Persistent homology (PH) summarizes the topological features of data that persist across scales, but it is insensitive to many geometric properties. Persistent Laplacians (PLs) address this limitation: for a pair of simplicial complexes in a filtration, the kernel of the persistent Laplacian recovers the persistent Betti number, while its non-zero spectrum carries additional geometric information. However, PLs are defined for individual pairs of filtration values, so analysing a whole filtration requires evaluating many operators that largely contain the same information, and the result is hard to summarize and interpret. In this paper, we show that this redundancy has a precise structure. For each pair of filtration values, we decompose the space of chains underlying the persistent up-Laplacian into the contributions inherited from the neighboring pairs and a remaining part, which we call the new persistent chain space. We show that the new persistent chain spaces of all earlier pairs form a direct sum decomposition of this space, so that every persistent up-Laplacian in the filtration is determined by them. Moreover, the new persistent chain spaces are non-trivial only at pairs given by the persistence pairing in the same and in the next homological dimension, so that the persistence pairing determines exactly where the persistent up-Laplacian can change. Experiments on several types of point clouds show that evaluating persistent Laplacians only at these pairs reduces the computation time, both for our implementation and for existing software.

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