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Why and how that deep learning works well on different tasks remains a mystery from a theoretical perspective. In this paper we draw a geometric picture of the deep learning system by finding its analogies with two existing geometric structures, the geometry of quantum computations and the geometry of the diffeomorphic template matching. In this framework, we give the geometric structures of different deep learning systems including convolutional neural networks, residual networks, recursive neural networks, recurrent neural networks and the equilibrium prapagation framework. We can also analysis the relationship between the geometrical structures and their performance of different networks in an algorithmic level so that the geometric framework may guide the design of the structures and algorithms of deep learning systems.
How does a 110-layer ResNet learn a high-complexity classifier using relatively few training examples and short training time? We present a theory towards explaining this in terms of Hierarchical Learning. We refer hierarchical learning as the learne
Classical linear metric learning methods have recently been extended along two distinct lines: deep metric learning methods for learning embeddings of the data using neural networks, and Bregman divergence learning approaches for extending learning E
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalizat
We show that a collection of Gaussian mixture models (GMMs) in $R^{n}$ can be optimally classified using $O(n)$ neurons in a neural network with two hidden layers (deep neural network), whereas in contrast, a neural network with a single hidden layer
We propose a novel framework, called Markov-Lipschitz deep learning (MLDL), to tackle geometric deterioration caused by collapse, twisting, or crossing in vector-based neural network transformations for manifold-based representation learning and mani