发表机构
ENS de Lyon, CNRS, Inria, Université Claude Bernard Lyon 1, LIP, UMR 5668（里昂高等师范学院、法国国家科学研究中心、法国国家信息与自动化研究所、里昂第一大学、计算与图像实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种条件，确保快速变换的蝴蝶分解取实部后仍保持Kronecker稀疏结构，并应用于DCT、DST和DHT，得到实值因子表示，利于低精度GPU实现。
AI 中文摘要
诸如离散余弦变换之类的快速变换可以表示为复值结构化稀疏矩阵（称为Kronecker稀疏因子）乘积的实部。更一般地，此类结构化稀疏矩阵的乘积出现在平方二进蝴蝶矩阵和君主矩阵中，已知这些矩阵能高效表示许多快速变换。然而，取此类乘积的实部或虚部是否保持这种结构，此前尚不清楚。在本信中，我们给出了因子上的一个条件以确保结构得以保持。该条件在离散余弦变换、离散正弦变换和离散Hartley变换的情形下均得到满足，从而我们能够提供这些变换作为实值Kronecker稀疏因子乘积的新表示，这为使用低精度数值格式的高效GPU实现开辟了新途径。
英文摘要
Fast transforms such as the Discrete Cosine Transform can be expressed as the real part of a product of complex-valued structured sparse matrices known as Kronecker-sparse factors. More generally, products of such structured sparse matrices appear in square dyadic butterfly and monarch matrices, which are known to represent many fast transforms efficiently. Yet, it has remained unclear whether taking the real or the imaginary part of such products preserves this structure. In this letter, we give a condition on the factors to ensure that the structure is preserved. This condition is satisfied in the case of the Discrete Cosine Transform, the Discrete Sine Transform and the Discrete Hartley Transform, allowing us to provide a new representation of these transforms as products of real-valued Kronecker-sparse factors, which opens new avenues for efficient GPU implementations using low-precision numerical format.
