Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and prove that the same form of common noise may force the convergence of the learning algorithm called `fictitious play', and this without any further potential or monotone structure. Several numerical examples are provided in order to support our theoretical analysis.

Time-dependent data often exhibit characteristics, such as non-stationarity and heavy-tailed errors, that would be inappropriate to model with the typical assumptions used in popular models. Thus, more flexible approaches are required to be able to accommodate such issues. To this end, we propose a Bayesian mixture of student-$t$ processes with an overall-local scale structure for the covariance. Moreover, we use a sequential Monte Carlo (SMC) sampler in order to perform online inference as data arrive in real-time. We demonstrate the superiority of our proposed approach compared to typical Gaussian process-based models on real-world data sets in order to prove the necessity of using mixtures of student-$t$ processes.

By now Bayesian methods are routinely used in practice for solving inverse problems. In inverse problems the parameter or signal of interest is observed only indirectly, as an image of a given map, and the observations are typically further corrupted with noise. Bayes offers a natural way to regularize these problems via the prior distribution and provides a probabilistic solution, quantifying the remaining uncertainty in the problem. However, the computational costs of standard, sampling based Bayesian approaches can be overly large in such complex models. Therefore, in practice variational Bayes is becoming increasingly popular. Nevertheless, the theoretical understanding of these methods is still relatively limited, especially in context of inverse problems. In our analysis we investigate variational Bayesian methods for Gaussian process priors to solve linear inverse problems. We consider both mildly and severely ill-posed inverse problems and work with the popular inducing variables variational Bayes approach proposed by Titsias in 2009. We derive posterior contraction rates for the variational posterior in general settings and show that the minimax estimation rate can be attained by correctly tunned procedures. As specific examples we consider a collection of inverse problems including the heat equation, Volterra operator and Radon transform and inducing variable methods based on population and empirical spectral features.

Recent language models have been improved by the addition of external memory. Nearest neighbor language models retrieve similar contexts to assist in word prediction. The addition of locality levels allows a model to learn how to weight neighbors based on their relative location to the current text in source documents, and have been shown to further improve model performance. Nearest neighbor models have been explored for controllable generation but have not examined the use of locality levels. We present a novel approach for this purpose and evaluate it using automatic and human evaluation on politeness, formality, supportiveness, and toxicity textual data. We find that our model is successfully able to control style and provides a better fluency-style trade-off than previous work.

This paper introduces the Gaussian multi-Graphical Model, a model to construct sparse graph representations of matrix- and tensor-variate data. We generalize prior work in this area by simultaneously learning this representation across several tensors that share axes, which is necessary to allow the analysis of multimodal datasets such as those encountered in multi-omics. Our algorithm uses only a single eigendecomposition per axis, achieving an order of magnitude speedup over prior work in the ungeneralized case. This allows the use of our methodology on large multi-modal datasets such as single-cell multi-omics data, which was challenging with previous approaches. We validate our model on synthetic data and five real-world datasets.

The fusion of causal models with deep learning introducing increasingly intricate data sets, such as the causal associations within images or between textual components, has surfaced as a focal research area. Nonetheless, the broadening of original causal concepts and theories to such complex, non-statistical data has been met with serious challenges. In response, our study proposes redefinitions of causal data into three distinct categories from the standpoint of causal structure and representation: definite data, semi-definite data, and indefinite data. Definite data chiefly pertains to statistical data used in conventional causal scenarios, while semi-definite data refers to a spectrum of data formats germane to deep learning, including time-series, images, text, and others. Indefinite data is an emergent research sphere inferred from the progression of data forms by us. To comprehensively present these three data paradigms, we elaborate on their formal definitions, differences manifested in datasets, resolution pathways, and development of research. We summarize key tasks and achievements pertaining to definite and semi-definite data from myriad research undertakings, present a roadmap for indefinite data, beginning with its current research conundrums. Lastly, we classify and scrutinize the key datasets presently utilized within these three paradigms.

Accurate estimation of treatment outcomes over time conditioning on the observed history is a fundamental problem in causal analysis and decision making in various applications (Mahar et al., 2021; Ye et al., 2023; Wang et al., 2023). For example, in medical domains, doctors are interested in knowing how a patient reacts to a treatment or multi-step treatments; in e-commerce, retailers are concerned about how future sales change if adjusting the price of an item. While randomized controlled trials (RCTs) are the gold standard for treatment outcome estimation, most often than not such trials are either too costly or even impractical to conduct. Therefore, utilizing available observed data (such as electronic health records (EHRs) and historical sales) for accurate treatment outcome estimation, has drawn increasing interest in the community. Compared to the well-studied i.i.d cases, treatment outcome estimation from time series observations not only finds more applications in the real world but also pose significant more challenges, due to the complex dynamics and the long-range dependencies in time series. Existing works along this endeavors explore various architectures with improved capacity and training strategies to alleviate time-dependent confounding. Recurrent marginal structural networks (RMSNs) (Lim, 2018), counterfactual recurrent networks (CRN) (Bica et al., 2020), and G-Net (Li et al., 2021) utilize architectures based on recurrent neural networks. To mitigate time-dependent confounding, they train proposed models with inverse probability of treatment weighting (IPTW), treatment invariant representation through gradient reversal, and G-computation respectively, in addition to the factual estimation loss on observed data. Causal Transformer (CT) (Melnychuk et al., 2022) further improves capturing long-range dependencies in the observational data

The COVID-19 pandemic underscored the importance of reliable, noninvasive diagnostic tools for robust public health interventions. In this work, we fused magnetic respiratory sensing technology (MRST) with machine learning (ML) to create a diagnostic platform for real-time tracking and diagnosis of COVID-19 and other respiratory diseases. The MRST precisely captures breathing patterns through three specific breath testing protocols: normal breath, holding breath, and deep breath. We collected breath data from both COVID-19 patients and healthy subjects in Vietnam using this platform, which then served to train and validate ML models. Our evaluation encompassed multiple ML algorithms, including support vector machines and deep learning models, assessing their ability to diagnose COVID-19. Our multi-model validation methodology ensures a thorough comparison and grants the adaptability to select the most optimal model, striking a balance between diagnostic precision with model interpretability. The findings highlight the exceptional potential of our diagnostic tool in pinpointing respiratory anomalies, achieving over 90% accuracy. This innovative sensor technology can be seamlessly integrated into healthcare settings for patient monitoring, marking a significant enhancement for the healthcare infrastructure.

Contemporary deep clustering approaches often rely on either contrastive or non-contrastive techniques to acquire effective representations for clustering tasks. Contrastive methods leverage negative pairs to achieve homogenous representations but can introduce class collision issues, potentially compromising clustering performance. On the contrary, non-contrastive techniques prevent class collisions but may produce non-uniform representations that lead to clustering collapse. In this work, we propose a novel end-to-end deep clustering approach named PIPCDR, designed to harness the strengths of both approaches while mitigating their limitations. PIPCDR incorporates a positive instance proximity loss and a cluster dispersion regularizer. The positive instance proximity loss ensures alignment between augmented views of instances and their sampled neighbors, enhancing within-cluster compactness by selecting genuinely positive pairs within the embedding space. Meanwhile, the cluster dispersion regularizer maximizes inter-cluster distances while minimizing within-cluster compactness, promoting uniformity in the learned representations. PIPCDR excels in producing well-separated clusters, generating uniform representations, avoiding class collision issues, and enhancing within-cluster compactness. We extensively validate the effectiveness of PIPCDR within an end-to-end Majorize-Minimization framework, demonstrating its competitive performance on moderate-scale clustering benchmark datasets and establishing new state-of-the-art results on large-scale datasets.