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The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. They are particularly useful in areas such as network analysis and geometry processing, as computation of a GW distance involves solving for registration between the objects which minimizes geometric distortion. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this article, we focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. Our goal is to understand the theoretical properties of this relaxed optimization problem, from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, we derive precise characterizations of how it fails the axioms of non-degeneracy and triangle inequality. These observations lead us to define a novel family of distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al.\ on robust versions of classical Wasserstein distance. We show that our new distances define true metrics, that they induce the same topology as the GW distances, and that they enjoy additional robustness to perturbations. These results provide a mathematically rigorous basis for using our robust partial GW distances in applications where outliers and partial matching are concerns.

In real-world data, information is stored in extremely large feature vectors. These variables are typically correlated due to complex interactions involving many features simultaneously. Such correlations qualitatively correspond to semantic roles and are naturally recognized by both the human brain and artificial neural networks. This recognition enables, for instance, the prediction of missing parts of an image or text based on their context. We present a method to detect these correlations in high-dimensional data represented as binary numbers. We estimate the binary intrinsic dimension of a dataset, which quantifies the minimum number of independent coordinates needed to describe the data, and is therefore a proxy of semantic complexity. The proposed algorithm is largely insensitive to the so-called curse of dimensionality, and can therefore be used in big data analysis. We test this approach identifying phase transitions in model magnetic systems and we then apply it to the detection of semantic correlations of images and text inside deep neural networks.

Over the past decade, studies of naturalistic language processing where participants are scanned while listening to continuous text have flourished. Using word embeddings at first, then large language models, researchers have created encoding models to analyze the brain signals. Presenting these models with the same text as the participants allows to identify brain areas where there is a significant correlation between the functional magnetic resonance imaging (fMRI) time series and the ones predicted by the models' artificial neurons. One intriguing finding from these studies is that they have revealed highly symmetric bilateral activation patterns, somewhat at odds with the well-known left lateralization of language processing. Here, we report analyses of an fMRI dataset where we manipulate the complexity of large language models, testing 28 pretrained models from 8 different families, ranging from 124M to 14.2B parameters. First, we observe that the performance of models in predicting brain responses follows a scaling law, where the fit with brain activity increases linearly with the logarithm of the number of parameters of the model (and its performance on natural language processing tasks). Second, although this effect is present in both hemispheres, it is stronger in the left than in the right hemisphere. Specifically, the left-right difference in brain correlation follows a scaling law with the number of parameters. This finding reconciles computational analyses of brain activity using large language models with the classic observation from aphasic patients showing left hemisphere dominance for language.

Volumetric video, the capture and display of three-dimensional (3D) imagery, has emerged as a revolutionary technology poised to transform the media landscape, enabling immersive experiences that transcend the limitations of traditional 2D video. One of the key challenges in this domain is the efficient delivery of these high-bandwidth, data-intensive volumetric video streams, which requires innovative transcoding and compression techniques. This research paper explores the state-of-the-art in volumetric video compression and delivery, with a focus on the potential of AI-driven solutions to address the unique challenges posed by this emerging medium.

This article examines how models of teacher digital maturity can be combined to produce a unified version that can be used to design diagnostic tools and methods. 11 models applicable to the field of compulsory education were identified through a literature review. The models and how their constituent dimensions contribute to the determination of maturity levels were analyzed. The summary highlights the diversity of the dimensions used and the fact that digital maturity is only partially taken into account. What's more, most of these models focus on the most recent maturity levels associated with innovative or pioneering teachers. The models tend to exclude teachers who are not digital users or who have a low level of digital use, but who are present in the French context. In the final part of the article, a proposal for a unified model of teachers' digital maturity, MUME, which addresses these two issues, is described, together with the preliminary results of a study aimed at designing a diagnostic method.

We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.

Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The LA is exact if the function is proportional to a normal density; its effectiveness therefore depends on the function's true shape. Here, we propose the use of the probabilistic numerical framework to develop a diagnostic for the LA and its underlying shape assumptions, modelling the function and its integral as a Gaussian process and devising a "test" by conditioning on a finite number of function values. The test is decidedly non-asymptotic and is not intended as a full substitute for numerical integration - rather, it is simply intended to test the feasibility of the assumptions underpinning the LA with as minimal computation. We discuss approaches to optimize and design the test, apply it to known sample functions, and highlight the challenges of high dimensions.

Image Quality of MRI brain scans is strongly influenced by within scanner head movements and the resulting image artifacts alter derived measures like brain volume and cortical thickness. Automated image quality assessment is key to controlling for confounding effects of poor image quality. In this study, we systematically test for the influence of image quality on univariate statistics and machine learning classification. We analyzed group effects of sex/gender on local brain volume and made predictions of sex/gender using logistic regression, while correcting for brain size. From three large publicly available datasets, two age and sex-balanced samples were derived to test the generalizability of the effect for pooled sample sizes of n=760 and n=1094. Results of the Bonferroni corrected t-tests over 3747 gray matter features showed a strong influence of low-quality data on the ability to find significant sex/gender differences for the smaller sample. Increasing sample size and more so image quality showed a stark increase in detecting significant effects in univariate group comparisons. For the classification of sex/gender using logistic regression, both increasing sample size and image quality had a marginal effect on the Area under the Receiver Operating Characteristic Curve for most datasets and subsamples. Our results suggest a more stringent quality control for univariate approaches than for multivariate classification with a leaning towards higher quality for classical group statistics and bigger sample sizes for machine learning applications in neuroimaging.

Underwriting is one of the important stages in an insurance company. The insurance company uses different factors to classify the policyholders. In this study, we apply several machine learning models such as nearest neighbour and logistic regression to the Actuarial Challenge dataset used by Qazvini (2019) to classify liability insurance policies into two groups: 1 - policies with claims and 2 - policies without claims.

Computer-generated holography (CGH) is a promising technology for augmented reality displays, such as head-mounted or head-up displays. However, its high computational demand makes it impractical for implementation. Recent efforts to integrate neural networks into CGH have successfully accelerated computing speed, demonstrating the potential to overcome the trade-off between computational cost and image quality. Nevertheless, deploying neural network-based CGH algorithms on computationally limited embedded systems requires more efficient models with lower computational cost, memory footprint, and power consumption. In this study, we developed a lightweight model for complex hologram generation by introducing neural network quantization. Specifically, we built a model based on tensor holography and quantized it from 32-bit floating-point precision (FP32) to 8-bit integer precision (INT8). Our performance evaluation shows that the proposed INT8 model achieves hologram quality comparable to that of the FP32 model while reducing the model size by approximately 70% and increasing the speed fourfold. Additionally, we implemented the INT8 model on a system-on-module to demonstrate its deployability on embedded platforms and high power efficiency.