This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $q\le p^*$ and $1/p+1/p^{*}=1$, we discuss properties of a width-independent capacity control, which only depends on the depth by a square root term. We further analyze the approximation properties of $L_{p,q}$ weight normalized deep neural networks. In particular, for an $L_{1,\infty}$ weight normalized network, the approximation error can be controlled by the $L_1$ norm of the output layer, and the corresponding generalization error only depends on the architecture by the square root of the depth.

Retrieving indexed documents, not by their topical content but their writing style opens the door for a number of applications in information retrieval (IR). One application is to retrieve textual content of a certain author X, where the queried IR system is provided beforehand with a set of reference texts of X. Authorship verification (AV), which is a research subject in the field of digital text forensics, is suitable for this purpose. The task of AV is to determine if two documents (i.e. an indexed and a reference document) have been written by the same author X. Even though AV represents a unary classification problem, a number of existing approaches consider it as a binary classification task. However, the underlying classification model of an AV method has a number of serious implications regarding its prerequisites, evaluability, and applicability. In our comprehensive literature review, we observed several misunderstandings regarding the differentiation of unary and binary AV approaches that require consideration. The objective of this paper is, therefore, to clarify these by proposing clear criteria and new properties that aim to improve the characterization of existing and future AV approaches. Given both, we investigate the applicability of eleven existing unary and binary AV methods as well as four generic unary classification algorithms on two self-compiled corpora. Furthermore, we highlight an important issue concerning the evaluation of AV methods based on fixed decision criterions, which has not been paid attention in previous AV studies.

We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

In this paper, we revisit the Empirical Risk Minimization problem in the non-interactive local model of differential privacy. In the case of constant or low dimensions ($p\ll n$), we first show that if the loss function is $(\infty, T)$-smooth, we can avoid a dependence of the sample complexity, to achieve error $\alpha$, on the exponential of the dimensionality $p$ with base $1/\alpha$ ({\em i.e.,} $\alpha^{-p}$), which answers a question in \cite{smith2017interaction}. Our approach is based on polynomial approximation. Then, we propose player-efficient algorithms with $1$-bit communication complexity and $O(1)$ computation cost for each player. The error bound is asymptotically the same as the original one. With some additional assumptions, we also give an efficient algorithm for the server. In the case of high dimensions ($n\ll p$), we show that if the loss function is a convex generalized linear function, the error can be bounded by using the Gaussian width of the constrained set, instead of $p$, which improves the one in \cite{smith2017interaction}.

A theoretical performance analysis of the graph neural network (GNN) is presented. For classification tasks, the neural network approach has the advantage in terms of flexibility that it can be employed in a data-driven manner, whereas Bayesian inference requires the assumption of a specific model. A fundamental question is then whether GNN has a high accuracy in addition to this flexibility. Moreover, whether the achieved performance is predominately a result of the backpropagation or the architecture itself is a matter of considerable interest. To gain a better insight into these questions, a mean-field theory of a minimal GNN architecture is developed for the graph partitioning problem. This demonstrates a good agreement with numerical experiments.

Let $\PP=\{ p_1, p_2, \ldots p_n \}$ and $\QQ = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\AA$ represent the $m\times n$ pairwise distance matrix with $\AA_{i,j} = d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling. We then show that in general, relative error approximation in sublinear time is impossible for distance matrices, even if one allows for bicriteria solutions. Additionally, we show that if $\PP = \QQ$ and $d$ is the squared Euclidean distance, which is not a metric but rather the square of a metric, then a relative error bicriteria solution can be found in sublinear time. Finally, we empirically compare our algorithm with the SVD and input sparsity time algorithms. Our algorithm is several hundred times faster than the SVD, and about $8$-$20$ times faster than input sparsity methods on real-world and and synthetic datasets of size $10^8$. Accuracy-wise, our algorithm is only slightly worse than that of the SVD (optimal) and input-sparsity time algorithms.

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decen- tralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as some applications to image aggregation.

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decen- tralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as some applications to image aggregation.

Convolutional long short-term memory (LSTM) networks have been widely used for action/gesture recognition, and different attention mechanisms have also been embedded into the LSTM or the convolutional LSTM (ConvLSTM) networks. Based on the previous gesture recognition architectures which combine the three-dimensional convolution neural network (3DCNN) and ConvLSTM, this paper explores the effects of attention mechanism in ConvLSTM. Several variants of ConvLSTM are evaluated: (a) Removing the convolutional structures of the three gates in ConvLSTM, (b) Applying the attention mechanism on the input of ConvLSTM, (c) Reconstructing the input and (d) output gates respectively with the modified channel-wise attention mechanism. The evaluation results demonstrate that the spatial convolutions in the three gates scarcely contribute to the spatiotemporal feature fusion, and the attention mechanisms embedded into the input and output gates cannot improve the feature fusion. In other words, ConvLSTM mainly contributes to the temporal fusion along with the recurrent steps to learn the long-term spatiotemporal features, when taking as input the spatial or spatiotemporal features. On this basis, a new variant of LSTM is derived, in which the convolutional structures are only embedded into the input-to-state transition of LSTM. The code of the LSTM variants is publicly available.

Convolutional long short-term memory (LSTM) networks have been widely used for action/gesture recognition, and different attention mechanisms have also been embedded into the LSTM or the convolutional LSTM (ConvLSTM) networks. Based on the previous gesture recognition architectures which combine the three-dimensional convolution neural network (3DCNN) and ConvLSTM, this paper explores the effects of attention mechanism in ConvLSTM. Several variants of ConvLSTM are evaluated: (a) Removing the convolutional structures of the three gates in ConvLSTM, (b) Applying the attention mechanism on the input of ConvLSTM, (c) Reconstructing the input and (d) output gates respectively with the modified channel-wise attention mechanism. The evaluation results demonstrate that the spatial convolutions in the three gates scarcely contribute to the spatiotemporal feature fusion, and the attention mechanisms embedded into the input and output gates cannot improve the feature fusion. In other words, ConvLSTM mainly contributes to the temporal fusion along with the recurrent steps to learn the long-term spatiotemporal features, when taking as input the spatial or spatiotemporal features. On this basis, a new variant of LSTM is derived, in which the convolutional structures are only embedded into the input-to-state transition of LSTM. The code of the LSTM variants is publicly available.