快速双线性卷积算法的推导与分析
Derivation and Analysis of Fast Bilinear Algorithms for Convolution
摘要：
卷积在信号处理、深度神经网络和数值求解器中的应用的流行推动了许多快速卷积算法的发展。在许多这些问题中，卷积是在TB或PB的数据上执行的，因此即使是常数的改进因子也可以显着减少计算时间。我们利用双线性算法的形式主义来描述和分析所有最流行的方法。这个统一的镜头允许我们研究不同的卷积变量之间的关系，以及推导出误差界和分析各种算法的成本。我们提供了新的推导，主要是利用矩阵和张量代数，来描述Winograd家族的卷积算法，以及一维和多维卷积之间的减少。我们提供了成本和误差界以及实验数值研究。我们对其中两种算法的实验表明，快速卷积算法不需要复杂的运算就能达到快速傅里叶变换（FFT）的精度。这些算法可用于多维输入的卷积问题或大于4的滤波器，扩展了基于Winograd的卷积算法的最新技术。
英文摘要：
The prevalence of convolution in applications within signal processing, deep neural networks, and numerical solvers has motivated the development of numerous fast convolution algorithms. In many of these problems, convolution is performed on terabytes or petabytes of data, so even constant factors of improvement can significantly reduce the computation time. We leverage the formalism of bilinear algorithms to describe and analyze all of the most popular approaches. This unified lens permits us to study the relationship between different variants of convolution as well as to derive error bounds and analyze the cost of the various algorithms. We provide new derivations, which predominantly leverage matrix and tensor algebra, to describe the Winograd family of convolution algorithms as well as reductions between 1D and multidimensional convolution. We provide cost and error bounds as well as experimental numerical studies. Our experiments for two of these algorithms, the overlap-add approach and Winograd convolution algorithm with polynomials of degree greater than one, show that fast convolution algorithms can rival the accuracy of the fast Fourier transform (FFT) without using complex arithmetic. These algorithms can be used for convolution problems with multidimensional inputs or for filters larger than size of four, extending the state-of-the-art in Winograd-based convolution algorithms.
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