Although linear and quadratic discriminant analysis are widely recognized classical methods, they can encounter significant challenges when dealing with non-Gaussian distributions or contaminated datasets. This is primarily due to their reliance on the Gaussian assumption, which lacks robustness. We first explain and review the classical methods to address this limitation and then present a novel approach that overcomes these issues. In this new approach, the model considered is an arbitrary Elliptically Symmetrical (ES) distribution per cluster with its own arbitrary scale parameter. This flexible model allows for potentially diverse and independent samples that may not follow identical distributions. By deriving a new decision rule, we demonstrate that maximum-likelihood parameter estimation and classification are simple, efficient, and robust compared to state-of-the-art methods.

Probabilistic principal component analysis (PPCA) is currently one of the most used statistical tools to reduce the ambient dimension of the data. From multidimensional scaling to the imputation of missing data, PPCA has a broad spectrum of applications ranging from science and engineering to quantitative finance. Despite this wide applicability in various fields, hardly any theoretical guarantees exist to justify the soundness of the maximum likelihood (ML) solution for this model. In fact, it is well known that the maximum likelihood estimation (MLE) can only recover the true model parameters up to a rotation. The main obstruction is posed by the inherent identifiability nature of the PPCA model resulting from the rotational symmetry of the parameterization. To resolve this ambiguity, we propose a novel approach using quotient topological spaces and in particular, we show that the maximum likelihood solution is consistent in an appropriate quotient Euclidean space. Furthermore, our consistency results encompass a more general class of estimators beyond the MLE. Strong consistency of the ML estimate and consequently strong covariance estimation of the PPCA model have also been established under a compactness assumption.

Causal intervention is an essential tool in causal inference. It is axiomatized under the rules of do-calculus in the case of structure causal models. We provide simple axiomatizations for families of probability distributions to be different types of interventional distributions. Our axiomatizations neatly lead to a simple and clear theory of causality that has several advantages: it does not need to make use of any modeling assumptions such as those imposed by structural causal models; it only relies on interventions on single variables; it includes most cases with latent variables and causal cycles; and more importantly, it does not assume the existence of an underlying true causal graph as we do not take it as the primitive object--in fact, a causal graph is derived as a by-product of our theory. We show that, under our axiomatizations, the intervened distributions are Markovian to the defined intervened causal graphs, and an observed joint probability distribution is Markovian to the obtained causal graph; these results are consistent with the case of structural causal models, and as a result, the existing theory of causal inference applies. We also show that a large class of natural structural causal models satisfy the theory presented here. We note that the aim of this paper is axiomatization of interventional families, which is subtly different from "causal modeling."

Moreover, models and develop generalization bounds for having generalization bounds not only contributes to the theoretical models based on the Wasserstein distance and understanding of GANs themselves, but also to the the total variation distance. Our first result on understanding of the structure of real-life datasets, if those the Wasserstein distance assumes the instance can be provably approximated by GAN-generated data. In space is bounded, while our second result takes addition, given that GANs are used for data-augmentation advantage of dimensionality reduction. Our results in fields such as medical image classification (see e.g. Frid-naturally apply to Wasserstein GANs and Adar et al., 2018), theoretical guarantees can substantiate Energy-Based GANs, and our bounds provide the soundness of such applications.

Segmentation has emerged as a fundamental field of computer vision and natural language processing, which assigns a label to every pixel/feature to extract regions of interest from an image/text. To evaluate the performance of segmentation, the Dice and IoU metrics are used to measure the degree of overlap between the ground truth and the predicted segmentation. In this paper, we establish a theoretical foundation of segmentation with respect to the Dice/IoU metrics, including the Bayes rule and Dice-/IoU-calibration, analogous to classification-calibration or Fisher consistency in classification. We prove that the existing thresholding-based framework with most operating losses are not consistent with respect to the Dice/IoU metrics, and thus may lead to a suboptimal solution. To address this pitfall, we propose a novel consistent ranking-based framework, namely RankDice/RankIoU, inspired by plug-in rules of the Bayes segmentation rule. Three numerical algorithms with GPU parallel execution are developed to implement the proposed framework in large-scale and high-dimensional segmentation. We study statistical properties of the proposed framework. We show it is Dice-/IoU-calibrated, and its excess risk bounds and the rate of convergence are also provided. The numerical effectiveness of RankDice/mRankDice is demonstrated in various simulated examples and Fine-annotated CityScapes, Pascal VOC and Kvasir-SEG datasets with state-of-the-art deep learning architectures.

Prediction algorithms, such as deep neural networks (DNNs), are used in many domain sciences to directly estimate internal parameters of interest in simulator-based models, especially in settings where the observations include images or complex high-dimensional data. In parallel, modern neural density estimators, such as normalizing flows, are becoming increasingly popular for uncertainty quantification, especially when both parameters and observations are high-dimensional. However, parameter inference is an inverse problem and not a prediction task; thus, an open challenge is to construct conditionally valid and precise confidence regions, with a guaranteed probability of covering the true parameters of the data-generating process, no matter what the (unknown) parameter values are, and without relying on large-sample theory. Many simulator-based inference (SBI) methods are indeed known to produce biased or overly confident parameter regions, yielding misleading uncertainty estimates. This paper presents WALDO, a novel method to construct confidence regions with finite-sample conditional validity by leveraging prediction algorithms or posterior estimators that are currently widely adopted in SBI. WALDO reframes the well-known Wald test statistic, and uses a computationally efficient regression-based machinery for classical Neyman inversion of hypothesis tests. We apply our method to a recent high-energy physics problem, where prediction with DNNs has previously led to estimates with prediction bias. We also illustrate how our approach can correct overly confident posterior regions computed with normalizing flows.

Recent digital advances have popularized predictive maintenance (PMx), offering enhanced efficiency, automation, accuracy, cost savings, and independence in maintenance. Yet, it continues to face numerous limitations such as poor explainability, sample inefficiency of data-driven methods, complexity of physics-based methods, and limited generalizability and scalability of knowledge-based methods. This paper proposes leveraging Digital Twins (DTs) to address these challenges and enable automated PMx adoption at larger scales. While we argue that DTs have this transformative potential, they have not yet reached the level of maturity needed to bridge these gaps in a standardized way. Without a standard definition for such evolution, this transformation lacks a solid foundation upon which to base its development. This paper provides a requirement-based roadmap supporting standardized PMx automation using DT technologies. A systematic approach comprising two primary stages is presented. First, we methodically identify the Informational Requirements (IRs) and Functional Requirements (FRs) for PMx, which serve as a foundation from which any unified framework must emerge. Our approach to defining and using IRs and FRs to form the backbone of any PMx DT is supported by the track record of IRs and FRs being successfully used as blueprints in other areas, such as for product development within the software industry. Second, we conduct a thorough literature review spanning fields to determine the ways in which these IRs and FRs are currently being used within DTs, enabling us to point to the specific areas where further research is warranted to support the progress and maturation of requirement-based PMx DTs.

Flow and bridge matching are a novel class of processes which encompass diffusion models. One of the main aspect of their increased flexibility is that these models can interpolate between arbitrary data distributions i.e. they generalize beyond generative modeling and can be applied to learning stochastic (and deterministic) processes of arbitrary transfer tasks between two given distributions. In this paper, we highlight that while flow and bridge matching processes preserve the information of the marginal distributions, they do \emph{not} necessarily preserve the coupling information unless additional, stronger optimality conditions are met. This can be problematic if one aims at preserving the original empirical pairing. We show that a simple modification of the matching process recovers this coupling by augmenting the velocity field (or drift) with the information of the initial sample point. Doing so, we lose the Markovian property of the process but preserve the coupling information between distributions. We illustrate the efficiency of our augmentation in learning mixture of image translation tasks.

There is a considerable body of work on data cleaning which employs various principles to rectify erroneous data and transform a dirty dataset into a cleaner one. One of prevalent approaches is probabilistic methods, including Bayesian methods. However, existing probabilistic methods often assume a simplistic distribution (e.g., Gaussian distribution), which is frequently underfitted in practice, or they necessitate experts to provide a complex prior distribution (e.g., via a programming language). This requirement is both labor-intensive and costly, rendering these methods less suitable for real-world applications. In this paper, we propose BClean, a Bayesian Cleaning system that features automatic Bayesian network construction and user interaction. We recast the data cleaning problem as a Bayesian inference that fully exploits the relationships between attributes in the observed dataset and any prior information provided by users. To this end, we present an automatic Bayesian network construction method that extends a structure learning-based functional dependency discovery method with similarity functions to capture the relationships between attributes. Furthermore, our system allows users to modify the generated Bayesian network in order to specify prior information or correct inaccuracies identified by the automatic generation process. We also design an effective scoring model (called the compensative scoring model) necessary for the Bayesian inference. To enhance the efficiency of data cleaning, we propose several approximation strategies for the Bayesian inference, including graph partitioning, domain pruning, and pre-detection. By evaluating on both real-world and synthetic datasets, we demonstrate that BClean is capable of achieving an F-measure of up to 0.9 in data cleaning, outperforming existing Bayesian methods by 2% and other data cleaning methods by 15%.

To facilitate reliable deployments of autonomous robots in the real world, Out-of-Distribution (OOD) detection capabilities are often required. A powerful approach for OOD detection is based on density estimation with Normalizing Flows (NFs). However, we find that prior work with NFs attempts to match the complex target distribution topologically with naive base distributions leading to adverse implications. In this work, we circumvent this topological mismatch using an expressive class-conditional base distribution trained with an information-theoretic objective to match the required topology. The proposed method enjoys the merits of wide compatibility with existing learned models without any performance degradation and minimum computation overhead while enhancing OOD detection capabilities. We demonstrate superior results in density estimation and 2D object detection benchmarks in comparison with extensive baselines. Moreover, we showcase the applicability of the method with a real-robot deployment.