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Title: End-to-end Learning of a Convolutional Neural Network via Deep Tensor Decomposition

Paper content:
In this paper we study the problem of learning the weights of a deep convolutional neural network.
We consider a network where convolutions are carried out over non-overlapping patches with a single kernel in each layer.
We develop an algorithm for simultaneously learning all the kernels from the training data.
Our approach dubbed Deep Tensor Decomposition (DeepTD) is based on a rank-1 tensor decomposition.
We theoretically investigate DeepTD under a realizable model for the training data where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels.
We show that DeepTD is data-efficient and provably works as soon as the sample size exceeds the total number of convolutional weights in the network.
Our numerical experiments demonstrate the effectiveness of DeepTD and verify our theoretical findings.
Deep neural network (DNN) architectures have led to state of the art performance in many domains including image recognition, natural language processing, recommendation systems, and video analysis (He et al. (2016) ; Krizhevsky et al. (2012) ; Van den Oord et al. (2013) ; Collobert & Weston (2008) ).
Convolutional neural networks (CNNs) are a class of deep, feed-forward neural networks with a specialized DNN architecture.
CNNs are responsible for some of the most significant performance gains of DNN architectures.
In particular, CNN architectures have led to striking performance improvements for image/object recognition tasks.
Convolutional neural networks, loosely inspired by the visual cortex of animals, construct increasingly higher level features (such as mouth and nose) from lower level features such as pixels.
An added advantage of CNNs which makes them extremely attractive for large-scale applications is their remarkable efficiency which can be attributed to: (1) intelligent utilization of parameters via weight-sharing, (2) their convolutional nature which exploits the local spatial structure of images/videos effectively, and (3) highly efficient matrix/vector multiplication involved in CNNs compared to fully-connected neural network architectures.Despite the wide empirical success of CNNs the reasons for the effectiveness of neural networks and CNNs in particular is still a mystery.
Recently there has been a surge of interest in developing more rigorous foundations for neural networks (Soltanolkotabi et al. (2017) FORMULA108 ).
Most of this existing literature however focus on learning shallow neural networks typically consisting of zero or one hidden layer.
In practical applications, depth seems to play a crucial role in constructing progressively higher-level features from pixels.
Indeed, state of the art Resnet models typically have hundreds of layers.
Furthermore, recent results suggest that increasing depth may substantially boost the expressive power of neural networks (Raghu et al. (2016) ; Cohen et al. (2016) ).;In
this paper, we propose an algorithm for approximately learning an arbitrarily deep CNN model with rigorous guarantees. Our
goal is to provide theoretical insights towards better understanding when training deep CNN architectures is computationally tractable and how much data is required for successful training. We
focus on a realizable model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels. We
use both labels and features in the training data to construct a tensor. Our
first insight is that, in the limit of infinite data this tensor converges to a population tensor which is approximately rank one and whose factors reveal the direction of the kernels. Our
second insight is that even with finite data this empirical tensor is still approximately rank one. We
show that the gap between the population and empirical tensors provably decreases with the increase in the size of the training data set and becomes negligible as soon as the size of the training data becomes proportional to the total numbers of the parameters in the planted CNN model. Combining
these insights we provide a tensor decomposition algorithm to learn the kernels from training data. We show that
our algorithm approximately learns the kernels (up to sign/scale ambiguities) as soon as the size of the training data is proportional to the total number of parameters of the planted CNN model. Our results
can be viewed as a first step towards provable end-to-end learning of practical deep CNN models. Extending the
connections between neural networks and tensors (Janzamin et al. (2015) ; Cohen et al. (2016) ; Zhong et al. (2017a) ), we show how tensor decomposition can be utilized to approximately learn deep networks despite the presence of nonlinearities and growing depth. While our focus
in this work is limited to tensors, we believe that our proposed algorithm may provide valuable insights for initializing local search methods (such as stochastic gradient descent) to enhance the quality and/or speed of CNN training.
In this paper we studied a multilayer CNN model with depth D. We assumed a non-overlapping structure where each layer has a single convolutional kernel and has stride length equal to the dimension of its kernel.
We establish a connection between approximating the CNN kernels and higher order tensor decompositions.
Based on this, we proposed an algorithm for simultaneously learning all kernels called the Deep Tensor Decomposition (DeepTD).
This algorithm builds a D-way tensor based on the training data and applies a rank one tensor factorization algorithm to this tensor to simultaneously estimate all of the convolutional kernels.
Assuming the input data is distributed i.i.d. according to a Gaussian model with corresponding output generated by a planted set of convolutional kernels, we prove DeepTD can approximately learn all kernels with a near minimal amount of training data.
A variety of numerical experiments complement our theoretical findings.

assistant: text: This paper proposes a learning method for a restricted case of deep convolutional networks, where the layers are limited to the non-overlapping case and have only one output channel per layer