Building Latent Spaces out of Sandwiches: Complementary Spectral Factorizations for Graph-Metric Embeddings

A feature-graph Laplacian defines a quadratic geometry on vectors indexed by its vertices. We study a capped-plus-residual spectral factorization that maps each vector into two filtered blocks. Their concatenation preserves the Laplacian Gram matrix for every spectral cut, so changing the cut cannot affect a downstream method that depends only on the complete Euclidean Gram geometry. This invariance is an instance of a squared-partition, or tight-frame, identity rather than a new metric construction. The cut instead controls a normalized allocation of each vector’s Dirichlet energy. We characterize the resulting balance curve and show that its slope changes recover the vector’s aggregated spectral-energy distribution on the positive Laplacian spectrum. We distinguish this exact characterization from the untested question of whether balance-curve features improve downstream applications beyond the canonical Laplacian embedding and simpler spectral summaries.