Principal component analysis (PCA) is one of the most widely used dimension reduction and multivariate statistical techniques. From a probabilistic perspective, PCA seeks a low-dimensional representation of data in the presence of independent identical Gaussian noise. Probabilistic PCA (PPCA) and its variants have been extensively studied for decades. Most of them assume the underlying noise follows a certain independent identical distribution. However, the noise in the real world is usually complicated and structured. To address this challenge, some non-linear variants of PPCA have been proposed. But those methods are generally difficult to interpret. To this end, we propose a powerful and intuitive PCA method (MN-PCA) through modeling the graphical noise by the matrix normal distribution, which enables us to explore the structure of noise in both the feature space and the sample space. MN-PCA obtains a low-rank representation of data and the structure of noise simultaneously. And it can be explained as approximating data over the generalized Mahalanobis distance. We develop two algorithms to solve this model: one maximizes the regularized likelihood, the other exploits the Wasserstein distance, which is more robust. Extensive experiments on various data demonstrate their effectiveness.

Phase Modulation on the Hypersphere (PMH) is a power efficient modulation scheme for the load-modulated multiple-input multiple-output (MIMO) transmitters with central power amplifiers (CP A). However, it is difficult to obtain the precise channel state information (CSI), and the traditional optimal maximum likelihood (ML) detection scheme incurs high complexity which increases exponentially with the number of antennas and the number of bits carried per antenna in the PMH modulation. To detect the PMH signals without knowing the prior CSI, we first propose a signal detection scheme, termed as the hypersphere clustering scheme based on the expectation maximization (EM) algorithm with maximum likelihood detection (HEM-ML). By leveraging machine learning, the proposed detection scheme can accurately obtain information of the channel from a few of the received symbols with little resource cost and achieve comparable detection results as that of the optimal ML detector. To further reduce the computational complexity in the ML detection in HEM-ML, we also propose the second signal detection scheme, termed as the hypersphere clustering scheme based on the EM algorithm with KD-tree detection (HEM-KD). The CSI obtained from the EM algorithm is used to build a spatial KD-tree receiver codebook and the signal detection problem can be transformed into a nearest neighbor search (NNS) problem. The detection complexity of HEM-KD is significantly reduced without any detection performance loss as compared to HEM-ML. Extensive simulation results verify the effectiveness of our proposed detection schemes. I NTRODUCTION The fifth generation (5G) wireless communication network is forecasted to provide over 1000 times higher capacity than the current system. In addition to dramatically expanding the available bandwidth, multiple-input multiple-output (MIMO) technology is playing a key role in improving the spectral efficiency (SE) and enhancing the throughput in the future wireless cellular communication systems [1]. This ambitious goal will however cause an inevitable energy consumption problem, thus limiting the number of the antennas at the base station (BS) and the user terminals in practice [2]. In the traditional design of the MIMO transceivers, each antenna is connected with one distinct radio frequency (RF) chain which includes a power amplifier (P A). This kind of structure enables the power consumption of the transmission to grow linearly with the number of the antennas. In addition, the use of Orthogonal Frequency Division Multiplexing (OFDM) signals in massive MIMO systems leads to a high peak-to-average power ratios (P APR) and exacerbates the costs of P As, thus reducing the power efficiency. On the other hand, to alleviate the effects of mutual coupling and correlated fading, the antennas should be set at least half of a wavelength apart from each other, which will inevitably cause the size problem [3].

In reinforcement learning (RL), agents often operate in partially observed and uncertain environments. Model-based RL suggests that this is best achieved by learning and exploiting a probabilistic model of the world. 'Active inference' is an emerging normative framework in cognitive and computational neuroscience that offers a unifying account of how biological agents achieve this. On this framework, inference, learning and action emerge from a single imperative to maximize the Bayesian evidence for a niched model of the world. However, implementations of this process have thus far been restricted to low-dimensional and idealized situations. Here, we present a working implementation of active inference that applies to high-dimensional tasks, with proof-of-principle results demonstrating efficient exploration and an order of magnitude increase in sample efficiency over strong model-free baselines. Our results demonstrate the feasibility of applying active inference at scale and highlight the operational homologies between active inference and current model-based approaches to RL.

In many real-world applications of machine learning, data are distributed across many clients and cannot leave the devices they are stored on. Furthermore, each client's data, computational resources and communication constraints may be very different. This setting is known as federated learning, in which privacy is a key concern. Differential privacy is commonly used to provide mathematical privacy guarantees. This work, to the best of our knowledge, is the first to consider federated, differentially private, Bayesian learning. We build on Partitioned Variational Inference (PVI) which was recently developed to support approximate Bayesian inference in the federated setting. We modify the client-side optimisation of PVI to provide an (${\epsilon}$, ${\delta}$)-DP guarantee. We show that it is possible to learn moderately private logistic regression models in the federated setting that achieve similar performance to models trained non-privately on centralised data.

Differential privacy is a cryptographically-motivated approach to privacy that has become a very active field of research over the last decade in theoretical computer science and machine learning. In this paradigm one assumes there is a trusted curator who holds the data of individuals in a database and the goal of privacy is to simultaneously protect individual data while allowing the release of global characteristics of the database. In this setting we introduce a general framework for parametric inference with differential privacy guarantees. We first obtain differentially private estimators based on bounded influence M-estimators by leveraging their gross-error sensitivity in the calibration of a noise term added to them in order to ensure privacy. We then show how a similar construction can also be applied to construct differentially private test statistics analogous to the Wald, score and likelihood ratio tests. We provide statistical guarantees for all our proposals via an asymptotic analysis. An interesting consequence of our results is to further clarify the connection between differential privacy and robust statistics. In particular, we demonstrate that differential privacy is a weaker stability requirement than infinitesimal robustness, and show that robust M-estimators can be easily randomized in order to guarantee both differential privacy and robustness towards the presence of contaminated data. We illustrate our results both on simulated and real data.

Learning models with discrete latent variables using stochastic gradient descent remains a challenge due to the high variance of gradient estimates. Modern variance reduction techniques mostly consider categorical distributions and have limited applicability when the number of possible outcomes becomes large. In this work, we consider models with latent permutations and propose control variates for the Plackett-Luce distribution. In particular, the control variates allow us to optimize black-box functions over permutations using stochastic gradient descent. To illustrate the approach, we consider a variety of causal structure learning tasks for continuous and discrete data. We show that our method outperforms competitive relaxation-based optimization methods and is also applicable to non-differentiable score functions.

In the area of credit risk analytics, current Bankruptcy Prediction Models (BPMs) struggle with (a) the availability of comprehensive and real-world data sets and (b) the presence of extreme class imbalance in the data (i.e., very few samples for the minority class) that degrades the performance of the prediction model. Moreover, little research has compared the relative performance of well-known BPM's on public datasets addressing the class imbalance problem. In this work, we apply eight classes of well-known BPMs, as suggested by a review of decades of literature, on a new public dataset named Freddie Mac Single-Family Loan-Level Dataset with resampling (i.e., adding synthetic minority samples) of the minority class to tackle class imbalance. Additionally, we apply some recent AI techniques (e.g., tree-based ensemble techniques) that demonstrate potentially better results on models trained with resampled data. In addition, from the analysis of 19 years (1999-2017) of data, we discover that models behave differently when presented with sudden changes in the economy (e.g., a global financial crisis) resulting in abrupt fluctuations in the national default rate. In summary, this study should aid practitioners/researchers in determining the appropriate model with respect to data that contains a class imbalance and various economic stages.

Bayesian neural networks (BNNs) introduce uncertainty estimation to deep networks by performing Bayesian inference on network weights. However, such models bring the challenges of inference, and further BNNs with weight uncertainty rarely achieve superior performance to standard models. In this paper, we investigate a new line of Bayesian deep learning by performing Bayesian reasoning on the structure of deep neural networks. Drawing inspiration from the neural architecture search, we define the network structure as gating weights on the redundant operations between computational nodes, and apply stochastic variational inference techniques to learn the structure distributions of networks. Empirically, the proposed method substantially surpasses the advanced deep neural networks across a range of classification and segmentation tasks. More importantly, our approach also preserves benefits of Bayesian principles, producing improved uncertainty estimation than the strong baselines including MC dropout and variational BNNs algorithms (e.g. noisy EK-FAC).

Gibbs sampling is a Markov chain Monte Carlo method that is often used for learning and inference on graphical models. Minibatching, in which a small random subset of the graph is used at each iteration, can help make Gibbs sampling scale to large graphical models by reducing its computational cost. In this paper, we propose a new auxiliary-variable minibatched Gibbs sampling method, {\it Poisson-minibatching Gibbs}, which both produces unbiased samples and has a theoretical guarantee on its convergence rate. In comparison to previous minibatched Gibbs algorithms, Poisson-minibatching Gibbs supports fast sampling from continuous state spaces and avoids the need for a Metropolis-Hastings correction on discrete state spaces. We demonstrate the effectiveness of our method on multiple applications and in comparison with both plain Gibbs and previous minibatched methods.

Markov chain Monte Carlo (MCMC) algorithm, a generic sampling method, is ubiquitous in modern statistics, especially in Bayesian fields. MCMC algorithms require only the evaluation of the target pointwise, up to a multiple constant, in order to sample from it. In Bayesian analysis, the object of main interest is the posterior, which is not in closed form in general, and MCMC has become a standard tool in this domain. However, MCMC is difficult to scale and its applications are limited when the observation size is very large, for it needs to sweep over the entire observations set in order to evaluate the likelihood function at each iteration. Recently, many methods have been proposed to better scale MCMC algorithms for big data sets and these can be roughly classified into two groups Bardenet et al. (2017): divide-and-conquer methods and subsampling-based methods. For divide-and-conquer methods, one splits the whole data set into subsets, runs MCMC over each subset to generate samples of parameters and combine these to produce an approximation of the true posterior. Depending on how MCMC is handled over the subsets, these methods can be further classified into two sub-categories.