Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.

/pdf/deep-residual-learning-for-image-recognition-b75jg61qrq.pdf

Deep Residual Learning for Image Recognition

We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.

Adam: A Method for Stochastic Optimization

We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation. To reduce overriding in the fully-connected layers we employed a recently-developed regularization method called "dropout" that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry.

/pdf/imagenet-classification-with-deep-convolutional-neural-1190x9a6b9.pdf

ImageNet Classification with Deep Convolutional Neural Networks

Learning to store information over extended time intervals by recurrent backpropagation takes a very long time, mostly because of insufficient, decaying error backflow. We briefly review Hochreiter's (1991) analysis of this problem, then address it by introducing a novel, efficient, gradient based method called long short-term memory (LSTM). Truncating the gradient where this does not do harm, LSTM can learn to bridge minimal time lags in excess of 1000 discrete-time steps by enforcing constant error flow through constant error carousels within special units. Multiplicative gate units learn to open and close access to the constant error flow. LSTM is local in space and time; its computational complexity per time step and weight is O. 1. Our experiments with artificial data involve local, distributed, real-valued, and noisy pattern representations. In comparisons with real-time recurrent learning, back propagation through time, recurrent cascade correlation, Elman nets, and neural sequence chunking, LSTM leads to many more successful runs, and learns much faster. LSTM also solves complex, artificial long-time-lag tasks that have never been solved by previous recurrent network algorithms.

Long short-term memory

In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.

Very Deep Convolutional Networks for Large-Scale Image Recognition

A time-delay neural network (TDNN) for phoneme recognition is discussed. By the use of two hidden layers in addition to an input and output layer it is capable of representing complex nonlinear decision surfaces. Three important properties of the TDNNs have been observed. First, it was able to invent without human interference meaningful linguistic abstractions in time and frequency such as formant tracking and segmentation. Second, it has learned to form alternate representations linking different acoustic events with the same higher level concept. In this fashion it can implement trading relations between lower level acoustic events leading to robust recognition performance despite considerable variability in the input speech. Third, the network is translation-invariant and does not rely on precise alignment or segmentation of the input. The TDNNs performance is compared with the best of hidden Markov models (HMMs) on a speaker-dependent phoneme-recognition task. The TDNN achieved a recognition of 98.5% compared to 93.7% for the HMM, i.e., a fourfold reduction in error. >

Phoneme recognition: neural networks vs. hidden Markov models vs. hidden Markov models

We introduce a new class of probabilistic latent variable model called the Implicit Mixture of Conditional Restricted Boltzmann Machines (imCRBM) for use in human pose tracking. Key properties of the imCRBM are as follows: (1) learning is linear in the number of training exemplars so it can be learned from large datasets; (2) it learns coherent models of multiple activities; (3) it automatically discovers atomic “movemes” and (4) it can infer transitions between activities, even when such transitions are not present in the training set. We describe the model and how it is learned and we demonstrate its use in the context of Bayesian filtering for multi-view and monocular pose tracking. The model handles difficult scenarios including multiple activities and transitions among activities. We report state-of-the-art results on the HumanEva dataset.

http://www.cs.toronto.edu/~hinton/absps/gwtaylor_cvpr2010.pdf

Dynamical binary latent variable models for 3D human pose tracking

Distributed representations are attractive for a number of reasons. They offer the possibility of representing concepts in a continuous space, they degrade gracefully with noise, and they can be processed in a parallel network of simple processing elements. However, the problem of representing nested structure in distributed representations has been for some time a prominent concern of both proponents and critics of connectionism (Fodor and Pylyshyn 1988; Smolensky 1990; Hinton 1990). The lack of connectionist representations for complex structure has held back progress in tackling higher-level cognitive tasks such as language understanding and reasoning.
In this thesis I review connectionist representations and propose a method for the distributed representation of nested structure, which I call "Holographic Reduced Representations" (HRRs). HRRs provide an implementation of Hinton's (1990) "reduced descriptions". HRRs use circular convolution to associate atomic items, which are represented by vectors. Arbitrary variable bindings, short sequences of various lengths, and predicates can be represented in a fixed-width vector. These representations are items in their own right, and can be used in constructing compositional structures. The noisy reconstructions extracted from convolution memories can be cleaned up by using a separate associative memory that has good reconstructive properties.
Circular convolution, which is the basic associative operator for HRRs, can be built into a recurrent neural network. The network can store and produce sequences. I show that neural network learning techniques can be used with circular convolution in order to learn representations for items and sequences.
One of the attractions of connectionist representations of compositional structures is the possibility of computing without decomposing structures. I show that it is possible to use dot-product comparisons of HRRs for nested structures to estimate the analogical similarity of the structures. This demonstrates how the surface form of connectionist representations can reflect underlying structural similarity and alignment.

Distributed representations and nested compositional structure

We describe a neurally-inspired, unsupervised learning algorithm that builds a non-linear generative model for pairs of face images from the same individual. Individuals are then recognized by finding the highest relative probability pair among all pairs that consist of a test image and an image whose identity is known. Our method compares favorably with other methods in the literature. The generative model consists of a single layer of rate-coded, non-linear feature detectors and it has the property that, given a data vector, the true posterior probability distribution over the feature detector activities can be inferred rapidly without iteration or approximation. The weights of the feature detectors are learned by comparing the correlations of pixel intensities and feature activations in two phases: When the network is observing real data and when it is observing reconstructions of real data generated from the feature activations.

/pdf/rate-coded-restricted-boltzmann-machines-for-face-w7p6vwo21o.pdf

Rate-coded Restricted Boltzmann Machines for Face Recognition

Recent work has sought to understand the behavior of neural networks by comparing representations between layers and between different trained models. We examine methods for comparing neural network representations based on canonical correlation analysis (CCA). We show that CCA belongs to a family of statistics for measuring multivariate similarity, but that neither CCA nor any other statistic that is invariant to invertible linear transformation can measure meaningful similarities between representations of higher dimension than the number of data points. We introduce a similarity index that measures the relationship between representational similarity matrices and does not suffer from this limitation. This similarity index is equivalent to centered kernel alignment (CKA) and is also closely connected to CCA. Unlike CCA, CKA can reliably identify correspondences between representations in networks trained from different initializations.

Geoffrey E. Hinton

Papers

Phoneme recognition: neural networks vs. hidden Markov models vs. hidden Markov models

Dynamical binary latent variable models for 3D human pose tracking

Distributed representations and nested compositional structure

Rate-coded Restricted Boltzmann Machines for Face Recognition

Similarity of Neural Network Representations Revisited