Hierarchical Zeta Regularization for Multi-Scale Tensor Aggregation: Convergence Theory and Spectral Justification

Description: We introduce a mathematically rigorous framework for aggregating hierarchical tensor sequences \(\{T^{(n)}\}_{n \geq 0}\) using weights derived from the analytic continuation of the Hurwitz zeta function evaluated at \(s = -1\). Specifically, we define the regularized sum \[S = \sum_{n=0}^{\infty} c_n T^{(n)}, \quad \text{where} \quad c_n = \zeta(-1, a_n) w_n,\] with scale parameters \(a_n = a_0 + \alpha n > 0\) and optional damping weights \(w_n\) (e.g., \(w_n = e^{-\lambda n}\)). Using the closed-form identity \(\zeta(-1, a) = -\frac{1}{2}(a^2 - a + \frac{1}{6})\), we establish: - Convergence conditions: Absolute convergence under polynomial decay (\(\|T^{(n)}\| = \mathcal{O}(n^{-p})\) with \(p > 3\)) or exponential damping (\(\|T^{(n)}\| = \mathcal{O}(e^{\mu n})\) with \(\lambda > \mu\)).- Spectral justification: A theoretical link to spectral zeta functions \(\zeta_L(s)\) and heat kernel asymptotics, grounding the weights in geometric and operator-theoretic principles.- Renormalization schemes: Practical counterterm decomposition to handle the inherent quadratic growth \(\zeta(-1, a_n) \sim -a_n^2/2\), inspired by quantum field theory.- Stability guarantees: Perturbation bounds showing Lipschitz continuity of the aggregation map.- Implementation guidance: Low-rank approximation strategies, spectral filter design, and parameter selection heuristics. This framework is domain-agnostic and applicable to multi-scale problems in graph representation learning, image pyramids, time-series modeling, and hierarchical machine learning. The approach replaces heuristic weighting (e.g., exponential smoothing) with a principled, geometry-aware alternative rooted in analytic number theory and spectral geometry.