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Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of results provide tight bounds on the average errors between empirical and population statistics of eigenvectors, fewer results are tight for entrywise analyses, which are critical for a number of problems such as community detection and ranking.

Following the recent trend in explicit neural memory structures, we present a new design of an external memory, wherein memories are stored in an Euclidean key space \mathbb R^ n . An LSTM controller performs read and write via specialized read and write heads. It can move a head by either providing a new address in the key space (aka random access) or moving from its previous position via a Lie group action (aka Lie access). In this way, the" L" and" R" instructions of a traditional Turing Machine are generalized to arbitrary elements of a fixed Lie group action. For this reason, we name this new model the Lie Access Neural Turing Machine, or LANTM.

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In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called sigmoid-ADMM pair ), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than that of deep ReLU nets by showing that ReLU activation function can be well approximated by deep sigmoid nets with two hidden layers and finitely many free parameters but not vice-verse. We also establish the global convergence of the proposed ADMM for the nonlinearly constrained formulation of the deep sigmoid nets training from arbitrary initial points to a Karush-Kuhn-Tucker (KKT) point at a rate of order O(1/k). Besides sigmoid activation, such a convergence theorem holds for a general class of smooth activations. Compared with the widely used stochastic gradient descent (SGD) algorithm for the deep ReLU nets training (called ReLU-SGD pair), the proposed sigmoid-ADMM pair is practically stable with respect to the algorithmic hyperparameters including the learning rate, initial schemes and the pro-processing of the input data. Moreover, we find that to approximate and learn simple but important functions the proposed sigmoid-ADMM pair numerically outperforms the ReLU-SGD pair.