Metric learning aims at finding a suitable distance metric over the input space, to improve the performance of distance-based learning algorithms. In high-dimensional settings, metric learning can also play the role of dimensionality reduction, by imposing a low-rank restriction to the learnt metric. In this paper, instead of training a low-rank metric on high-dimensional data, we consider a randomly compressed version of the data, and train a full-rank metric there. We give theoretical guarantees on the error of distance-based metric learning, with respect to the random compression, which do not depend on the ambient dimension. Our bounds do not make any explicit assumptions, aside from i.i.d. data from a bounded support, and automatically tighten when benign geometrical structures are present. Experimental results on both synthetic and real data sets support our theoretical findings in high-dimensional settings.

In optimization-based approaches to inverse problems and to statistical estimation, it is common to augment criteria that enforce data fidelity with a regularizer that promotes desired structural properties in the solution. The choice of a suitable regularizer is typically driven by a combination of prior domain information and computational considerations. Convex regularizers are attractive computationally but they are limited in the types of structure they can promote. On the other hand, nonconvex regularizers are more flexible in the forms of structure they can promote and they have showcased strong empirical performance in some applications, but they come with the computational challenge of solving the associated optimization problems. In this paper, we seek a systematic understanding of the power and the limitations of convex regularization by investigating the following questions: Given a distribution, what is the optimal regularizer for data drawn from the distribution? What properties of a data source govern whether the optimal regularizer is convex? We address these questions for the class of regularizers specified by functionals that are continuous, positively homogeneous, and positive away from the origin. We say that a regularizer is optimal for a data distribution if the Gibbs density with energy given by the regularizer maximizes the population likelihood (or equivalently, minimizes cross-entropy loss) over all regularizer-induced Gibbs densities. As the regularizers we consider are in one-to-one correspondence with star bodies, we leverage dual Brunn-Minkowski theory to show that a radial function derived from a data distribution is akin to a ``computational sufficient statistic'' as it is the key quantity for identifying optimal regularizers and for assessing the amenability of a data source to convex regularization.

We propose an innovative and generic methodology to analyse individual and collective behaviour through individual trajectory data. The work is motivated by the analysis of GPS trajectories of fishing vessels collected from regulatory tracking data in the context of marine biodiversity conservation and ecosystem-based fisheries management. We build a low-dimensional latent representation of trajectories using convolutional neural networks as non-linear mapping. This is done by training a conditional variational auto-encoder taking into account covariates. The posterior distributions of the latent representations can be linked to the characteristics of the actual trajectories. The latent distributions of the trajectories are compared with the Bhattacharyya coefficient, which is well-suited for comparing distributions. Using this coefficient, we analyse the variation of the individual behaviour of each vessel during time. For collective behaviour analysis, we build proximity graphs and use an extension of the stochastic block model for multiple networks. This model results in a clustering of the individuals based on their set of trajectories. The application to French fishing vessels enables us to obtain groups of vessels whose individual and collective behaviours exhibit spatio-temporal patterns over the period 2014-2018.

The software for clustering students according to their educational achievements using fuzzy logic was developed in Python using the Google Colab cloud service. In the process of analyzing educational data, the problems of Data Mining are solved, since only some characteristics of the educational process are obtained from a large sample of data. Data clustering was performed using the classic K-Means method, which is characterized by simplicity and high speed. Cluster analysis was performed in the space of two features using the machine learning library scikit-learn (Python). The obtained clusters are described by fuzzy triangular membership functions, which allowed to correctly determine the membership of each student to a certain cluster. Creation of fuzzy membership functions is done using the scikit-fuzzy library. The development of fuzzy functions of objects belonging to clusters is also useful for educational purposes, as it allows a better understanding of the principles of using fuzzy logic. As a result of processing test educational data using the developed software, correct results were obtained. It is shown that the use of fuzzy membership functions makes it possible to correctly determine the belonging of students to certain clusters, even if such clusters are not clearly separated. Due to this, it is possible to more accurately determine the recommended level of difficulty of tasks for each student, depending on his previous evaluations.

Markov chain Monte Carlo (MCMC) methods are simulated by local exploration of complex statistical distributions, and while bypassing the cumbersome requirement of a specific analytical expression for the target, this stochastic exploration of an uncertain parameter space comes at the expense of a large number of samples, and this computational complexity increases with parameter dimensionality. Although at the exploration level, some methods are proposed to accelerate the convergence of the algorithm, such as tempering, Hamiltonian Monte Carlo, Rao-redwellization, and scalable methods for better performance, it cannot avoid the stochastic nature of this exploration. We consider the target distribution as a mapping where the infinite-dimensional Eulerian space of the parameters consists of a number of deterministic submanifolds and propose a generalized energy metric, termed weighted Riesz energy, where a number of points is generated through pairwise interactions, to discretize rectifiable submanifolds. We study the properties of the point, called Riesz particle, and embed it into sequential MCMC, and we find that there will be higher acceptance rates with fewer evaluations, we validate it through experimental comparative analysis from a linear Gaussian state-space model with synthetic data and a non-linear stochastic volatility model with real-world data.

We consider the problem of learning causal Directed Acyclic Graphs (DAGs) using combinations of observational and interventional experimental data. Current methods tailored to this setting assume that interventions either destroy parent-child relations of the intervened (target) nodes or only alter such relations without modifying the parent sets, even when the intervention targets are unknown. We relax this assumption by proposing a Bayesian method for causal discovery from general interventions, which allow for modifications of the parent sets of the unknown targets. Even in this framework, DAGs and general interventions may be identifiable only up to some equivalence classes. We provide graphical characterizations of such interventional Markov equivalence and devise compatible priors for Bayesian inference that guarantee score equivalence of indistinguishable structures. We then develop a Markov Chain Monte Carlo (MCMC) scheme to approximate the posterior distribution over DAGs, intervention targets and induced parent sets. Finally, we evaluate the proposed methodology on both simulated and real protein expression data.

Additive friction stir deposition (AFSD) is a novel solid-state additive manufacturing technique that circumvents issues of porosity, cracking, and properties anisotropy that plague traditional powder bed fusion and directed energy deposition approaches. However, correlations between process parameters, thermal profiles, and resulting microstructure in AFSD remain poorly understood. This hinders process optimization for properties. This work employs a framework combining supervised machine learning (SML) and physics-informed neural networks (PINNs) to predict peak temperature distribution in AFSD from process parameters. Eight regression algorithms were implemented for SML modeling, while four PINNs leveraged governing equations for transport, wave propagation, heat transfer, and quantum mechanics. Across multiple statistical measures, ensemble techniques like gradient boosting proved superior for SML, with lowest MSE of 165.78. The integrated ML approach was also applied to classify deposition quality from process factors, with logistic regression delivering robust accuracy. By fusing data-driven learning and fundamental physics, this dual methodology provides comprehensive insights into tailoring microstructure through thermal management in AFSD. The work demonstrates the power of bridging statistical and physics-based modeling for elucidating AM process-property relationships.

Accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, the classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easily interpretable. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. Additionally to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for the tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Some simulations and real insurance data will be discussed to illustrate the applicability of these models.

Many approaches for optimizing decision making systems rely on gradient based methods requiring informative feedback from the environment. However, in the case where such feedback is sparse or uninformative, such approaches may result in poor performance. Derivative-free approaches such as Bayesian Optimization mitigate the dependency on the quality of gradient feedback, but are known to scale poorly in the high-dimension setting of complex decision making systems. This problem is exacerbated if the system requires interactions between several actors cooperating to accomplish a shared goal. To address the dimensionality challenge, we propose a compact multi-layered architecture modeling the dynamics of actor interactions through the concept of role. We introduce Hessian-aware Bayesian Optimization to efficiently optimize the multi-layered architecture parameterized by a large number of parameters, and give the first improved regret bound in additive high-dimensional Bayesian Optimization since Mutny & Krause (2018). Our approach shows strong empirical results under malformed or sparse reward.

We present GIFT (Generative Interpretable Fine-tuning Transformers) for fine-tuning pretrained (often large) Transformer models at downstream tasks in a parameter-efficient way with built-in interpretability. Our GIFT is a deep parameter-residual learning method, which addresses two problems in fine-tuning a pretrained Transformer model: Where to apply the parameter-efficient fine-tuning (PEFT) to be extremely lightweight yet sufficiently expressive, and How to learn the PEFT to better exploit the knowledge of the pretrained model in a direct way? For the former, we select the final projection (linear) layer in the multi-head self-attention of a Transformer model, and verify its effectiveness. For the latter, in contrast to the prior art that directly introduce new model parameters (often in low-rank approximation form) to be learned in fine-tuning with downstream data, we propose a method for learning to generate the fine-tuning parameters. Our GIFT is a hyper-Transformer which take as input the pretrained parameters of the projection layer to generate its fine-tuning parameters using a proposed Parameter-to-Cluster Attention (PaCa). The PaCa results in a simple clustering-based forward explainer that plays the role of semantic segmentation in testing. In experiments, our proposed GIFT is tested on the VTAB benchmark and the fine-grained visual classification (FGVC) benchmark. It obtains significantly better performance than the prior art. Our code is available at https://github.com/savadikarc/gift