Herz versus Fefferman: Symmetric and asymmetric Bochner--Riesz theory

What the math gives to ML

The paper identifies a sharp failure mode for spectral cutoff operators on spaces whose two ends have different effective dimensions. Besides the usual Bochner–Riesz smoothness threshold controlled by the larger dimension, an asymmetric interaction term proportional to the dimension gap appears; this warns against applying equally sharp spectral filters across heterogeneous regions. A transferable neural-network analogue is to make graph spectral convolutions use a data-adaptive smoothness exponent estimated from local spectral geometry rather than a globally fixed cutoff. The main opportunity is improved stability and accuracy for graph neural networks on graphs with heterogeneous local structure, although the original theorem does not directly prove a neural-network guarantee.