More Like this

1. Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks

   Liu, Hao ; Chen, Minshuo ; Zhao, Tuo ; Liao, Wenjing ( July 2021 , International Conference on Machine Learning)

   null (Ed.)

   Most of existing statistical theories on deep neural networks have sample complexities cursed by the data dimension and therefore cannot well explain the empirical success of deep learning on high-dimensional data. To bridge this gap, we propose to exploit the low-dimensional structures of the real world datasets and establish theoretical guarantees of convolutional residual networks (ConvResNet) in terms of function approximation and statistical recovery for binary classification problem. Specifically, given the data lying on a 𝑑-dimensional manifold isometrically embedded in ℝ^𝐷, we prove that if the network architecture is properly chosen, ConvResNets can (1) approximate Besov functions on manifolds with arbitrary accuracy, and (2) learn a classifier by minimizing the empirical logistic risk, which gives an excess risk in the order of 𝑛−2s/(2s+d), where 𝑠 is a smoothness parameter. This implies that the sample complexity depends on the intrinsic dimension 𝑑, instead of the data dimension 𝐷. Our results demonstrate that ConvResNets are adaptive to low-dimensional structures of data sets. 

   more » « less
2. Efficient approximation of deep ReLU networks for functions on low dimensional manifolds

   Chen, Minshuo ; Jiang, Haoming ; Liao, Wenjing ; Zhao, Tuo ( December 2019 , Conference on Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

   more » « less
3. Efficient approximation of deep ReLU networks for functions on low dimensional manifolds

   Chen, Minshuo ; Jiang, Haomin ; Liao, Wenjing ; Zhao, Tuo ( December 2019 , Conference on Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

   more » « less
4. Efficient Approximation of Deep ReLU Networks for Functions on Low Dimensional Manifolds

   Chen, Minshuo ; Jiang, Haoming ; Liao, Wenjing ; Zhao, Tuo ( December 2019 , Advances in Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

   more » « less
5. Surface Networks

   Kostrikov, Ilya ; Jiang, Zhongshi ; Panozzo, Daniele ; Zorin, Denis ; Bruna, Joan ( June 2018 , IEEE Conference on Computer Vision and Pattern Recognition)

   We study data-driven representations for three- dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curva- ture directions — this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models Surface Networks (SN). We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs. 

   more » « less