分级卷积网络：无需尺度信号实现尺度不变性
Graded convolutional networks: scale invariance without the scale and robustness to unknown blur
一篇很硬核的视觉理论论文：用加权球面的 retraction 替代尺度归一化，模糊下 MNIST 准确率从 66.2% 拉到 88.4%，做尺度不变特征的研究者可以看看。
论文将图像的高斯 jet 视为分级向量空间：膨胀 λ 会使 k 阶导数乘以 λ^-k，局部分级卷积等价于齐次微分算子。作者提出在无尺度信号时通过映射到加权球面的 retraction 来构造膨胀不变特征，且该 retraction 在未知高斯模糊下仅随尺度变化。在 MNIST 两像素模糊下，基于 retraction 的网络保持 88.4% 准确率，而尺度归一化导数网络降至 66.2%；在 Lindeberg 的高斯导数网络中引入 retraction 后准确率从 91.5% 提升到 95.1%。
Graded convolutional networks: scale invariance without the scale and robustness to unknown blur
The Gaussian jet of an image is a graded vector space: a dilation by $λ$ multiplies derivatives of order $k$ by $λ^{-k}$. Local graded convolutions are exactly homogeneous differential operators, and a local feature is dilation-invariant if and only if it factors through the scale-normalized jet or, without the scale, through the retraction onto the weighted sphere of the grading. Under unknown Gaussian blur the retraction changes only by a change of scale, whereas the scale-normalized jet acquires a grading factor that no radial normalization removes. At the zero jet the retraction separates image families that radial normalizations and continuous classifiers cannot. Networks built on the retraction are universal for invariant features. On MNIST such a network keeps 88.4% accuracy under a two-pixel blur, against 66.2% for the same network on scale-normalized derivatives; in Lindeberg's Gaussian derivative network the retraction raises accuracy at that blur from 91.5% to 95.1%.