This is the Hadamard package for Macaulay2 which computes the Hadamard
product of projective subvarieties.

这是针对 Macaulay2 的 Hadamard 软件包，用于计算射影子变量的 Hadamard 积。

哈达玛

Hadamard

The visual explanation of learned representation of models helps to
understand the fundamentals of learning. The attentional models of previous
works used to visualize the attended regions over an image or text using their
learned weights to confirm their intended mechanism. Kim et al. (2016) show
that the Hadamard product in multimodal deep networks, which is well-known for
the joint function of visual question answering tasks, implicitly performs an
attentional mechanism for visual inputs. In this work, we extend their work to
show that the Hadamard product in multimodal deep networks performs not only
for visual inputs but also for textual inputs simultaneously using the proposed
gradient-based visualization technique. The attentional effect of Hadamard
product is visualized for both visual and textual inputs by analyzing the two
inputs and an output of the Hadamard product with the proposed method and
compared with learned attentional weights of a visual question answering model.

本文拓展了 Kim et al. (2016) 的工作，提出了一种基于梯度的可视化技术，证明了多模式深度网络中的 Hadamard 乘积不仅适用于视觉输入，同时适用于文本输入，并可通过该技术可视化 Hadamard 乘积对视觉和文本输入的注意力机制。

多模深度网络中 Hadamard 乘积的可视化解释

Visual Explanations from Hadamard Product in Multimodal Deep Networks

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。