Title: Locality-Aware Multiresolution Graph Spectral Filtering to Mitigate Oversmoothing and Oversquashing.
Primary area: learning on graphs and other geometries & topologies
Keywords: Graph Kearning, Deep learning, Clustering.

Abstract: Real-world graphs demonstrate region-specific heterophily: some regions are smooth and suitable for low-pass averaging, whereas others are sharp and necessitate high-pass contrast.  However, spectral GNNs that utilize a global eigenbasis or high-order polynomial surrogates for filtering rely upon dense, non-orthogonal, non-local bases that blend signals from distant, semantically unrelated regions. This results in small clusters connected to large homogeneous hubs becoming indistinguishable and overshadowed by continuous global mixing, exacerbating oversmoothing. To address these limitations, in this paper, we present Hierarchical Multi-view Haar (HMH), a locality-first, multiresolution framework for learning frequency signal region-by-region.HLH estimates diffusion neighborhoods and constructs a diffusion barrier to identify incompatible neighbors that are weakly connected under diffusion (i.e., connected by long, low-weight paths), preventing incompatible clusters from being merged together.A lightweight MLP layer is applied on the bottom-K eigenspace of a multi-view Laplacian (which combines topology and feature affinity), generating soft node scores for coarsening the graph into a hierarchy.Rather than a full graph-wide eigendecomposition, at each level, we create a strictly local, degree-aware, block-sparse orthonormal Haar basis that consists of a low-pass scaling wavelet and a high-pass inter-/intra-cluster wavelet. This sparsity enables near-linear operations per layer while preserving strict locality. The multi-scale unpooling layer combines coarse and fine signals to preserve small, high-contrast local structures while maintaining global context. The hierarchy reduces effective path lengths for message passing, preventing oversquashing. Empirically, HMH attains state-of-the-art performance with linear scalability, enhancing node classification by up to 3\% on heterophilous benchmarks and up to 7\% on graph classification tasks.