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Neural networks are high-dimensional nonlinear dynamical systems that process information through the coordinated activity of many interconnected units. Understanding how biological and machine-learning networks function and learn requires knowledge of the structure of this coordinated activity, information contained in cross-covariances between units. Although dynamical mean field theory (DMFT) has elucidated several features of random neural networks -- in particular, that they can generate chaotic activity -- existing DMFT approaches do not support the calculation of cross-covariances. We solve this longstanding problem by extending the DMFT approach via a two-site cavity method. This reveals, for the first time, several spatial and temporal features of activity coordination, including the effective dimension, defined as the participation ratio of the spectrum of the covariance matrix. Our results provide a general analytical framework for studying the structure of collective activity in random neural networks and, more broadly, in high-dimensional nonlinear dynamical systems with quenched disorder.

Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces to extract hidden and useful information or facilitate visual understanding and interpretation of the data. However, few of them take into consideration the potential cluster information contained implicitly in the high-dimensional data. In this paper, we propose LaptSNE, a new graph-layout nonlinear dimensionality reduction method based on t-SNE, one of the best techniques for visualizing high-dimensional data as 2D scatter plots. Specifically, LaptSNE leverages the eigenvalue information of the graph Laplacian to shrink the potential clusters in the low-dimensional embedding when learning to preserve the local and global structure from high-dimensional space to low-dimensional space. It is nontrivial to solve the proposed model because the eigenvalues of normalized symmetric Laplacian are functions of the decision variable. We provide a majorization-minimization algorithm with convergence guarantee to solve the optimization problem of LaptSNE and show how to calculate the gradient analytically, which may be of broad interest when considering optimization with Laplacian-composited objective. We evaluate our method by a formal comparison with state-of-the-art methods, both visually and via established quantitative measurements. The results demonstrate the superiority of our method over baselines such as t-SNE and UMAP. We also extend our method to spectral clustering and establish an accurate and parameter-free clustering algorithm, which provides us high reliability and convenience in real applications.

We propose a novel inference procedure for linear combinations of high-dimensional regression coefficients in generalized estimating equations, which have been widely used for correlated data analysis for decades. Our estimator, obtained via constructing a system of projected estimating equations, is shown to be asymptotically normally distributed under certain regularity conditions. We also introduce a data-driven cross-validation procedure to select the tuning parameter for estimating the projection direction, which is not addressed in the existing procedures. We demonstrate the robust finite-sample performance, especially in estimation bias and confidence interval coverage, of the proposed method via extensive simulations, and apply the method to gene expression data on riboflavin production with Bacillus subtilis.

This paper proposes a supervised dimension reduction methodology for tensor data which has two advantages over most image-based prognostic models. First, the model does not require tensor data to be complete which expands its application to incomplete data. Second, it utilizes time-to-failure (TTF) to supervise the extraction of low-dimensional features which makes the extracted features more effective for the subsequent prognostic. Besides, an optimization algorithm is proposed for parameter estimation and closed-form solutions are derived under certain distributions.

This paper presents a multi-layer motion planning and control architecture for autonomous racing, capable of avoiding static obstacles, performing active overtakes, and reaching velocities above 75 $m/s$. The used offline global trajectory generation and the online model predictive controller are highly based on optimization and dynamic models of the vehicle, where the tires and camber effects are represented in an extended version of the basic Pacejka Magic Formula. The proposed single-track model is identified and validated using multi-body motorsport libraries which allow simulating the vehicle dynamics properly, especially useful when real experimental data are missing. The fundamental regularization terms and constraints of the controller are tuned to reduce the rate of change of the inputs while assuring an acceptable velocity and path tracking. The motion planning strategy consists of a Fren\'et-Frame-based planner which considers a forecast of the opponent produced by a Kalman filter. The planner chooses the collision-free path and velocity profile to be tracked on a 3 seconds horizon to realize different goals such as following and overtaking. The proposed solution has been applied on a Dallara AV-21 racecar and tested at oval race tracks achieving lateral accelerations up to 25 $m/s^{2}$.

Causal effect estimation from observational data is a challenging problem, especially with high dimensional data and in the presence of unobserved variables. The available data-driven methods for tackling the problem either provide an estimation of the bounds of a causal effect (i.e. nonunique estimation) or have low efficiency. The major hurdle for achieving high efficiency while trying to obtain unique and unbiased causal effect estimation is how to find a proper adjustment set for confounding control in a fast way, given the huge covariate space and considering unobserved variables. In this paper, we approach the problem as a local search task for finding valid adjustment sets in data. We establish the theorems to support the local search for adjustment sets, and we show that unique and unbiased estimation can be achieved from observational data even when there exist unobserved variables. We then propose a data-driven algorithm that is fast and consistent under mild assumptions. We also make use of a frequent pattern mining method to further speed up the search of minimal adjustment sets for causal effect estimation. Experiments conducted on extensive synthetic and real-world datasets demonstrate that the proposed algorithm outperforms the state-of-the-art criteria/estimators in both accuracy and time-efficiency.

Data visualisation helps understanding data represented by multiple variables, also called features, stored in a large matrix where individuals are stored in lines and variable values in columns. These data structures are frequently called multidimensional spaces.In this paper, we illustrate ways of employing the visual results of multidimensional projection algorithms to understand and fine-tune the parameters of their mathematical framework. Some of the common mathematical common to these approaches are Laplacian matrices, Euclidian distance, Cosine distance, and statistical methods such as Kullback-Leibler divergence, employed to fit probability distributions and reduce dimensions. Two of the relevant algorithms in the data visualisation field are t-distributed stochastic neighbourhood embedding (t-SNE) and Least-Square Projection (LSP). These algorithms can be used to understand several ranges of mathematical functions including their impact on datasets. In this article, mathematical parameters of underlying techniques such as Principal Component Analysis (PCA) behind t-SNE and mesh reconstruction methods behind LSP are adjusted to reflect the properties afforded by the mathematical formulation. The results, supported by illustrative methods of the processes of LSP and t-SNE, are meant to inspire students in understanding the mathematics behind such methods, in order to apply them in effective data analysis tasks in multiple applications.

A data-driven model augmentation framework, referred to as Weakly-coupled Integrated Inference and Machine Learning (IIML), is presented to improve the predictive accuracy of physical models. In contrast to parameter calibration, this work seeks corrections to the structure of the model by a) inferring augmentation fields that are consistent with the underlying model, and b) transforming these fields into corrective model forms. The proposed approach couples the inference and learning steps in a weak sense via an alternating optimization approach. This coupling ensures that the augmentation fields remain learnable and maintain consistent functional relationships with local modeled quantities across the training dataset. An iterative solution procedure is presented in this paper, removing the need to embed the augmentation function during the inference process. This framework is used to infer an augmentation introduced within a Polymer electrolyte membrane fuel cell (PEMFC) model using a small amount of training data (from only 14 training cases.) These training cases belong to a dataset consisting of high-fidelity simulation data obtained from a high-fidelity model of a first generation Toyota Mirai. All cases in this dataset are characterized by different inflow and outflow conditions on the same geometry. When tested on 1224 different configurations, the inferred augmentation significantly improves the predictive accuracy for a wide range of physical conditions. Predictions and available data for the current density distribution are also compared to demonstrate the predictive capability of the model for quantities of interest which were not involved in the inference process. The results demonstrate that the weakly-coupled IIML framework offers sophisticated and robust model augmentation capabilities without requiring extensive changes to the numerical solver.

A key requirement for the success of supervised deep learning is a large labeled dataset - a condition that is difficult to meet in medical image analysis. Self-supervised learning (SSL) can help in this regard by providing a strategy to pre-train a neural network with unlabeled data, followed by fine-tuning for a downstream task with limited annotations. Contrastive learning, a particular variant of SSL, is a powerful technique for learning image-level representations. In this work, we propose strategies for extending the contrastive learning framework for segmentation of volumetric medical images in the semi-supervised setting with limited annotations, by leveraging domain-specific and problem-specific cues. Specifically, we propose (1) novel contrasting strategies that leverage structural similarity across volumetric medical images (domain-specific cue) and (2) a local version of the contrastive loss to learn distinctive representations of local regions that are useful for per-pixel segmentation (problem-specific cue). We carry out an extensive evaluation on three Magnetic Resonance Imaging (MRI) datasets. In the limited annotation setting, the proposed method yields substantial improvements compared to other self-supervision and semi-supervised learning techniques. When combined with a simple data augmentation technique, the proposed method reaches within 8% of benchmark performance using only two labeled MRI volumes for training, corresponding to only 4% (for ACDC) of the training data used to train the benchmark.

State-of-the-art Convolutional Neural Network (CNN) benefits a lot from multi-task learning (MTL), which learns multiple related tasks simultaneously to obtain shared or mutually related representations for different tasks. The most widely-used MTL CNN structure is based on an empirical or heuristic split on a specific layer (e.g., the last convolutional layer) to minimize different task-specific losses. However, this heuristic sharing/splitting strategy may be harmful to the final performance of one or multiple tasks. In this paper, we propose a novel CNN structure for MTL, which enables automatic feature fusing at every layer. Specifically, we first concatenate features from different tasks according to their channel dimension, and then formulate the feature fusing problem as discriminative dimensionality reduction. We show that this discriminative dimensionality reduction can be done by 1x1 Convolution, Batch Normalization, and Weight Decay in one CNN, which we refer to as Neural Discriminative Dimensionality Reduction (NDDR). We perform ablation analysis in details for different configurations in training the network. The experiments carried out on different network structures and different task sets demonstrate the promising performance and desirable generalizability of our proposed method.