Invariance of Weight Distributions in Rectified MLPs

An interesting approach to analyzing neural networks that has received renewed attention is to examine the equivalent kernel of the neural network. This is based on the fact that a fully connected feedforward network with one hidden layer, a certain weight distribution, an activation function, and an infinite number of neurons can be viewed as a mapping into a Hilbert space. We derive the equivalent kernels of MLPs with ReLU or Leaky ReLU activations for all rotationally-invariant weight distributions, generalizing a previous result that required Gaussian weight distributions. Additionally, the Central Limit Theorem is used to show that for certain activation functions, kernels corresponding to layers with weight distributions having $0$ mean and finite absolute third moment are asymptotically universal, and are well approximated by the kernel corresponding to layers with spherical Gaussian weights. In deep networks, as depth increases the equivalent kernel approaches a pathological fixed point, which can be used to argue why training randomly initialized networks can be difficult. Our results also have implications for weight initialization.Comment: ICML 201

Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case

Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices that appear in research related to deep neural networks (DNNs), such as the dynamical isometry, Fisher information matrix, and training dynamics. FPT suits these researches because the DNN's parameter-Jacobian and input-Jacobian are polynomials of layerwise Jacobians. However, the critical assumption of asymptotic freenss of the layerwise Jacobian has not been proven completely so far. The asymptotic freeness assumption plays a fundamental role when propagating spectral distributions through the layers. Haar distributed orthogonal matrices are essential for achieving dynamical isometry. In this work, we prove asymptotic freeness of layerwise Jacobians of multilayer perceptron (MLP) in this case. A key of the proof is an invariance of the MLP. Considering the orthogonal matrices that fix the hidden units in each layer, we replace each layer's parameter matrix with itself multiplied by the orthogonal matrix, and then the MLP does not change. Furthermore, if the original weights are Haar orthogonal, the Jacobian is also unchanged by this replacement. Lastly, we can replace each weight with a Haar orthogonal random matrix independent of the Jacobian of the activation function using this key fact.Comment: Any comments are welcomed. In v4, we changed notations for readabilit

Machine learning and deep learning for emotion recognition

Ús de diferents tècniques de deep learning per al reconeixement d'emocions a partir d'imatges i videos. Les diferents tècniques s'apliquen, es valoren i comparen amb l'objectiu de fer-les servir conjuntament en una aplicació final.Outgoin

Modeling the Influence of Data Structure on Learning in Neural Networks: The Hidden Manifold Model

Understanding the reasons for the success of deep neural networks trained using stochastic gradient-based methods is a key open problem for the nascent theory of deep learning. The types of data where these networks are most successful, such as images or sequences of speech, are characterized by intricate correlations. Yet, most theoretical work on neural networks does not explicitly model training data or assumes that elements of each data sample are drawn independently from some factorized probability distribution. These approaches are, thus, by construction blind to the correlation structure of real-world datasets and their impact on learning in neural networks. Here, we introduce a generative model for structured datasets that we call the hidden manifold model. The idea is to construct high-dimensional inputs that lie on a lower-dimensional manifold, with labels that depend only on their position within this manifold, akin to a single-layer decoder or generator in a generative adversarial network. We demonstrate that learning of the hidden manifold model is amenable to an analytical treatment by proving a "Gaussian equivalence property"(GEP), and we use the GEP to show how the dynamics of two-layer neural networks trained using one-pass stochastic gradient descent is captured by a set of integro-differential equations that track the performance of the network at all times. This approach permits us to analyze in detail how a neural network learns functions of increasing complexity during training, how its performance depends on its size, and how it is impacted by parameters such as the learning rate or the dimension of the hidden manifold