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The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques which have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.

Biological organisms have acquired sophisticated body shapes for walking or climbing through million-year evolutionary processes. In contrast, the components of locomoting soft robots, such as legs and arms, are designed in trial-and-error loops guided by a priori knowledge and experience, which leaves considerable room for improvement. Here, we present optimized soft robots that performed a specific locomotion task without any a priori assumptions or knowledge of the layout and shapes of the limbs by fully exploiting the computational capabilities for topology optimization. The only requirements introduced were a design domain and a periodically acting pneumatic actuator. The freeform shape of a soft body was derived from iterative updates in a gradient-based topology optimization that incorporated complex physical phenomena, such as large deformations, contacts, material viscosity, and fluid-structure interactions, in transient problems. The locomotion tasks included a horizontal movement on flat ground (walking) and a vertical movement between two walls (climbing). Without any human intervention, optimized soft robots have limbs and exhibit locomotion similar to those of biological organisms. Linkage-like structures were formed for the climbing task to assign different movements to multiple legs with limited degrees of freedom in the actuator. We fabricated the optimized design using 3D printing and confirmed the performance of these robots. This study presents a new and efficient strategy for designing soft robots and other bioinspired systems, suggesting that a purely mathematical process can produce shapes reminiscent of nature's long-term evolution.

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

We present a rotation equivariant, quasi-monolithic graph neural network framework for the reduced-order modeling of fluid-structure interaction systems. With the aid of an arbitrary Lagrangian-Eulerian formulation, the system states are evolved temporally with two sub-networks. The movement of the mesh is reduced to the evolution of several coefficients via complex-valued proper orthogonal decomposition, and the prediction of these coefficients over time is handled by a single multi-layer perceptron. A finite element-inspired hypergraph neural network is employed to predict the evolution of the fluid state based on the state of the whole system. The structural state is implicitly modeled by the movement of the mesh on the solid-fluid interface; hence it makes the proposed framework quasi-monolithic. The effectiveness of the proposed framework is assessed on two prototypical fluid-structure systems, namely the flow around an elastically-mounted cylinder, and the flow around a hyperelastic plate attached to a fixed cylinder. The proposed framework tracks the interface description and provides stable and accurate system state predictions during roll-out for at least 2000 time steps, and even demonstrates some capability in self-correcting erroneous predictions. The proposed framework also enables direct calculation of the lift and drag forces using the predicted fluid and mesh states, in contrast to existing convolution-based architectures. The proposed reduced-order model via graph neural network has implications for the development of physics-based digital twins concerning moving boundaries and fluid-structure interactions.

This work addresses the approximation of the mean curvature flow of thin structures for which classical phase field methods are not suitable. By thin structures we mean either structures of higher codimension, typically filaments, or surfaces (including non orientables surfaces) that are not boundaries of a set. We propose a novel approach which consists in plugging into the classical Allen--Cahn equation a penalization term localized around the skeleton of the evolving set. This ensures that a minimal thickness is preserved during the evolution process. The numerical efficacy of our approach is illustrated with accurate approximations of the evolution by mean curvature flow of filament structures. Furthermore, we show the seamless adaptability of our approach to compute numerical approximations of solutions to the Steiner and Plateau problems in three dimensions.

The elusive nature of gradient-based optimization in neural networks is tied to their loss landscape geometry, which is poorly understood. However recent work has brought solid evidence that there is essentially no loss barrier between the local solutions of gradient descent, once accounting for weight-permutations that leave the network's computation unchanged. This raises questions for approximate inference in Bayesian neural networks (BNNs), where we are interested in marginalizing over multiple points in the loss landscape. In this work, we first extend the formalism of marginalized loss barrier and solution interpolation to BNNs, before proposing a matching algorithm to search for linearly connected solutions. This is achieved by aligning the distributions of two independent approximate Bayesian solutions with respect to permutation matrices. We build on the results of Ainsworth et al. (2023), reframing the problem as a combinatorial optimization one, using an approximation to the sum of bilinear assignment problem. We then experiment on a variety of architectures and datasets, finding nearly zero marginalized loss barriers for linearly connected solutions.

Classification of unlabeled data is usually achieved by supervised learning from labeled samples. Although there exist many sophisticated supervised machine learning methods that can predict the missing labels with a high level of accuracy, they often lack the required transparency in situations where it is important to provide interpretable results and meaningful measures of confidence. Body fluid classification of forensic casework data is the case in point. We develop a new Biclustering Dirichlet Process for Class-assignment with Random Matrices (BDP-CaRMa), with a three-level hierarchy of clustering, and a model-based approach to classification that adapts to block structure in the data matrix. As the class labels of some observations are missing, the number of rows in the data matrix for each class is unknown. BDP-CaRMa handles this and extends existing biclustering methods by simultaneously biclustering multiple matrices each having a randomly variable number of rows. We demonstrate our method by applying it to the motivating problem, which is the classification of body fluids based on mRNA profiles taken from crime scenes. The analyses of casework-like data show that our method is interpretable and produces well-calibrated posterior probabilities. Our model can be more generally applied to other types of data with a similar structure to the forensic data.

Heavy tails is a common feature of filtering distributions that results from the nonlinear dynamical and observation processes as well as the uncertainty from physical sensors. In these settings, the Kalman filter and its ensemble version - the ensemble Kalman filter (EnKF) - that have been designed under Gaussian assumptions result in degraded performance. t-distributions are a parametric family of distributions whose tail-heaviness is modulated by a degree of freedom $\nu$. Interestingly, Cauchy and Gaussian distributions correspond to the extreme cases of a t-distribution for $\nu = 1$ and $\nu = \infty$, respectively. Leveraging tools from measure transport (Spantini et al., SIAM Review, 2022), we present a generalization of the EnKF whose prior-to-posterior update leads to exact inference for t-distributions. We demonstrate that this filter is less sensitive to outlying synthetic observations generated by the observation model for small $\nu$. Moreover, it recovers the Kalman filter for $\nu = \infty$. For nonlinear state-space models with heavy-tailed noise, we propose an algorithm to estimate the prior-to-posterior update from samples of joint forecast distribution of the states and observations. We rely on a regularized expectation-maximization (EM) algorithm to estimate the mean, scale matrix, and degree of freedom of heavy-tailed \textit{t}-distributions from limited samples (Finegold and Drton, arXiv preprint, 2014). Leveraging the conditional independence of the joint forecast distribution, we regularize the scale matrix with an $l1$ sparsity-promoting penalization of the log-likelihood at each iteration of the EM algorithm. By sequentially estimating the degree of freedom at each analysis step, our filter can adapt its prior-to-posterior update to the tail-heaviness of the data. We demonstrate the benefits of this new ensemble filter on challenging filtering problems.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.