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

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

We show that matrix completion with trace-norm regularization can be significantly hurt when entries of the matrix are sampled non-uniformly. We introduce a weighted version of the trace-norm regularizer that works well also with non-uniform sampling. Our experimental results demonstrate that the weighted trace-norm regularization indeed yields significant gains on the (highly non-uniformly sampled) Netflix dataset.

Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm

We conduct an in-depth exploration of different strategies for doing event detection in videos using convolutional neural networks (CNNs) trained for image classification. We study different ways of performing spatial and temporal pooling, feature normalization, choice of CNN layers as well as choice of classifiers. Making judicious choices along these dimensions led to a very significant increase in performance over more naive approaches that have been used till now. We evaluate our approach on the challenging TRECVID MED'14 dataset with two popular CNN architectures pretrained on ImageNet. On this MED'14 dataset, our methods, based entirely on image-trained CNN features, can outperform several state-of-the-art non-CNN models. Our proposed late fusion of CNN- and motion-based features can further increase the mean average precision (mAP) on MED'14 from 34.95% to 38.74%. The fusion approach achieves the state-of-the-art classification performance on the challenging UCF-101 dataset.

/pdf/exploiting-image-trained-cnn-architectures-for-unconstrained-4nqy3znovw.pdf

Exploiting Image-trained CNN Architectures for Unconstrained Video Classification

We propose a Deep Belief Network architecture for learning a joint representation of multimodal data. The model denes a probability distribution over the space of multimodal inputs and allows sampling from the conditional distributions over each data modality. This makes it possible for the model to create a multimodal representation even when some data modalities are missing. Our experimental results on bi-modal data consisting of images and text show that the Multimodal DBN can learn a good generative model of the joint space of image and text inputs that is useful for lling in missing data so it can be used both for image annotation and image retrieval. We further demonstrate that using the representation discovered by the Multimodal DBN our model can significantly outperform SVMs and LDA on discriminative tasks.

http://www.cs.toronto.edu/~nitish/icml2012/paper.pdf

Learning Representations for Multimodal Data with Deep Belief Nets

We revisit the choice of SGD for training deep neural networks by reconsidering the appropriate geometry in which to optimize the weights. We argue for a geometry invariant to rescaling of weights that does not affect the output of the network, and suggest Path-SGD, which is an approximate steepest descent method with respect to a path-wise regularizer related to max-norm regularization. Path-SGD is easy and efficient to implement and leads to empirical gains over SGD and Ada-Grad.

/pdf/path-sgd-path-normalized-optimization-in-deep-neural-53nekr7k3c.pdf

Path-SGD: path-normalized optimization in deep neural networks

While Boltzmann Machines have been successful at unsupervised learning and density modeling of images and speech data, they can be very sensitive to noise in the data. In this paper, we introduce a novel model, the Robust Boltzmann Machine (RoBM), which allows Boltzmann Machines to be robust to corruptions. In the domain of visual recognition, the RoBM is able to accurately deal with occlusions and noise by using multiplicative gating to induce a scale mixture of Gaussians over pixels. Image denoising and in-painting correspond to posterior inference in the RoBM. Our model is trained in an unsupervised fashion with unlabeled noisy data and can learn the spatial structure of the occluders. Compared to standard algorithms, the RoBM is significantly better at recognition and denoising on several face databases.

/pdf/robust-boltzmann-machines-for-recognition-and-denoising-32m46na907.pdf

Ruslan Salakhutdinov

Papers

Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm

Exploiting Image-trained CNN Architectures for Unconstrained Video Classification

Learning Representations for Multimodal Data with Deep Belief Nets

Path-SGD: path-normalized optimization in deep neural networks

Robust Boltzmann Machines for recognition and denoising