Convolutional neural networks (CNNs) are currently state-of-the-art for
various classification tasks, but are computationally expensive. Propagating
through the convolutional layers is very slow, as each kernel in each layer
must sequentially calculate many dot products for a single forward and backward
propagation which equates to $\mathcal{O}(N^{2}n^{2})$ per kernel per layer
where the inputs are $N \times N$ arrays and the kernels are $n \times n$
arrays. Convolution can be efficiently performed as a Hadamard product in the
frequency domain. The bottleneck is the transformation which has a cost of
$\mathcal{O}(N^{2}\log_2 N)$ using the fast Fourier transform (FFT). However,
the increase in efficiency is less significant when $N\gg n$ as is the case in
CNNs. We mitigate this by using the "overlap-and-add" technique reducing the
computational complexity to $\mathcal{O}(N^2\log_2 n)$ per kernel. This method
increases the algorithm's efficiency in both the forward and backward
propagation, reducing the training and testing time for CNNs. Our empirical
results show our method reduces computational time by a factor of up to 16.3
times the traditional convolution implementation for a 8 $\times$ 8 kernel and
a 224 $\times$ 224 image.

使用 “重叠相加” 技术和快速傅里叶变换来减少卷积神经网络中卷积层的计算时间，实现了更高效的向前和向后传播。