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

We start with a collection of item embeddings and a graph connecting their feature coordinates. The question is how to use that feature graph to construct a latent representation of the items while retaining information about each item’s variation across the graph. Motivated by a two-block spectral construction in our implementation, we split the graph Laplacian into capped and residual parts, transform each embedding through their square roots, and concatenate the results. The two blocks change with the spectral cut, but we show that the inner products and distances of the complete representation do not: its Gram matrix is XLX⊤at every cut. The cut therefore cannot tune a method that uses only this complete Euclidean geometry. It does change how each item’s Laplacian energy is allocated between the blocks. We characterize this allocation as a balance curve and show that the complete curve determines the item’s aggregated spectral-energy weights. The Gram identity is an instance of established positive-semidefinite factorization and tight-frame algebra; the contribution here is to clarify what this particular construction does and does not provide when building a latent representation from embeddings and a feature graph. Whether its blockwise information improves a downstream task remains untested.