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Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.

Forecasts for key macroeconomic variables are almost always made simultaneously by the same organizations, presented together, and used together in policy analyses and decision-makings. It is therefore important to know whether the forecasters are skillful enough to forecast the future values of those variables. Here a method for joint evaluation of skill in directional forecasts of multiple variables is introduced. The method is simple to use and does not rely on complicated assumptions required by the conventional statistical methods for measuring accuracy of directional forecast. The data on GDP growth and inflation forecasts of three organizations from Thailand, namely, the Bank of Thailand, the Fiscal Policy Office, and the Office of the National Economic and Social Development Council as well as the actual data on GDP growth and inflation of Thailand between 2001 and 2021 are employed in order to demonstrate how the method could be used to evaluate the skills of forecasters in practice. The overall results indicate that these three organizations are somewhat skillful in forecasting the direction-of-changes of GDP growth and inflation when no band and a band of +/- 1 standard deviation of the forecasted outcome are considered. However, when a band of +/- 0.5% of the forecasted outcome is introduced, the skills in forecasting the direction-of-changes of GDP growth and inflation of these three organizations are, at best, little better than intelligent guess work.

This paper develops a flexible and computationally efficient multivariate volatility model, which allows for dynamic conditional correlations and volatility spillover effects among financial assets. The new model has desirable properties such as identifiability and computational tractability for many assets. A sufficient condition of the strict stationarity is derived for the new process. Two quasi-maximum likelihood estimation methods are proposed for the new model with and without low-rank constraints on the coefficient matrices respectively, and the asymptotic properties for both estimators are established. Moreover, a Bayesian information criterion with selection consistency is developed for order selection, and the testing for volatility spillover effects is carefully discussed. The finite sample performance of the proposed methods is evaluated in simulation studies for small and moderate dimensions. The usefulness of the new model and its inference tools is illustrated by two empirical examples for 5 stock markets and 17 industry portfolios, respectively.

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some $4$-dimensional manifolds without and with boundary.

The human cerebral cortex has many bumps and grooves called gyri and sulci. Even though there is a high inter-individual consistency for the main cortical folds, this is not the case when we examine the exact shapes and details of the folding patterns. Because of this complexity, characterizing the cortical folding variability and relating them to subjects' behavioral characteristics or pathologies is still an open scientific problem. Classical approaches include labeling a few specific patterns, either manually or semi-automatically, based on geometric distances, but the recent availability of MRI image datasets of tens of thousands of subjects makes modern deep-learning techniques particularly attractive. Here, we build a self-supervised deep-learning model to detect folding patterns in the cingulate region. We train a contrastive self-supervised model (SimCLR) on both Human Connectome Project (1101 subjects) and UKBioBank (21070 subjects) datasets with topological-based augmentations on the cortical skeletons, which are topological objects that capture the shape of the folds. We explore several backbone architectures (convolutional network, DenseNet, and PointNet) for the SimCLR. For evaluation and testing, we perform a linear classification task on a database manually labeled for the presence of the "double-parallel" folding pattern in the cingulate region, which is related to schizophrenia characteristics. The best model, giving a test AUC of 0.76, is a convolutional network with 6 layers, a 10-dimensional latent space, a linear projection head, and using the branch-clipping augmentation. This is the first time that a self-supervised deep learning model has been applied to cortical skeletons on such a large dataset and quantitatively evaluated. We can now envisage the next step: applying it to other brain regions to detect other biomarkers.

Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.

SDRDPy is a desktop application that allows experts an intuitive graphic and tabular representation of the knowledge extracted by any supervised descriptive rule discovery algorithm. The application is able to provide an analysis of the data showing the relevant information of the data set and the relationship between the rules, data and the quality measures associated for each rule regardless of the tool where algorithm has been executed. All of the information is presented in a user-friendly application in order to facilitate expert analysis and also the exportation of reports in different formats.

The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.