shortcuts. Because they operate at a single resolution, single-layer traversal is susceptible to local minima traps and requires fixed initialization to succeed, making it impractical. We introduce *Hierarchical Manifold Traversal Networks (HMTNs)*, which approximate a $d$-dimensional manifold with a stack of traversal graphs at geometrically increasing resolutions. A cross-layer map provides a strong initialization for each finer layer from the converged coarse layer. Importantly, in HMTN the hierarchy depth grows only logarithmically in the number of training samples $N$ and depends on the intrinsic dimension $d$ rather than the ambient dimension $D$. Our empirical results on time-series gravitational-wave manifolds show that, in the random-initialization regime, HMTN achieves a $134\times$ improvement in MSE relative to single-layer traversal at only $3.9\times$ the multiplication cost. On patch denoising, gains range from $1.6\times$ on MNIST, which has the lowest scale separation, to $6$–$9\times$ on CIFAR-10 and CelebA, whose scale separation is substantially larger. HMTN also offers a favorable accuracy–compute trade-off against classical search-based and learning-based alternatives: it requires a median of $5.7\times$ fewer multiplications than exhaustive nearest-neighbor search, and nonlinear autoencoders need $25\times$ more multiplications to match its accuracy. Training HMTN is a one-time CPU cost of 9-32 minutes on our gravitational-wave benchmarks.