深度学习随机矩阵理论模型_v0.1.pptx

深度学习的随机矩阵理论模型;;神经网络将许多单一的神经元连接在一起 一个神经元的输出作为另一个神经元的输入 多层神经网络模型可以理解为多个非线性函数“嵌套” 多层神经网络层数可以无限叠加 具有无限建模能力, 可以拟合任意函数 ;Sigmoid Tanh Rectified linear units (ReLU) ;层数逐年增加;Features are learned rather than hand-crafted More layers capture more invariances More data to train deeper networks More computing (GPUs) Better regularization: Dropout New nonlinearities Max pooling, Rectified linear units (ReLU) Theoretical understanding of deep networks remains shallow;Experimental Neuroscience uncovered: Neural architecture of Retina/LGN/V1/V2/V3/ etc Existence of neurons with weights and activation functions (simple cells) Pooling neurons (complex cells) All these features are somehow present in Deep Learning systems;Olshausen and Field demonstrated that receptive fields learned from image patches. Olshausen and Field showed that optimization process can drive learning image representations. ;Olshausen-Field representations bear strong resemblance to defined mathematical objects from harmonic analysis wavelets, ridgelets, curvelets. Harmonic analysis: long history of developing optimal representations via optimization Research in 1990s: Wavelets etc are optimal sparsifying transforms for certain classes of images ;Class prediction rule can be viewed as function f(x) of high-dimensional argument Curse of Dimensionality Traditional theoretical obstacle to high-dimensional approximation Functions of high dimensional x can wiggle in too many dimensions to be learned from finite datasets;Approximation theory Perceptrons and multilayer feedforward networks are universal approximators: Cybenko ’89, Hornik ’89, Hornik ’91, Barron ‘93 Optimization theory No spurious local optima for linear networks: Baldi Hornik ’89 Stuck in local minima: Brady ‘89 Stuck in local minima, but convergence guarantees for linearly separable data: Gori Tesi ‘92 Manifold of spurious local optima: Frasconi ’97;Invariance, stability, and learning theory Scattering networks: Bruna ’11, Bruna ’13, Mallat ’13 Defor