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Cointegration analysis was developed for non-stationary linear processes that exhibit stationary relationships between coordinates. Estimation of the cointegration relationships in a multi-dimensional cointegrated process typically proceeds in two steps. First the rank is estimated, then the cointegration matrix is estimated, conditionally on the estimated rank (reduced rank regression). The asymptotics of the estimator is usually derived under the assumption of knowing the true rank. In this paper, we quantify the asymptotic bias and find the asymptotic distributions of the cointegration estimator in case of misspecified rank. Furthermore, we suggest a new class of weighted reduced rank estimators that allow for more flexibility in settings where rank selection is hard. We show empirically that a proper choice of weights can lead to increased predictive performance when there is rank uncertainty. Finally, we illustrate the estimators on empirical EEG data from a psychological experiment on visual processing.

The pursuit of long-term autonomy mandates that robotic agents must continuously adapt to their changing environments and learn to solve new tasks. Continual learning seeks to overcome the challenge of catastrophic forgetting, where learning to solve new tasks causes a model to forget previously learnt information. Prior-based continual learning methods are appealing for robotic applications as they are space efficient and typically do not increase in computational complexity as the number of tasks grows. Despite these desirable properties, prior-based approaches typically fail on important benchmarks and consequently are limited in their potential applications compared to their memory-based counterparts. We introduce Bayesian adaptive moment regularization (BAdam), a novel prior-based method that better constrains parameter growth, leading to lower catastrophic forgetting. Our method boasts a range of desirable properties for robotic applications such as being lightweight and task label-free, converging quickly, and offering calibrated uncertainty that is important for safe real-world deployment. Results show that BAdam achieves state-of-the-art performance for prior-based methods on challenging single-headed class-incremental experiments such as Split MNIST and Split FashionMNIST, and does so without relying on task labels or discrete task boundaries.

Invariance describes transformations that do not alter data's underlying semantics. Neural networks that preserve natural invariance capture good inductive biases and achieve superior performance. Hence, modern networks are handcrafted to handle well-known invariances (ex. translations). We propose a framework to learn novel network architectures that capture data-dependent invariances via pruning. Our learned architectures consistently outperform dense neural networks on both vision and tabular datasets in both efficiency and effectiveness. We demonstrate our framework on multiple deep learning models across 3 vision and 40 tabular datasets.

Many modern datasets, from areas such as neuroimaging and geostatistics, come in the form of a random sample of tensor-valued data which can be understood as noisy observations of a smooth multidimensional random function. Most of the traditional techniques from functional data analysis are plagued by the curse of dimensionality and quickly become intractable as the dimension of the domain increases. In this paper, we propose a framework for learning continuous representations from a sample of multidimensional functional data that is immune to several manifestations of the curse. These representations are constructed using a set of separable basis functions that are defined to be optimally adapted to the data. We show that the resulting estimation problem can be solved efficiently by the tensor decomposition of a carefully defined reduction transformation of the observed data. Roughness-based regularization is incorporated using a class of differential operator-based penalties. Relevant theoretical properties are also established. The advantages of our method over competing methods are demonstrated in a simulation study. We conclude with a real data application in neuroimaging.

Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.

Humans have a natural instinct to identify unknown object instances in their environments. The intrinsic curiosity about these unknown instances aids in learning about them, when the corresponding knowledge is eventually available. This motivates us to propose a novel computer vision problem called: `Open World Object Detection', where a model is tasked to: 1) identify objects that have not been introduced to it as `unknown', without explicit supervision to do so, and 2) incrementally learn these identified unknown categories without forgetting previously learned classes, when the corresponding labels are progressively received. We formulate the problem, introduce a strong evaluation protocol and provide a novel solution, which we call ORE: Open World Object Detector, based on contrastive clustering and energy based unknown identification. Our experimental evaluation and ablation studies analyze the efficacy of ORE in achieving Open World objectives. As an interesting by-product, we find that identifying and characterizing unknown instances helps to reduce confusion in an incremental object detection setting, where we achieve state-of-the-art performance, with no extra methodological effort. We hope that our work will attract further research into this newly identified, yet crucial research direction.

The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork. We propose a subnetwork selection strategy that aims to maximally preserve the model's predictive uncertainty. Empirically, our approach is effective compared to ensembles and less expressive posterior approximations over full networks.

Graph Convolutional Networks (GCNs) and their variants have experienced significant attention and have become the de facto methods for learning graph representations. GCNs derive inspiration primarily from recent deep learning approaches, and as a result, may inherit unnecessary complexity and redundant computation. In this paper, we reduce this excess complexity through successively removing nonlinearities and collapsing weight matrices between consecutive layers. We theoretically analyze the resulting linear model and show that it corresponds to a fixed low-pass filter followed by a linear classifier. Notably, our experimental evaluation demonstrates that these simplifications do not negatively impact accuracy in many downstream applications. Moreover, the resulting model scales to larger datasets, is naturally interpretable, and yields up to two orders of magnitude speedup over FastGCN.

This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way---just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. See related articles at //explained.ai

The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. Experiments on two machine translation tasks show these models to be superior in quality while being more parallelizable and requiring significantly less time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-to-German translation task, improving over the existing best results, including ensembles by over 2 BLEU. On the WMT 2014 English-to-French translation task, our model establishes a new single-model state-of-the-art BLEU score of 41.8 after training for 3.5 days on eight GPUs, a small fraction of the training costs of the best models from the literature. We show that the Transformer generalizes well to other tasks by applying it successfully to English constituency parsing both with large and limited training data.