发表机构
NVIDIA（英伟达）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究循环二元卷积中用哈达玛变换替代DFT产生的误差，通过确定精确误差抵消、分析误差算子秩及零空间维度、得出期望误差表达式等，揭示该误差有结构、可预测且受对齐控制，除特定子空间滤波器外会使输出能量加倍。
AI 中文摘要
二元卷积和循环卷积都可以分别使用哈达玛变换和快速傅里叶变换（FFT）计算的离散傅里叶变换（DFT）在\(O(N\log N)\)时间内完成。哈达玛变换因其实值符号翻转而更受青睐，但其替代DFT会引入代数误差。我们给出了三个互补的结果来刻画这种误差。首先，我们确定了精确的误差抵消：两个输入和两个输出位置普遍无误差，且输出的任何重新排序都无法消除此误差。其次，误差算子几乎是满秩的，而其零空间只有对数维。第三，期望误差由单个对齐标量控制，通过对随机滤波器求平均得到封闭形式的表达式。一般来说，除了通用零误差子空间中的滤波器不会产生误差外，替代误差会使输出能量渐近加倍。总的来说，这些结果表明替代误差是有结构的、可预测的且受对齐控制。
英文摘要
Dyadic and circular convolution can both be computed in $O(N\log N)$ time using the Hadamard transform and the FFT-computed discrete Fourier transform (DFT), respectively. The Hadamard transform is preferable for its real-valued sign flips, yet its substitution for the DFT introduces algebraic error. We present three complementary results that characterize this error. First, we identify exact error cancellation: two input and two output positions are universally error-free, and no reordering of the output can eliminate this error. Second, the error operator is nearly full rank, while its null space has only logarithmic dimension. Third, the expected error is governed by a single alignment scalar, with a closed-form expression obtained by averaging over random filters. In general, the substitution error asymptotically doubles the output energy, except for filters in the universal zero-error subspace, which incur no error. Collectively, these results show that the substitution error is structured, predictable, and governed by alignment.
