Extracting Dual Analytic Geometries of Linear Transformations to Achieve Efficient Computation

TL;DR

A novel framework for fast integral operations by uncovering hidden geometries in the row and column structures of the underlying operators by using the Questionnaire algorithm, an iterative procedure that constructs adaptive hierarchical partition trees, revealing latent multiscale organizations and exposing local low-rank structures within the data.

Abstract

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 \texttt{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 \texttt{\texttt{Butterfly}} algorithm, which exploits the learned hierarchical low-rank structure; and (2) Adaptive \texttt{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 \texttt{Questionnaire Factorization and Fast Transform (QFFT)}. Unlike classical approaches that rely on prior knowledge of the underlying geometry, \texttt{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.

Figures (18)

• Figure 1: Skeletonization
• Figure 2: Visualization of the space-filling curve at each tree level. Data points are uniformly sampled from the square $[-1, 1] \times [-1, 1]$. Each subplot corresponds to a level of the partition tree. The title of each subplot indicates the tree level. The colors and the indices $1, 2, 3, \ldots$ denote the left-to-right ordering of the nodes at that level. The yellow line connects the nodes according to the space-filling curve traversal.
• Figure 3: Approximation of the DST-IV transform using Butterfly and eGHWT. Left: the input signal $y(x) = 5\sin(\pi (k_1+0.5)x) + 10\sin( \pi (k_2+0.5)x) + 5\sin( \pi (k3+0.5)x))$, where $k_1 = 10$, $k_2 = 500$, and $k_3 = 1000$. Right: Comparison of the transform using the direct multiplication, Butterfly with $\varepsilon = 10^{-11}$, and eGHWT with $\varepsilon = 5\times 10^{-2}$.
• Figure 4: Comparison of the initial cosine similarity matrix $W_K$ (top-left) and $W_X$ (bottom-left) with the learned multiscale correlation affinity $W_{\mathcal{T}_X}^{\mathrm{corr}}$(top-right) and $W_{\mathcal{T}_K}^{\mathrm{corr}}$ (bottom-right) among of the kernel $\sin(\pi (k-0.5) \frac{(i-0.5)}{N})$, where $X = \{x_i\}_{i=1}^{N}$ and $K = \{k-0.5\}_{k=1}^N$, for $i = 1,\ldots,N$, $k = 1\,\ldots,N$, and $N = 256$.
• Figure 5: Analysis of the DST-IV kernel for 10 iterations of Algorithm \ref{['alg:questionnaire']}, showing (a) reorganization, (b) space embedding, and (c) frequency embedding.

Theorems & Definitions (8)

• Definition 2.1: Multiscale Partition Tree Structure $T_X$
• Lemma 2.2
• Lemma 3.1
• Definition 3.2: Tree Affinities
• Remark 3.3
• Remark 3.4
• Definition 3.5: Coupled Geometry
• Remark 3.6