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Digital sensors can lead to noisy results under many circumstances. To be able to remove the undesired noise from images, proper noise modeling and an accurate noise parameter estimation is crucial. In this project, we use a Poisson-Gaussian noise model for the raw-images captured by the sensor, as it fits the physical characteristics of the sensor closely. Moreover, we limit ourselves to the case where observed (noisy), and ground-truth (noise-free) image pairs are available. Using such pairs is beneficial for the noise estimation and is not widely studied in literature. Based on this model, we derive the theoretical maximum likelihood solution, discuss its practical implementation and optimization. Further, we propose two algorithms based on variance and cumulant statistics. Finally, we compare the results of our methods with two different approaches, a CNN we trained ourselves, and another one taken from literature. The comparison between all these methods shows that our algorithms outperform the others in terms of MSE and have good additional properties.

Obtaining first-order regret bounds -- regret bounds scaling not as the worst-case but with some measure of the performance of the optimal policy on a given instance -- is a core question in sequential decision-making. While such bounds exist in many settings, they have proven elusive in reinforcement learning with large state spaces. In this work we address this gap, and show that it is possible to obtain regret scaling as $\widetilde{\mathcal{O}}(\sqrt{d^3 H^3 \cdot V_1^\star \cdot K} + d^{3.5}H^3\log K )$ in reinforcement learning with large state spaces, namely the linear MDP setting. Here $V_1^\star$ is the value of the optimal policy and $K$ is the number of episodes. We demonstrate that existing techniques based on least squares estimation are insufficient to obtain this result, and instead develop a novel robust self-normalized concentration bound based on the robust Catoni mean estimator, which may be of independent interest.

The simulation of human neurons and neurotransmission mechanisms has been realized in deep neural networks based on the theoretical implementations of activation functions. However, recent studies have reported that the threshold potential of neurons exhibits different values according to the locations and types of individual neurons, and that the activation functions have limitations in terms of representing this variability. Therefore, this study proposes a simple yet effective activation function that facilitates different thresholds and adaptive activations according to the positions of units and the contexts of inputs. Furthermore, the proposed activation function mathematically exhibits a more generalized form of Swish activation function, and thus we denoted it as Adaptive SwisH (ASH). ASH highlights informative features that exhibit large values in the top percentiles in an input, whereas it rectifies low values. Most importantly, ASH exhibits trainable, adaptive, and context-aware properties compared to other activation functions. Furthermore, ASH represents general formula of the previously studied activation function and provides a reasonable mathematical background for the superior performance. To validate the effectiveness and robustness of ASH, we implemented ASH into many deep learning models for various tasks, including classification, detection, segmentation, and image generation. Experimental analysis demonstrates that our activation function can provide the benefits of more accurate prediction and earlier convergence in many deep learning applications.

We consider inverse problems in Hilbert spaces under correlated Gaussian noise and use a Bayesian approach to find their regularised solution. We focus on mildly ill-posed inverse problems with the noise being generalised derivative of fractional Brownian motion, using a novel wavelet - based approach we call vaguelette-vaguelette. It allows us to apply sequence space methods without assuming that all operators are simultaneously diagonalisable. The results are proved for more general bases and covariance operators. Our primary aim is to study the posterior contraction rate in such inverse problems over Sobolev classes of true functions, comparing it to the derived minimax rate. Secondly, we study the effect of plugging in a consistent estimator of variances in sequence space on the posterior contraction rate, for instance where there are repeated observations. This result is also applied to the problem where the forward operator is observed with error. Thirdly, we show that an adaptive empirical Bayes posterior distribution contracts at the optimal rate, in the minimax sense, under a condition on prior smoothness, with a plugged in maximum marginal likelihood estimator of the prior scale. These theoretical results are illustrated on simulated data.

One of the most important features of financial time series data is volatility. There are often structural changes in volatility over time, and an accurate estimation of the volatility of financial time series requires careful identification of change-points. A common approach to modeling the volatility of time series data is the well-known GARCH model. Although the problem of change-point estimation of volatility dynamics derived from the GARCH model has been considered in the literature, these approaches rely on parametric assumptions of the conditional error distribution, which are often violated in financial time series. This may lead to inaccuracies in change-point detection resulting in unreliable GARCH volatility estimates. This paper introduces a novel change-point detection algorithm based on a semiparametric GARCH model. The proposed method retains the structural advantages of the GARCH process while incorporating the flexibility of nonparametric conditional error distribution. The approach utilizes a penalized likelihood derived from a semiparametric GARCH model and an efficient binary segmentation algorithm. The results show that in terms of change-point estimation and detection accuracy, the semiparametric method outperforms the commonly used Quasi-MLE (QMLE) and other variations of GARCH models in wide-ranging scenarios.

Variational Bayes methods are a scalable estimation approach for many complex state space models. However, existing methods exhibit a trade-off between accurate estimation and computational efficiency. This paper proposes a variational approximation that mitigates this trade-off. This approximation is based on importance densities that have been proposed in the context of efficient importance sampling. By directly conditioning on the observed data, the proposed method produces an accurate approximation to the exact posterior distribution. Because the steps required for its calibration are computationally efficient, the approach is faster than existing variational Bayes methods. The proposed method can be applied to any state space model that has a closed-form measurement density function and a state transition distribution that belongs to the exponential family of distributions. We illustrate the method in numerical experiments with stochastic volatility models and a macroeconomic empirical application using a high-dimensional state space model.

High-dimensional matrix-variate time series data are becoming widely available in many scientific fields, such as economics, biology, and meteorology. To achieve significant dimension reduction while preserving the intrinsic matrix structure and temporal dynamics in such data, Wang et al. (2017) proposed a matrix factor model that is shown to provide effective analysis. In this paper, we establish a general framework for incorporating domain or prior knowledge in the matrix factor model through linear constraints. The proposed framework is shown to be useful in achieving parsimonious parameterization, facilitating interpretation of the latent matrix factor, and identifying specific factors of interest. Fully utilizing the prior-knowledge-induced constraints results in more efficient and accurate modeling, inference, dimension reduction as well as a clear and better interpretation of the results. In this paper, constrained, multi-term, and partially constrained factor models for matrix-variate time series are developed, with efficient estimation procedures and their asymptotic properties. We show that the convergence rates of the constrained factor loading matrices are much faster than those of the conventional matrix factor analysis under many situations. Simulation studies are carried out to demonstrate the finite-sample performance of the proposed method and its associated asymptotic properties. We illustrate the proposed model with three applications, where the constrained matrix-factor models outperform their unconstrained counterparts in the power of variance explanation under the out-of-sample 10-fold cross-validation setting.

This paper considers the estimation and inference of the low-rank components in high-dimensional matrix-variate factor models, where each dimension of the matrix-variates ($p \times q$) is comparable to or greater than the number of observations ($T$). We propose an estimation method called $\alpha$-PCA that preserves the matrix structure and aggregates mean and contemporary covariance through a hyper-parameter $\alpha$. We develop an inferential theory, establishing consistency, the rate of convergence, and the limiting distributions, under general conditions that allow for correlations across time, rows, or columns of the noise. We show both theoretical and empirical methods of choosing the best $\alpha$, depending on the use-case criteria. Simulation results demonstrate the adequacy of the asymptotic results in approximating the finite sample properties. The $\alpha$-PCA compares favorably with the existing ones. Finally, we illustrate its applications with a real numeric data set and two real image data sets. In all applications, the proposed estimation procedure outperforms previous methods in the power of variance explanation using out-of-sample 10-fold cross-validation.

Spike-and-slab and horseshoe regression are arguably the most popular Bayesian variable selection approaches for linear regression models. However, their performance can deteriorate if outliers and heteroskedasticity are present in the data, which are common features in many real-world statistics and machine learning applications. In this work, we propose a Bayesian nonparametric approach to linear regression that performs variable selection while accounting for outliers and heteroskedasticity. Our proposed model is an instance of a Dirichlet process scale mixture model with the advantage that we can derive the full conditional distributions of all parameters in closed form, hence producing an efficient Gibbs sampler for posterior inference. Moreover, we present how to extend the model to account for heavy-tailed response variables. The performance of the model is tested against competing algorithms on synthetic and real-world datasets.

The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.