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Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats and therefore we do not have one specific data structure. We call such data in general non-standard data. To achieve this, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing depth functions for non-standard data using formal concept analysis, we give a systematic basis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data and therefore increases the spaces centrality is currently discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.

In practical applications, effectively segmenting cracks in large-scale computed tomography (CT) images holds significant importance for understanding the structural integrity of materials. However, classical methods and Machine Learning algorithms often incur high computational costs when dealing with the substantial size of input images. Hence, a robust algorithm is needed to pre-detect crack regions, enabling focused analysis and reducing computational overhead. The proposed approach addresses this challenge by offering a streamlined method for identifying crack regions in CT images with high probability. By efficiently identifying areas of interest, our algorithm allows for a more focused examination of potential anomalies within the material structure. Through comprehensive testing on both semi-synthetic and real 3D CT images, we validate the efficiency of our approach in enhancing crack segmentation while reducing computational resource requirements.

Although the multi-jointed underactuated manipulator is highly dexterous, its grasping capacity does not match that of the parallel jaw gripper. This work introduces a fractal gripper to enhance the grasping capacity of multi-joint underactuated manipulators, preserving their passive clamping features. We describe in detail the working principle and manufacturing process of the fractal gripper. This work, inspired by the 'Fractal Vise' structure, resulted in the invention of a fractal gripper with mode switching capabilities. The fractal gripper inherits the inherent adaptive properties of the fractal structure and realizes the self-resetting function by integrating spring into the original design, thereby enhancing the efficiency of object grasping tasks. The fractal gripper prevents object damage by distributing pressure evenly and applying it at multiple points through its fractal structure during closure. Objects of various shapes are effectively grasped by the fractal gripper, which ensures a safe and secure grasp. The superior performance was provided by the force distribution characteristics of the fractal gripper. By applying the flexible polymer PDMS, which possesses superior elasticity, to the fractal structure's wrapping surface, potential scratching during grasping is effectively prevented, thus protecting the object's geometric surface. Grab experiments with objects of diverse shapes and sizes confirm fractal gripper multi-scale adaptability and superior grasping stability.

This paper explores the role of generalized continuum mechanics, and the feasibility of model-free data-driven computing approaches thereof, in solids undergoing failure by strain localization. Specifically, we set forth a methodology for capturing material instabilities using data-driven mechanics without prior information regarding the failure mode. We show numerically that, in problems involving strain localization, the standard data-driven framework for Cauchy/Boltzmann continua fails to capture the length scale of the material, as expected. We address this shortcoming by formulating a generalized data-driven framework for micromorphic continua that effectively captures both stiffness and length-scale information, as encoded in the material data, in a model-free manner. These properties are exhibited systematically in a one-dimensional softening bar problem and further verified through selected plane-strain problems.

Randomized matrix algorithms have become workhorse tools in scientific computing and machine learning. To use these algorithms safely in applications, they should be coupled with posterior error estimates to assess the quality of the output. To meet this need, this paper proposes two diagnostics: a leave-one-out error estimator for randomized low-rank approximations and a jackknife resampling method to estimate the variance of the output of a randomized matrix computation. Both of these diagnostics are rapid to compute for randomized low-rank approximation algorithms such as the randomized SVD and randomized Nystr\"om approximation, and they provide useful information that can be used to assess the quality of the computed output and guide algorithmic parameter choices.

The ability to dynamically adjust the computational load of neural models during inference is crucial for on-device processing scenarios characterised by limited and time-varying computational resources. A promising solution is presented by early-exit architectures, in which additional exit branches are appended to intermediate layers of the encoder. In self-attention models for automatic speech recognition (ASR), early-exit architectures enable the development of dynamic models capable of adapting their size and architecture to varying levels of computational resources and ASR performance demands. Previous research on early-exiting ASR models has relied on pre-trained self-supervised models, fine-tuned with an early-exit loss. In this paper, we undertake an experimental comparison between fine-tuning pre-trained backbones and training models from scratch with the early-exiting objective. Experiments conducted on public datasets reveal that early-exit models trained from scratch not only preserve performance when using fewer encoder layers but also exhibit enhanced task accuracy compared to single-exit or pre-trained models. Furthermore, we explore an exit selection strategy grounded in posterior probabilities as an alternative to the conventional frame-based entropy approach. Results provide insights into the training dynamics of early-exit architectures for ASR models, particularly the efficacy of training strategies and exit selection methods.

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Brain simulation builds dynamical models to mimic the structure and functions of the brain, while brain-inspired computing (BIC) develops intelligent systems by learning from the structure and functions of the brain. The two fields are intertwined and should share a common programming framework to facilitate each other's development. However, none of the existing software in the fields can achieve this goal, because traditional brain simulators lack differentiability for training, while existing deep learning (DL) frameworks fail to capture the biophysical realism and complexity of brain dynamics. In this paper, we introduce BrainPy, a differentiable brain simulator developed using JAX and XLA, with the aim of bridging the gap between brain simulation and BIC. BrainPy expands upon the functionalities of JAX, a powerful AI framework, by introducing complete capabilities for flexible, efficient, and scalable brain simulation. It offers a range of sparse and event-driven operators for efficient and scalable brain simulation, an abstraction for managing the intricacies of synaptic computations, a modular and flexible interface for constructing multi-scale brain models, and an object-oriented just-in-time compilation approach to handle the memory-intensive nature of brain dynamics. We showcase the efficiency and scalability of BrainPy on benchmark tasks, highlight its differentiable simulation for biologically plausible spiking models, and discuss its potential to support research at the intersection of brain simulation and BIC.

Inferring parameters of a latent variable model can be a daunting task when the conditional distribution of the latent variables given the observed ones is intractable. Variational approaches prove to be computationally efficient but, possibly, lack theoretical guarantees on the estimates, while sampling based solutions are quite the opposite. Starting from already available variational approximations, we define a first Monte Carlo EM algorithm to obtain maximum likelihood estimators, focusing on the Poisson log-normal model which provides a generic framework for the analysis of multivariate count data. We then extend this algorithm to the case of a composite likelihood in order to be able to handle higher dimensional count data.

Full waveform inversion (FWI) is a large-scale nonlinear ill-posed problem for which computationally expensive Newton-type methods can become trapped in undesirable local minima, particularly when the initial model lacks a low-wavenumber component and the recorded data lacks low-frequency content. A modification to the Gauss-Newton (GN) method is proposed to address these issues. The standard GN system for multisource multireceiver FWI is reformulated into an equivalent matrix equation form, with the solution becoming a diagonal matrix rather than a vector as in the standard system. The search direction is transformed from a vector to a matrix by relaxing the diagonality constraint, effectively adding a degree of freedom to the subsurface offset axis. The relaxed system can be explicitly solved with only the inversion of two small matrices that deblur the data residual matrix along the source and receiver dimensions, which simplifies the inversion of the Hessian matrix. When used to solve the extended source FWI objective function, the Extended GN (EGN) method integrates the benefits of both model and source extension. The EGN method effectively combines the computational effectiveness of the reduced FWI method with the robustness characteristics of extended formulations and offers a promising solution for addressing the challenges of FWI. It bridges the gap between these extended formulations and the reduced FWI method, enhancing inversion robustness while maintaining computational efficiency. The robustness and stability of the EGN algorithm for waveform inversion are demonstrated numerically.