Extracting Dual Analytic Geometries of Linear Transformations to Achieve Efficient Computation

We propose a novel framework for fast integral operations by uncovering hidden geometries in the row and column structures of the underlying operators. This is accomplished through the $\mathtt{\text{Questionnaire}}$ algorithm, an iterative procedure that constructs adaptive hierarchical partition trees, revealing latent multiscale organizations and exposing local low-rank structures within the data. Guided by these geometries, we employ two complementary techniques: (1) The $\mathtt{\text{Butterfly}}$ algorithm, which exploits the learned hierarchical low-rank structure; and (2) Adaptive $\mathtt{\text{eGHWT}}$, best tilings in both space and frequency using all levels of the generalized Haar–Walsh wavelet packets. These techniques enable efficient matrix factorization and multiplication. We coin our algorithms as $\mathtt{\text{Questionnaire}}\mathtt{\text{Factorization}}\mathtt{\text{and}}\mathtt{\text{Fast}}\mathtt{\text{Transform}}\mathtt{\text{(QFFT)}}$. Unlike classical approaches that rely on prior knowledge of the underlying geometry, $\mathtt{\text{QFFT}}$ is fully data-driven and applicable to matrices arising from irregular or unknown distributions. Even when the rows and columns both appear mutually orthogonal, our framework identifies the intrinsic ordering of orthogonal vectors that reveal hidden sparsity of the kernel. We demonstrate the effectiveness of our approach on matrices associated with heterogeneous operators and families of orthogonal polynomials. The resulting compressed representations reduce storage complexity from $\mathcal{O}(N^2)$ to $\mathcal{O}(N\log N)$, enabling fast computation and scalable implementation.