Local clustering aims to identify specific substructures within a large graph without requiring full knowledge of the entire graph. These substructures are typically small compared to the overall graph, enabling the problem to be approached by finding a sparse solution to a linear system associated with the graph Laplacian. In this work, we first propose a method for identifying specific local clusters when very few labeled data is given, which we term semi-supervised local clustering. We then extend this approach to the unsupervised setting when no prior information on labels is available. The proposed methods involve randomly sampling the graph, applying diffusion through local cluster extraction, then examining the overlap among the results to find each cluster. We establish the co-membership conditions for any pair of nodes and rigorously prove the correctness of our methods. Additionally, we conduct extensive experiments to demonstrate that the proposed methods achieve state-of-the-arts results in the low-label rates regime.

Integrating various data modalities brings valuable insights into underlying phenomena. Multimodal factor analysis (FA) uncovers shared axes of variation underlying different simple data modalities, where each sample is represented by a vector of features. However, FA is not suited for structured data modalities, such as text or single cell sequencing data, where multiple data points are measured per each sample and exhibit a clustering structure. To overcome this challenge, we introduce FACTM, a novel, multi-view and multi-structure Bayesian model that combines FA with correlated topic modeling and is optimized using variational inference. Additionally, we introduce a method for rotating latent factors to enhance interpretability with respect to binary features. On text and video benchmarks as well as real-world music and COVID-19 datasets, we demonstrate that FACTM outperforms other methods in identifying clusters in structured data, and integrating them with simple modalities via the inference of shared, interpretable factors.

We propose a novel method for density estimation that leverages an estimated score function to debias kernel density estimation (SD-KDE). In our approach, each data point is adjusted by taking a single step along the score function with a specific choice of step size, followed by standard KDE with a modified bandwidth. The step size and modified bandwidth are chosen to remove the leading order bias in the KDE. Our experiments on synthetic tasks in 1D, 2D and on MNIST, demonstrate that our proposed SD-KDE method significantly reduces the mean integrated squared error compared to the standard Silverman KDE, even with noisy estimates in the score function. These results underscore the potential of integrating score-based corrections into nonparametric density estimation.

We present a framework for adaptive-stepsize MCMC sampling based on time-rescaled Langevin dynamics, in which the stepsize variation is dynamically driven by an additional degree of freedom. Our approach augments the phase space by an additional variable which in turn defines a time reparameterization. The use of an auxiliary relaxation equation allows accumulation of a moving average of a local monitor function and provides for precise control of the timestep while circumventing the need to modify the drift term in the physical system. Our algorithm is straightforward to implement and can be readily combined with any off-the-peg fixed-stepsize Langevin integrator. As a particular example, we consider control of the stepsize by monitoring the norm of the log-posterior gradient, which takes inspiration from the Adam optimizer, the stepsize being automatically reduced in regions of steep change of the log posterior and increased on plateaus, improving numerical stability and convergence speed. As in Adam, the stepsize variation depends on the recent history of the gradient norm, which enhances stability and improves accuracy compared to more immediate control approaches. We demonstrate the potential benefit of this method--both in accuracy and in stability--in numerical experiments including Neal's funnel and a Bayesian neural network for classification of MNIST data.

We propose a parametric integral probability metric (IPM) to measure the discrepancy between two probability measures. The proposed IPM leverages a specific parametric family of discriminators, such as single-node neural networks with ReLU activation, to effectively distinguish between distributions, making it applicable in high-dimensional settings. By optimizing over the parameters of the chosen discriminator class, the proposed IPM demonstrates that its estimators have good convergence rates and can serve as a surrogate for other IPMs that use smooth nonparametric discriminator classes. We present an efficient algorithm for practical computation, offering a simple implementation and requiring fewer hyperparameters. Furthermore, we explore its applications in various tasks, such as covariate balancing for causal inference and fair representation learning. Across such diverse applications, we demonstrate that the proposed IPM provides strong theoretical guarantees, and empirical experiments show that it achieves comparable or even superior performance to other methods.

Markedly increased computational power and data acquisition have led to growing interest in data-driven inverse dynamics problems. These seek to answer a fundamental question: What can we learn from time series measurements of a complex dynamical system? For small systems interacting with external environments, the effective dynamics are inherently stochastic, making it crucial to properly manage noise in data. Here, we explore this for systems obeying Langevin dynamics and, using currents, we construct a learning framework for stochastic modeling. Currents have recently gained increased attention for their role in bounding entropy production (EP) from thermodynamic uncertainty relations (TURs). We introduce a fundamental relationship between the cumulant currents there and standard machine-learning loss functions. Using this, we derive loss functions for several key thermodynamic functions directly from the system dynamics without the (common) intermediate step of deriving a TUR. These loss functions reproduce results derived both from TURs and other methods. More significantly, they open a path to discover new loss functions for previously inaccessible quantities. Notably, this includes access to per-trajectory entropy production, even if the observed system is driven far from its steady-state. We also consider higher order estimation. Our method is straightforward and unifies dynamic inference with recent approaches to entropy production estimation. Taken altogether, this reveals a deep connection between diffusion models in machine learning and entropy production estimation in stochastic thermodynamics.

Accurate kinetic analysis of [$^{18}$F]FDG distribution in dynamic positron emission tomography (PET) requires anatomically constrained modelling of image-derived input functions (IDIFs). Traditionally, IDIFs are obtained from the aorta, neglecting anatomical variations and complex vascular contributions. This study proposes a multi-organ segmentation-based approach that integrates IDIFs from the aorta, portal vein, pulmonary artery, and ureters. Using high-resolution CT segmentations of the liver, lungs, kidneys, and bladder, we incorporate organ-specific blood supply sources to improve kinetic modelling. Our method was evaluated on dynamic [$^{18}$F]FDG PET data from nine patients, resulting in a mean squared error (MSE) reduction of $13.39\%$ for the liver and $10.42\%$ for the lungs. These initial results highlight the potential of multiple IDIFs in improving anatomical modelling and fully leveraging dynamic PET imaging. This approach could facilitate the integration of tracer kinetic modelling into clinical routine.

We develop a new approximative estimation method for conditional Shapley values obtained using a linear regression model. We develop a new estimation method and outperform existing methodology and implementations. Compared to the sequential method in the shapr-package (i.e fit one and one model), our method runs in minutes and not in hours. Compared to the iterative method in the shapr-package, we obtain better estimates in less than or almost the same amount of time. When the number of covariates becomes too large, one can still fit thousands of regression models at once using our method. We focus on a linear regression model, but one can easily extend the method to accommodate several types of splines that can be estimated using multivariate linear regression due to linearity in the parameters.

Functional survival models are key tools for analyzing time-to-event data with complex predictors, such as functional or high-dimensional inputs. Despite their predictive strength, these models often lack interpretability, which limits their value in practical decision-making and risk analysis. This study investigates two key survival models: the Functional Survival Tree (FST) and the Functional Random Survival Forest (FRSF). It introduces novel methods and tools to enhance the interpretability of FST models and improve the explainability of FRSF ensembles. Using both real and simulated datasets, the results demonstrate that the proposed approaches yield efficient, easy-to-understand decision trees that accurately capture the underlying decision-making processes of the model ensemble.

This paper investigates regularized stochastic gradient descent (SGD) algorithms for estimating nonlinear operators from a Polish space to a separable Hilbert space. We assume that the regression operator lies in a vector-valued reproducing kernel Hilbert space induced by an operator-valued kernel. Two significant settings are considered: an online setting with polynomially decaying step sizes and regularization parameters, and a finite-horizon setting with constant step sizes and regularization parameters. We introduce regularity conditions on the structure and smoothness of the target operator and the input random variables. Under these conditions, we provide a dimension-free convergence analysis for the prediction and estimation errors, deriving both expectation and high-probability error bounds. Our analysis demonstrates that these convergence rates are nearly optimal. Furthermore, we present a new technique for deriving bounds with high probability for general SGD schemes, which also ensures almost-sure convergence. Finally, we discuss potential extensions to more general operator-valued kernels and the encoder-decoder framework.