The one-class kernel spectral regression (OC-KSR), the regression-based formulation of the kernel null-space approach has been found to be an effective Fisher criterion-based methodology for one-class classification (OCC), achieving state-of-the-art performance in one-class classification while providing relatively high robustness against data corruption. This work extends the OC-KSR methodology to a multi-task setting where multiple one-class problems share information for improved performance. By viewing the multi-task structure learning problem as one of compositional function learning, first, the OC-KSR method is extended to learn multiple tasks' structure \textit{linearly} by posing it as an instantiation of the separable kernel learning problem in a vector-valued reproducing kernel Hilbert space where an output kernel encodes tasks' structure while another kernel captures input similarities. Next, a non-linear structure learning mechanism is proposed which captures multiple tasks' relationships \textit{non-linearly} via an output kernel. The non-linear structure learning method is then extended to a sparse setting where different tasks compete in an output composition mechanism, leading to a sparse non-linear structure among multiple problems. Through extensive experiments on different data sets, the merits of the proposed multi-task kernel null-space techniques are verified against the baseline as well as other existing multi-task one-class learning techniques.

Two main obstacles preventing the widespread adoption of variational Bayesian neural networks are the high parameter overhead that makes them infeasible on large networks, and the difficulty of implementation, which can be thought of as "programming overhead." MC dropout [Gal and Ghahramani, 2016] is popular because it sidesteps these obstacles. Nevertheless, dropout is often harmful to model performance when used in networks with batch normalization layers [Li et al., 2018], which are an indispensable part of modern neural networks. We construct a general variational family for ensemble-based Bayesian neural networks that encompasses dropout as a special case. We further present two specific members of this family that work well with batch normalization layers, while retaining the benefits of low parameter and programming overhead, comparable to non-Bayesian training. Our proposed methods improve predictive accuracy and achieve almost perfect calibration on a ResNet-18 trained with ImageNet.

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In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model fit as it quantifies the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through $k$-fold partitioning or leave-$p$-out subsampling. We show that the marginal likelihood is formally equivalent to exhaustive leave-$p$-out cross-validation averaged over all values of $p$ and all held-out test sets when using the log posterior predictive probability as the scoring rule. Moreover, the log posterior predictive is the only coherent scoring rule under data exchangeability. This offers new insight into the marginal likelihood and cross-validation and highlights the potential sensitivity of the marginal likelihood to the setting of the prior. We suggest an alternative approach using aggregate cross-validation following a preparatory training phase. Our work has connections to prequential analysis and intrinsic Bayes factors but is motivated through a different course.

Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Acharya et al. 2016 showed that there is a single estimator that is competitive for all symmetric properties. This work proved that computing the distribution that approximately maximizes \emph{profile likelihood (PML)}, i.e. the probability of observed frequency of frequencies, and returning the value of the property on this distribution is sample competitive with respect to a broad class of estimators of symmetric properties. Further, they showed that even computing an approximation of the PML suffices to achieve such a universal plug-in estimator. Unfortunately, prior to this work there was no known polynomial time algorithm to compute an approximate PML and it was open to obtain a polynomial time universal plug-in estimator through the use of approximate PML. In this paper we provide a algorithm (in number of samples) that, given $n$ samples from a distribution, computes an approximate PML distribution up to a multiplicative error of $\exp(n^{2/3} \mathrm{poly} \log(n))$ in time nearly linear in $n$. Generalizing work of Acharya et al. 2016 on the utility of approximate PML we show that our algorithm provides a nearly linear time universal plug-in estimator for all symmetric functions up to accuracy $\epsilon = \Omega(n^{-0.166})$. Further, we show how to extend our work to provide efficient polynomial-time algorithms for computing a $d$-dimensional generalization of PML (for constant $d$) that allows for universal plug-in estimation of symmetric relationships between distributions.

We analyze a new robust method for the reconstruction of probability distributions of observed data in the presence of output outliers. It is based on a so-called gradient conjugate prior (GCP) network which outputs the parameters of a prior. By rigorously studying the dynamics of the GCP learning process, we derive an explicit formula for correcting the obtained variance of the marginal distribution and removing the bias caused by outliers in the training set. Assuming a Gaussian (input-dependent) ground truth distribution contaminated with a proportion $\varepsilon$ of outliers, we show that the fitted mean is in a $c e^{-1/\varepsilon}$-neighborhood of the ground truth mean and the corrected variance is in a $b\varepsilon$-neighborhood of the ground truth variance, whereas the uncorrected variance of the marginal distribution can even be infinite. We explicitly find $b$ as a function of the output of the GCP network, without a priori knowledge of the outliers (possibly input-dependent) distribution. Experiments with synthetic and real-world data sets indicate that the GCP network fitted with a standard optimizer outperforms other robust methods for regression.

When concept drift is detected during classification in a data stream, a common remedy is to retrain a framework's classifier. However, this loses useful information if the classifier has learnt the current concept well, and this concept will recur again in the future. Some frameworks retain and reuse classifiers, but it can be time-consuming to select an appropriate classifier to reuse. These frameworks rarely match the accuracy of state-of-the-art ensemble approaches. For many data stream tasks, speed is important: fast, accurate frameworks are needed for time-dependent applications. We propose the Enhanced Concept Profiling Framework (ECPF), which aims to recognise recurring concepts and reuse a classifier trained previously, enabling accurate classification immediately following a drift. The novelty of ECPF is in how it uses similarity of classifications on new data, between a new classifier and existing classifiers, to quickly identify the best classifier to reuse. It always trains both a new classifier and a reused classifier, and retains the more accurate classifier when concept drift occurs. Finally, it creates a copy of reused classifiers, so a classifier well-suited for a recurring concept will not be impacted by being trained on a different concept. In our experiments, ECPF classifies significantly more accurately than a state-of-the-art classifier reuse framework (Diversity Pool) and a state-of-the-art ensemble technique (Adaptive Random Forest) on synthetic datasets with recurring concepts. It classifies real-world datasets five times faster than Diversity Pool, and six times faster than Adaptive Random Forest and is not significantly less accurate than either.

A new approach for blind channel equalization and decoding, using variational autoencoders (VAEs), is introduced. We first consider the reconstruction of uncoded data symbols transmitted over a noisy linear intersymbol interference (ISI) channel, with an unknown impulse response, without using pilot symbols. We derive an approximated maximum likelihood estimate to the channel parameters and reconstruct the transmitted data. We demonstrate significant and consistent improvements in the error rate of the reconstructed symbols, compared to existing blind equalization methods such as constant modulus, thus enabling faster channel acquisition. The VAE equalizer uses a fully convolutional neural network with a small number of free parameters. These results are extended to blind equalization over a noisy nonlinear ISI channel with unknown parameters. We then consider coded communication using low-density parity-check (LDPC) codes transmitted over a noisy linear or nonlinear ISI channel. The goal is to reconstruct the transmitted message from the channel observations corresponding to a transmitted codeword, without using pilot symbols. We demonstrate substantial improvements compared to expectation maximization (EM) using turbo equalization. Furthermore, in our simulations we demonstrate a relatively small gap between the performance of the new unsupervised equalization method and that of the fully channel informed (non-blind) turbo equalizer.

Bayesian networks are a central tool in machine learning and artificial intelligence, and make use of conditional independencies to impose structure on joint distributions. However, they are generally not as expressive as deep learning models and inference is hard and slow. In contrast, deep probabilistic models such as sum-product networks (SPNs) capture joint distributions in a tractable fashion, but use little interpretable structure. Here, we extend the notion of SPNs towards conditional distributions, which combine simple conditional models into high-dimensional ones. As shown in our experiments, the resulting conditional SPNs can be naturally used to impose structure on deep probabilistic models, allow for mixed data types, while maintaining fast and efficient inference.

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as an activation function not operating element-wise. We present the results of empirical experiments to back up the theoretical justification.