In this tutorial, we present a compact and holistic discussion of Deep Learning with a focus on Convolutional Neural Networks (CNNs) and supervised regression. While there are numerous books and articles on the individual topics we cover, comprehensive and detailed tutorials that address Deep Learning from a foundational yet rigorous and accessible perspective are rare. Most resources on CNNs are either too advanced, focusing on cutting-edge architectures, or too narrow, addressing only specific applications like image classification.This tutorial not only summarizes the most relevant concepts but also provides an in-depth exploration of each, offering a complete yet agile set of ideas. Moreover, we highlight the powerful synergy between learning theory, statistic, and machine learning, which together underpin the Deep Learning and CNN frameworks. We aim for this tutorial to serve as an optimal resource for students, professors, and anyone interested in understanding the foundations of Deep Learning. Upon acceptance we will provide an accompanying repository under \href{https://github.com/neoglez/deep-learning-tutorial}{https://github.com/neoglez/deep-learning-tutorial} Keywords: Tutorial, Deep Learning, Convolutional Neural Networks, Machine Learning.

We study supervised classification for datasets with a very large number of input variables. The na\"ive Bayes classifier is attractive for its simplicity, scalability and effectiveness in many real data applications. When the strong na\"ive Bayes assumption of conditional independence of the input variables given the target variable is not valid, variable selection and model averaging are two common ways to improve the performance. In the case of the na\"ive Bayes classifier, the resulting weighting scheme on the models reduces to a weighting scheme on the variables. Here we focus on direct estimation of variable weights in such a weighted na\"ive Bayes classifier. We propose a sparse regularization of the model log-likelihood, which takes into account prior penalization costs related to each input variable. Compared to averaging based classifiers used up until now, our main goal is to obtain parsimonious robust models with less variables and equivalent performance. The direct estimation of the variable weights amounts to a non-convex optimization problem for which we propose and compare several two-stage algorithms. First, the criterion obtained by convex relaxation is minimized using several variants of standard gradient methods. Then, the initial non-convex optimization problem is solved using local optimization methods initialized with the result of the first stage. The various proposed algorithms result in optimization-based weighted na\"ive Bayes classifiers, that are evaluated on benchmark datasets and positioned w.r.t. to a reference averaging-based classifier.

Image-goal navigation enables a robot to reach the location where a target image was captured, using visual cues for guidance. However, current methods either rely heavily on data and computationally expensive learning-based approaches or lack efficiency in complex environments due to insufficient exploration strategies. To address these limitations, we propose Bayesian Embodied Image-goal Navigation Using Gaussian Splatting, a novel method that formulates ImageNav as an optimal control problem within a model predictive control framework. BEINGS leverages 3D Gaussian Splatting as a scene prior to predict future observations, enabling efficient, real-time navigation decisions grounded in the robot's sensory experiences. By integrating Bayesian updates, our method dynamically refines the robot's strategy without requiring extensive prior experience or data. Our algorithm is validated through extensive simulations and physical experiments, showcasing its potential for embodied robot systems in visually complex scenarios.

Anxiety is endemic to every person, with an occurrence rate of approximately 20% [World Health Organization, 2017]. Between 2020 and 2022, over one in six people (17.2% or 3.4 million people) aged 16 to 85 years experienced an anxiety disorder [Australian Bureau of Statistics]. Anxiety is caused by changes in the situation, nervousness and common symptoms, including sweating, trembling and excessive worrying, which affect a person's daily life. Anxiety disorders encompass a range of conditions, such as generalised anxiety disorder (GAD), panic disorder (PD), social anxiety disorder (SAD), obsessive-compulsive disorder (OCD), various phobia-related disorders, physical pain related protective behaviour [Li et al., 2020, 2021] and depression [Ghosh and Anwar, 2021]. Current clinical approaches for diagnosing these disorders often suffer from limitations in accuracy and objectivity, relying heavily on self-reports, patient histories and clinical observations. These methods can be subjective and may not capture the nuanced neural and behavioural patterns associated with anxiety, leading to potential misdiagnoses. Recent research has shown promising results in using machine learning techniques to detect anxiety through physiological analysis [Abd-Alrazaq et al., 2023], such as respiration, electrocardiogram (ECG), photoplethysmography (PPG), electrodermal response (EDA) and electroencephalography (EEG), to identify patterns associated with anxiety states [Abd-Alrazaq et al., 2023].

This paper presents P2U-SLAM, a visual Simultaneous Localization And Mapping (SLAM) system with a wide Field of View (FoV) camera, which utilizes pose uncertainty and point uncertainty. While the wide FoV enables considerable repetitive observations of historical map points for matching cross-view features, the data properties of the historical map points and the poses of historical keyframes have changed during the optimization process. The neglect of data property changes triggers the absence of a partial information matrix in optimization and leads to the risk of long-term positioning performance degradation. The purpose of our research is to reduce the risk of the wide field of view visual input to the SLAM system. Based on the conditional probability model, this work reveals the definite impact of the above data properties changes on the optimization process, concretizes it as point uncertainty and pose uncertainty, and gives a specific mathematical form. P2U-SLAM respectively embeds point uncertainty and pose uncertainty into the tracking module and local mapping, and updates these uncertainties after each optimization operation including local mapping, map merging, and loop closing. We present an exhaustive evaluation in 27 sequences from two popular public datasets with wide-FoV visual input. P2U-SLAM shows excellent performance compared with other state-of-the-art methods. The source code will be made publicly available at https://github.com/BambValley/P2U-SLAM.

We are motivated primarily by the adaptation of text-to-speech synthesis models; however we argue that more generic parameter-efficient fine-tuning (PEFT) is an appropriate framework to do such adaptation. Nevertheless, catastrophic forgetting remains an issue with PEFT, damaging the pre-trained model's inherent capabilities. We demonstrate that existing Bayesian learning techniques can be applied to PEFT to prevent catastrophic forgetting as long as the parameter shift of the fine-tuned layers can be calculated differentiably. In a principled series of experiments on language modeling and speech synthesis tasks, we utilize established Laplace approximations, including diagonal and Kronecker-factored approaches, to regularize PEFT with the low-rank adaptation (LoRA) and compare their performance in pre-training knowledge preservation. Our results demonstrate that catastrophic forgetting can be overcome by our methods without degrading the fine-tuning performance, and using the Kronecker-factored approximation produces a better preservation of the pre-training knowledge than the diagonal ones.

Recent advancements in unsupervised domain adaptation (UDA) and semi-supervised learning (SSL), particularly incorporating causality, have led to significant methodological improvements in these learning problems. However, a formal theory that explains the role of causality in the generalization performance of UDA/SSL is still lacking. In this paper, we consider the UDA/SSL scenarios where we access $m$ labelled source data and $n$ unlabelled target data as training instances under different causal settings with a parametric probabilistic model. We study the learning performance (e.g., excess risk) of prediction in the target domain from an information-theoretic perspective. Specifically, we distinguish two scenarios: the learning problem is called causal learning if the feature is the cause and the label is the effect, and is called anti-causal learning otherwise. We show that in causal learning, the excess risk depends on the size of the source sample at a rate of $O(\frac{1}{m})$ only if the labelling distribution between the source and target domains remains unchanged. In anti-causal learning, we show that the unlabelled data dominate the performance at a rate of typically $O(\frac{1}{n})$. These results bring out the relationship between the data sample size and the hardness of the learning problem with different causal mechanisms.

Multivariate time series data suffer from the problem of missing values, which hinders the application of many analytical methods. To achieve the accurate imputation of these missing values, exploiting inter-correlation by employing the relationships between sequences (i.e., a network) is as important as the use of temporal dependency, since a sequence normally correlates with other sequences. Moreover, exploiting an adequate network depending on time is also necessary since the network varies over time. However, in real-world scenarios, we normally know neither the network structure nor when the network changes beforehand. Here, we propose a missing value imputation method for multivariate time series, namely MissNet, that is designed to exploit temporal dependency with a state-space model and inter-correlation by switching sparse networks. The network encodes conditional independence between features, which helps us understand the important relationships for imputation visually. Our algorithm, which scales linearly with reference to the length of the data, alternatively infers networks and fills in missing values using the networks while discovering the switching of the networks. Extensive experiments demonstrate that MissNet outperforms the state-of-the-art algorithms for multivariate time series imputation and provides interpretable results.

Causal inference has been a pivotal challenge across diverse domains such as medicine and economics, demanding a complicated integration of human knowledge, mathematical reasoning, and data mining capabilities. Recent advancements in natural language processing (NLP), particularly with the advent of large language models (LLMs), have introduced promising opportunities for traditional causal inference tasks. This paper reviews recent progress in applying LLMs to causal inference, encompassing various tasks spanning different levels of causation. We summarize the main causal problems and approaches, and present a comparison of their evaluation results in different causal scenarios. Furthermore, we discuss key findings and outline directions for future research, underscoring the potential implications of integrating LLMs in advancing causal inference methodologies.

The paper presents a novel approach for unsupervised techniques in the field of clustering. A new method is proposed to enhance existing literature models using the proper Bayesian bootstrap to improve results in terms of robustness and interpretability. Our approach is organized in two steps: k-means clustering is used for prior elicitation, then proper Bayesian bootstrap is applied as resampling method in an ensemble clustering approach. Results are analyzed introducing measures of uncertainty based on Shannon entropy. The proposal provides clear indication on the optimal number of clusters, as well as a better representation of the clustered data. Empirical results are provided on simulated data showing the methodological and empirical advances obtained.