Discrete Fourier transforms provide a significant speedup in the computation of convolutions in deep learning. In this work, we demonstrate that, beyond its advantages for efficient computation, the spectral domain also provides a powerful representation in which to model and train convolutional neural networks (CNNs). We employ spectral representations to introduce a number of innovations to CNN design. First, we propose spectral pooling, which performs dimensionality reduction by truncating the representation in the frequency domain. This approach preserves considerably more information per parameter than other pooling strategies and enables flexibility in the choice of pooling output dimensionality. This representation also enables a new form of stochastic regularization by randomized modification of resolution. We show that these methods achieve competitive results on classification and approximation tasks, without using any dropout or max-pooling. Finally, we demonstrate the effectiveness of complex-coefficient spectral parameterization of convolutional filters. While this leaves the underlying model unchanged, it results in a representation that greatly facilitates optimization. We observe on a variety of popular CNN configurations that this leads to significantly faster convergence during training.

We propose rectified factor networks (RFNs) to efficiently construct very sparse, non-linear, high-dimensional representations of the input. RFN models identify rare and small events in the input, have a low interference between code units, have a small reconstruction error, and explain the data covariance structure. RFN learning is a generalized alternating minimization algorithm derived from the posterior regularization method which enforces non-negative and normalized posterior means. We proof convergence and correctness of the RFN learning algorithm. On benchmarks, RFNs are compared to other unsupervised methods like autoencoders, RBMs, factor analysis, ICA, and PCA. In contrast to previous sparse coding methods, RFNs yield sparser codes, capture the data's covariance structure more precisely, and have a significantly smaller reconstruction error. We test RFNs as pretraining technique for deep networks on different vision datasets, where RFNs were superior to RBMs and autoencoders. On gene expression data from two pharmaceutical drug discovery studies, RFNs detected small and rare gene modules that revealed highly relevant new biological insights which were so far missed by other unsupervised methods.

Corrupting the input and hidden layers of deep neural networks (DNNs) with multiplicative noise, often drawn from the Bernoulli distribution (or 'dropout'), provides regularization that has significantly contributed to deep learning's success. However, understanding how multiplicative corruptions prevent overfitting has been difficult due to the complexity of a DNN's functional form. In this paper, we show that when a Gaussian prior is placed on a DNN's weights, applying multiplicative noise induces a Gaussian scale mixture, which can be reparameterized to circumvent the problematic likelihood function. Analysis can then proceed by using a type-II maximum likelihood procedure to derive a closed-form expression revealing how regularization evolves as a function of the network's weights. Results show that multiplicative noise forces weights to become either sparse or invariant to rescaling. We find our analysis has implications for model compression as it naturally reveals a weight pruning rule that starkly contrasts with the commonly used signal-to-noise ratio (SNR). While the SNR prunes weights with large variances, seeing them as noisy, our approach recognizes their robustness and retains them. We empirically demonstrate our approach has a strong advantage over the SNR heuristic and is competitive to retraining with soft targets produced from a teacher model.

Syntactic constituency parsing is a fundamental problem in natural language processing and has been the subject of intensive research and engineering for decades. As a result, the most accurate parsers are domain specific, complex, and inefficient. In this paper we show that the domain agnostic attention-enhanced sequence-to-sequence model achieves state-of-the-art results on the most widely used syntactic constituency parsing dataset, when trained on a large synthetic corpus that was annotated using existing parsers. It also matches the performance of standard parsers when trained only on a small human-annotated dataset, which shows that this model is highly data-efficient, in contrast to sequence-to-sequence models without the attention mechanism. Our parser is also fast, processing over a hundred sentences per second with an unoptimized CPU implementation.

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 AdaGrad.

Pre-training is crucial for learning deep neural networks. Most of existing pre-training methods train simple models (e.g., restricted Boltzmann machines) and then stack them layer by layer to form the deep structure. This layer-wise pre-training has found strong theoretical foundation and broad empirical support. However, it is not easy to employ such method to pre-train models without a clear multi-layer structure,e.g., recurrent neural networks (RNNs). This paper presents a new pre-training approach based on knowledge transfer learning. In contrast to the layer-wise approach which trains model components incrementally, the new approach trains the entire model as a whole but with an easier objective function. This is achieved by utilizing soft targets produced by a prior trained model (teacher model). Compared to the conventional layer-wise methods, this new method does not care about the model structure, so can be used to pre-train very complex models. Experiments on a speech recognition task demonstrated that with this approach, complex RNNs can be well trained with a weaker deep neural network (DNN) model. Furthermore, the new method can be combined with conventional layer-wise pre-training to deliver additional gains.

There has been a lot of recent interest in designing neural network models to estimate a distribution from a set of examples. We introduce a simple modification for autoencoder neural networks that yields powerful generative models. Our method masks the autoencoder's parameters to respect autoregressive constraints: each input is reconstructed only from previous inputs in a given ordering. Constrained this way, the autoencoder outputs can be interpreted as a set of conditional probabilities, and their product, the full joint probability. We can also train a single network that can decompose the joint probability in multiple different orderings. Our simple framework can be applied to multiple architectures, including deep ones. Vectorized implementations, such as on GPUs, are simple and fast. Experiments demonstrate that this approach is competitive with state-of-the-art tractable distribution estimators. At test time, the method is significantly faster and scales better than other autoregressive estimators.

In this work we study the properties of deep neural networks (DNN) with random weights. We formally prove that these networks perform a distance-preserving embedding of the data. Based on this we then draw conclusions on the size of the training data and the networks' structure. A longer version of this paper with more results and details can be found in (Giryes et al., 2015). In particular, we formally prove in the longer version that DNN with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data.

We introduce two Python frameworks to train neural networks on large datasets: Blocks and Fuel. Blocks is based on Theano, a linear algebra compiler with CUDA-support. It facilitates the training of complex neural network models by providing parametrized Theano operations, attaching metadata to Theano's symbolic computational graph, and providing an extensive set of utilities to assist training the networks, e.g. training algorithms, logging, monitoring, visualization, and serialization. Fuel provides a standard format for machine learning datasets. It allows the user to easily iterate over large datasets, performing many types of pre-processing on the fly.

Inspired by recent successes of deep learning in computer vision, we propose a novel framework for encoding time series as different types of images, namely, Gramian Angular Summation/Difference Fields (GASF/GADF) and Markov Transition Fields (MTF). This enables the use of techniques from computer vision for time series classification and imputation. We used Tiled Convolutional Neural Networks (tiled CNNs) on 20 standard datasets to learn high-level features from the individual and compound GASF-GADF-MTF images. Our approaches achieve highly competitive results when compared to nine of the current best time series classification approaches. Inspired by the bijection property of GASF on 0/1 rescaled data, we train Denoised Auto-encoders (DA) on the GASF images of four standard and one synthesized compound dataset. The imputation MSE on test data is reduced by 12.18%-48.02% when compared to using the raw data. An analysis of the features and weights learned via tiled CNNs and DAs explains why the approaches work.