CUR 分解是一種具高度可解釋性的資料分析工具，其目標為利用原始資料矩陣中少數的行與列來進行近似，從而在保留原始資料結構的同時，加速大規模矩陣的運算效率。本文旨在改進 Xu 等人所提出之基於能量的自適應CUR 分解方法，提出一種基於快速傅立葉轉換的自適應 CUR 分解演算法。首先，我們回顧 CUR 分解的基本架構及其所涉及的行選取問題。接著，我們運用快速傅立葉轉換的頻域特性，結合 Wang 與 Zhang 所提出的自適應取樣方法，提出一種高效的行選取流程，進而提升 CUR 分解在取樣品質與計算效率上的表現，這也是本文最主要的貢獻。最後，我們透過數值模擬實驗驗證本方法於滿秩與非滿秩影像矩陣上的適用性，結果顯示其在精確度與效能方面皆具顯著優勢。;CUR decomposition approximates a data matrix using a small number of its actual columns and rows, making it a highly interpretable data analysis tool. Its primary objective is to accelerate large-scale matrix computations while preserving the structural information of the original data. This thesis aims to improve the energy-based adaptive CUR decomposition method proposed by Xu et al. by introducing an adaptive CUR decomposition algorithm based on the Fast Fourier Transform (FFT). We first review the fundamental framework of CUR decomposition and the associated column selection problem. Then, leveraging the frequency-domain characteristics of FFT and integrating the adaptive sampling strategy proposed by Wang and Zhang, we design an efficient column selection procedure to enhance both the sampling quality and computational efficiency of CUR decomposition. This contribution constitutes the core innovation of this work. Finally, we validate the applicability of the proposed method through numerical experiments on both full-rank and non-full-rank image matrices, with results demonstrating significant improvements in both accuracy and performance.
