arXiv2026
We generalize the Calderon-Zygmund methodology for operator analysis and the Rokhlin-Greengard fast multipole algorithms, in which the domain and range of an operator are jointly organized. Our methods analyze unstructured kernels by uncovering the joint hidden geometries of their domain and range. The main tool is the Questionnaire algorithm, an iterative procedure that constructs adaptive bi-hierarchical partition trees that reveal latent multiscale organization and local low-rank structure in the kernel. When the geometry is unknown, the iteration starts from any initial affinity, even a random one, and, with a stochastic flip perturbation and a spectral space-filling curve that fixes the left-right order of every tree, reorganizes the matrix into an image of the operator's "phase space": the joint space of its sources and sensors, in duality, as positions and frequencies are for the Fourier transform. This ordering gives the phase space an analytic structure on which any classical transform can be carried out; the resulting compression is direct evidence of that structure. Guided by these geometries, we obtain efficient factorizations and fast matrix-vector products in two ways: (1) the Butterfly algorithm, which exploits the learned bi-hierarchical localized low-rank structure; and (2) the adaptive eGHWT, which selects best tilings in space and sequency from all levels of generalized Haar-Walsh wavelet packets. We call the resulting algorithms the Questionnaire Factorization and Fast Transform (QFFT). Unlike classical approaches requiring prior knowledge of the geometry, QFFT is fully data-driven and applies to matrices from irregular or unknown distributions. We demonstrate it on heterogeneous oscillatory operators and on orthogonal eigenvector expansions with high multiplicity and random sampling, reducing storage from O(N^2) to O(N log N) and enabling fast, scalable computation.