This perspective article is meant to be a self-contained introduction to and review of transfer operators such as the Koopman operator and the Perron-Frobenius operator as well as an overview of different applications. We will first introduce the required foundations and then show how transfer operators can not only be used to analyze highly nonlinear dynamical systems but also complex networks. In particular, we will focus on relationships between transfer operators for continuous-time stochastic processes defined on a continuous state space--whether they be reversible, non-reversible but time-homogeneous, or time-inhomogeneous--and their discrete counterparts associated with random walks on undirected, directed, and time-evolving graphs. Transfer operators play an important role in an increasing number of research fields. A few exemplary applications are illustrated in Figure 1.

A fundamental problem in supervised learning is to find a good set of features or distance measures. If the new set of features is of lower dimensionality and can be obtained by a simple transformation of the original data, they can make the model understandable, reduce overfitting, and even help to detect distribution drift. We propose a supervised dimensionality reduction method Gradient Boosting Mapping (GBMAP), where the outputs of weak learners -- defined as one-layer perceptrons -- define the embedding. We show that the embedding coordinates provide better features for the supervised learning task, making simple linear models competitive with the state-of-the-art regressors and classifiers. We also use the embedding to find a principled distance measure between points. The features and distance measures automatically ignore directions irrelevant to the supervised learning task. We also show that we can reliably detect out-of-distribution data points with potentially large regression or classification errors. GBMAP is fast and works in seconds for dataset of million data points or hundreds of features. As a bonus, GBMAP provides a regression and classification performance comparable to the state-of-the-art supervised learning methods.

This study evaluates the performance of various supervised machine learning models in analyzing highly correlated neural signaling data from the Adolescent Brain Cognitive Development (ABCD) Study, with a focus on predicting obsessive-compulsive disorder scales. We simulated a dataset to mimic the correlation structures commonly found in imaging data and evaluated logistic regression, elastic networks, random forests, and XG-Boost on their ability to handle multicollinearity and accurately identify predictive features. Our study aims to guide the selection of appropriate machine learning methods for processing neuroimaging data, highlighting models that best capture underlying signals in high feature correlations and prioritize clinically relevant features associated with Obsessive-Compulsive Disorder (OCD).

Out-of-distribution (OOD) detection is critical when deploying machine learning models in the real world. Outlier exposure methods, which incorporate auxiliary outlier data in the training process, can drastically improve OOD detection performance compared to approaches without advanced training strategies. We introduce Hopfield Boosting, a boosting approach, which leverages modern Hopfield energy (MHE) to sharpen the decision boundary between the in-distribution and OOD data. Hopfield Boosting encourages the model to concentrate on hard-to-distinguish auxiliary outlier examples that lie close to the decision boundary between in-distribution and auxiliary outlier data. Our method achieves a new state-of-the-art in OOD detection with outlier exposure, improving the FPR95 metric from 2.28 to 0.92 on CIFAR-10 and from 11.76 to 7.94 on CIFAR-100.

The increasing complexity of Artificial Intelligence models poses challenges to interpretability, particularly in the healthcare sector. This study investigates the impact of deep learning model complexity and Explainable AI (XAI) efficacy, utilizing four ResNet architectures (ResNet-18, 34, 50, 101). Through methodical experimentation on 4,369 lung X-ray images of COVID-19-infected and healthy patients, the research evaluates models' classification performance and the relevance of corresponding XAI explanations with respect to the ground-truth disease masks. Results indicate that the increase in model complexity is associated with a decrease in classification accuracy and AUC-ROC scores (ResNet-18: 98.4%, 0.997; ResNet-101: 95.9%, 0.988). Notably, in eleven out of twelve statistical tests performed, no statistically significant differences occurred between XAI quantitative metrics - Relevance Rank Accuracy and the proposed Positive Attribution Ratio - across trained models. These results suggest that increased model complexity does not consistently lead to higher performance or relevance of explanations for models' decision-making processes.

Sum-of-norms clustering is a popular convexification of K-means clustering. We show that, if the dataset is made of a large number of independent random variables distributed according to the uniform measure on the union of two disjoint balls of unit radius, and if the balls are sufficiently close to one another, then sum-of-norms clustering will typically fail to recover the decomposition of the dataset into two clusters. As the dimension tends to infinity, this happens even when the distance between the centers of the two balls is taken to be as large as 2 2. In order to show this, we introduce and analyze a continuous version of sum-of-norms clustering, where the dataset is replaced by a general measure. In particular, we state and prove a local-global characterization of the clustering that seems to be new even in the case of discrete datapoints.

With the advancement of neural networks, diverse methods for neural Granger causality have emerged, which demonstrate proficiency in handling complex data, and nonlinear relationships. However, the existing framework of neural Granger causality has several limitations. It requires the construction of separate predictive models for each target variable, and the relationship depends on the sparsity on the weights of the first layer, resulting in challenges in effectively modeling complex relationships between variables as well as unsatisfied estimation accuracy of Granger causality. Moreover, most of them cannot grasp full-time Granger causality. To address these drawbacks, we propose a Jacobian Regularizer-based Neural Granger Causality (JRNGC) approach, a straightforward yet highly effective method for learning multivariate summary Granger causality and full-time Granger causality by constructing a single model for all target variables. Specifically, our method eliminates the sparsity constraints of weights by leveraging an input-output Jacobian matrix regularizer, which can be subsequently represented as the weighted causal matrix in the post-hoc analysis. Extensive experiments show that our proposed approach achieves competitive performance with the state-of-the-art methods for learning summary Granger causality and full-time Granger causality while maintaining lower model complexity and high scalability.

With the rise of Large Language Models (LLMs), the novel metric "Brainscore" emerged as a means to evaluate the functional similarity between LLMs and human brain/neural systems. Our efforts were dedicated to mining the meaning of the novel score by constructing topological features derived from both human fMRI data involving 190 subjects, and 39 LLMs plus their untrained counterparts. Subsequently, we trained 36 Linear Regression Models and conducted thorough statistical analyses to discern reliable and valid features from our constructed ones. Our findings reveal distinctive feature combinations conducive to interpreting existing brainscores across various brain regions of interest (ROIs) and hemispheres, thereby significantly contributing to advancing interpretable machine learning (iML) studies. The study is enriched by our further discussions and analyses concerning existing brainscores. To our knowledge, this study represents the first attempt to comprehend the novel metric brainscore within this interdisciplinary domain.

Decentralized learning (DL) faces increased vulnerability to privacy breaches due to sophisticated attacks on machine learning (ML) models. Secure aggregation is a computationally efficient cryptographic technique that enables multiple parties to compute an aggregate of their private data while keeping their individual inputs concealed from each other and from any central aggregator. To enhance communication efficiency in DL, sparsification techniques are used, selectively sharing only the most crucial parameters or gradients in a model, thereby maintaining efficiency without notably compromising accuracy. However, applying secure aggregation to sparsified models in DL is challenging due to the transmission of disjoint parameter sets by distinct nodes, which can prevent masks from canceling out effectively. This paper introduces CESAR, a novel secure aggregation protocol for DL designed to be compatible with existing sparsification mechanisms. CESAR provably defends against honest-but-curious adversaries and can be formally adapted to counteract collusion between them. We provide a foundational understanding of the interaction between the sparsification carried out by the nodes and the proportion of the parameters shared under CESAR in both colluding and non-colluding environments, offering analytical insight into the working and applicability of the protocol. Experiments on a network with 48 nodes in a 3-regular topology show that with random subsampling, CESAR is always within 0.5% accuracy of decentralized parallel stochastic gradient descent (D-PSGD), while adding only 11% of data overhead. Moreover, it surpasses the accuracy on TopK by up to 0.3% on independent and identically distributed (IID) data.

Data-driven, machine learning (ML) models of atomistic interactions are often based on flexible and non-physical functions that can relate nuanced aspects of atomic arrangements into predictions of energies and forces. As a result, these potentials are as good as the training data (usually results of so-called ab initio simulations) and we need to make sure that we have enough information for a model to become sufficiently accurate, reliable and transferable. The main challenge stems from the fact that descriptors of chemical environments are often sparse high-dimensional objects without a well-defined continuous metric. Therefore, it is rather unlikely that any ad hoc method of choosing training examples will be indiscriminate, and it will be easy to fall into the trap of confirmation bias, where the same narrow and biased sampling is used to generate train- and test- sets. We will demonstrate that classical concepts of statistical planning of experiments and optimal design can help to mitigate such problems at a relatively low computational cost. The key feature of the method we will investigate is that they allow us to assess the informativeness of data (how much we can improve the model by adding/swapping a training example) and verify if the training is feasible with the current set before obtaining any reference energies and forces -- a so-called off-line approach. In other words, we are focusing on an approach that is easy to implement and doesn't require sophisticated frameworks that involve automated access to high-performance computational (HPC).