The input tensor shape is 6 3 3. The corresponding weight matrix has f = 20 rows (one row per filter) and 24 columns (c κ1 κ2), as for the corresponding feature matrix, it has 24 rows and 4 columns, the 4 here is the number of convolution windows (i.e., number of pixels/entries in each of the output feature maps). After multiplying those matrices, we reshape them to the desired shape to obtain the desired output feature maps. In this section, we provide more details pertaining to our method. A.1 Method Preliminaries Our layer-wise compression technique hinges upon the insight that any linear layer may be cast as a matrix multiplication, which enables us to rely on SVD as compression subroutine. Focusing on convolutions we show how such a layer can be recast as matrix multiplication. Similar approaches have been used by Denton et al. (2014); Idelbayev and Carreira-Perpinán (2020); Wen et al. (2017) among others. The equivalence of Y and Y can be easily established via an appropriate reshaping operation since p= p1p2. Equipped with the notion of correspondence between convolution and matrix multiplication our goal is to decompose the layer via its matrix operator W Rf cκ1κ2. To this end, we compute the j-rank approximation of W using SVD and factor it into a pair of smaller matrices U Rf j and V Rj cκ1κ2.

From optimal transport to robust dimensionality reduction, a plethora of machine learning applications can be cast into the min-max optimization problems over Riemannian manifolds. Though many min-max algorithms have been analyzed in the Euclidean setting, it has proved elusive to translate these results to the Riemannian case. Zhang et al. have recently shown that geodesic convex concave Riemannian problems always admit saddle-point solutions. Inspired by this result, we study whether a performance gap between Riemannian and optimal Euclidean space convex-concave algorithms is necessary. We answer this question in the negative--we prove that the Riemannian corrected extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically stronglyconvex-concave case, matching the Euclidean result. Our results also extend to the stochastic or non-smooth case where RCEG and Riemanian gradient ascent descent (RGDA) achieve near-optimal convergence rates up to factors depending on curvature of the manifold.

Please refer to Table 5. Table 5: Architecture of E4-Net on Mnist-rot classification, p means dropout rate. The hyperparameters we use in this architecture are kernel size k = 5, reduction ratio r = 1, and the number of slices s = 2. In the large model, we increase the channel dimension to 24, the number of slices to 12, the reduction ratio to 2, and keep other hyperparameters the same. We take ResNet-18 [2], which is composed of an initial convolution layer, followed by 4 stage Res-Blocks and one final classification layer.

Convolutional neural networks (CNNs) have dominated the field of Computer Vision and achieved great success due to their built-in translation equivariance. Group equivariant CNNs (G-CNNs) that incorporate more equivariance can significantly improve the performance of conventional CNNs. However, G-CNNs are faced with two major challenges: spatial-agnostic problem and expensive computational cost. In this work, we propose a general framework of previous equivariant models, which includes G-CNNs and equivariant self-attention layers as special cases.

Uncertainty estimation is essential to make neural networks trustworthy in realworld applications. Extensive research efforts have been made to quantify and reduce predictive uncertainty. However, most existing works are designed for unimodal data, whereas multi-view uncertainty estimation has not been sufficiently investigated. Therefore, we propose a new multi-view classification framework for better uncertainty estimation and out-of-domain sample detection, where we associate each view with an uncertainty-aware classifier and combine the predictions of all the views in a principled way.

Network embedding plays a significant role in a variety of applications. To capture the topology of the network, most of the existing network embedding algorithms follow a sampling training procedure, which maximizes the similarity (e.g., embedding vectors' dot product) between positively sampled node pairs and minimizes the similarity between negatively sampled node pairs in the embedding space. Typically, close node pairs function as positive samples while distant node pairs are usually considered as negative samples. However, under different or even competing sampling strategies, some methods champion sampling distant node pairs as positive samples to encapsulate longer distance information in link prediction, whereas others advocate adding close nodes into the negative sample set to boost the performance of node recommendation. In this paper, we seek to understand the intrinsic relationships between these competing strategies. To this end, we identify two properties (discrimination and monotonicity) that given any node pair proximity distribution, node embeddings should embrace. Moreover, we quantify the empirical error of the trained similarity score w.r.t. the sampling strategy, which leads to an important finding that the discrimination property and the monotonicity property for all node pairs can not be satisfied simultaneously in real-world applications. Guided by such analysis, a simple yet novel model (SENSEI) is proposed, which seamlessly fulfills the discrimination property and the partial monotonicity within the top-K ranking list. Extensive experiments show that SENSEI outperforms the state-of-the-arts in plain network embedding.

Finding neural network weights that generalize well from small datasets is difficult. A promising approach is to learn a weight initialization such that a small number of weight changes results in low generalization error. We show that this form of meta-learning can be improved by letting the learning algorithm decide which weights to change, i.e., by learning where to learn. We find that patterned sparsity emerges from this process, with the pattern of sparsity varying on a problem-byproblem basis.