Yang Pan

Erwin Riegler, Alex Bühler, Yang Pan et al.

We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\varepsilon$ is no larger than $2^{b(\varepsilon)}$ with $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. This result improves Lemma IX.4 in Perekrestenko et al. as it shows that the rate at which $b(\varepsilon)$ tends to infinity as $\varepsilon$ tends to zero equals the rectifiability parameter $m$, which can be much smaller than the ambient dimension. We extend this result to countably $m$-rectifiable measures and show that this rate still equals the rectifiability parameter $m$ provided that, among other technical assumptions, the measure decays exponentially on the individual components of the countably $m$-rectifiable support set.

Guoqiang Zhong, Pan Yang, Sijiang Wang et al.

Hashing-based methods seek compact and efficient binary codes that preserve the neighborhood structure in the original data space. For most existing hashing methods, an image is first encoded as a vector of hand-crafted visual feature, followed by a hash projection and quantization step to get the compact binary vector. Most of the hand-crafted features just encode the low-level information of the input, the feature may not preserve the semantic similarities of images pairs. Meanwhile, the hashing function learning process is independent with the feature representation, so the feature may not be optimal for the hashing projection. In this paper, we propose a supervised hashing method based on a well designed deep convolutional neural network, which tries to learn hashing code and compact representations of data simultaneously. The proposed model learn the binary codes by adding a compact sigmoid layer before the loss layer. Experiments on several image data sets show that the proposed model outperforms other state-of-the-art methods.