Data assimilation is a vital component in modern global medium-range weather forecasting systems to obtain the best estimation of the atmospheric state by combining the short-term forecast and observations. Recently, AI-based data assimilation approaches have attracted increasing attention for their significant advantages over traditional techniques in terms of computational consumption.

We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.

Kolmogorov-Arnold Networks (KANs) relocate learnable nonlinearities from nodes to edges, demonstrating remarkable capabilities in scientific machine learning and interpretable modeling. However, current KAN implementations suffer from fundamental inefficiencies due to redundancy in high-dimensional spline parameter spaces, where numerous distinct parameterisations yield functionally equivalent behaviors. This redundancy manifests as a "nuisance space" in the model's Jacobian, leading to susceptibility to overfitting and poor generalization. We introduce Projective Kolmogorov-Arnold Networks (P-KANs), a novel training framework that guides edge function discovery towards interpretable functional representations through entropy-minimisation techniques from signal analysis and sparse dictionary learning. Rather than constraining functions to predetermined spaces, our approach maintains spline space flexibility while introducing "gravitational" terms that encourage convergence towards optimal functional representations. Our key insight recognizes that optimal representations can be identified through entropy analysis of projection coefficients, compressing edge functions to lower-parameter projective spaces (Fourier, Chebyshev, Bessel). P-KANs demonstrate superior performance across multiple domains, achieving up to 80% parameter reduction while maintaining representational capacity, significantly improved robustness to noise compared to standard KANs, and successful application to industrial automated fiber placement prediction. Our approach enables automatic discovery of mixed functional representations where different edges converge to different optimal spaces, providing both compression benefits and enhanced interpretability for scientific machine learning applications.

Functional data analysis (FDA) and ensemble learning can be powerful tools for analyzing complex environmental time series. Recent literature has highlighted the key role of diversity in enhancing accuracy and reducing variance in ensemble methods.This paper introduces Randomized Spline Trees (RST), a novel algorithm that bridges these two approaches by incorporating randomized functional representations into the Random Forest framework. RST generates diverse functional representations of input data using randomized B-spline parameters, creating an ensemble of decision trees trained on these varied representations. We provide a theoretical analysis of how this functional diversity contributes to reducing generalization error and present empirical evaluations on six environmental time series classification tasks from the UCR Time Series Archive. Results show that RST variants outperform standard Random Forests and Gradient Boosting on most datasets, improving classification accuracy by up to 14\%. The success of RST demonstrates the potential of adaptive functional representations in capturing complex temporal patterns in environmental data. This work contributes to the growing field of machine learning techniques focused on functional data and opens new avenues for research in environmental time series analysis.

Managing the emotional aspect remains a challenge in automatic music generation. Prior works aim to learn various emotions at once, leading to inadequate modeling. This paper explores the disentanglement of emotions in piano performance generation through a two-stage framework. The first stage focuses on valence modeling of lead sheet, and the second stage addresses arousal modeling by introducing performance-level attributes. To further capture features that shape valence, an aspect less explored by previous approaches, we introduce a novel functional representation of symbolic music. This representation aims to capture the emotional impact of major-minor tonality, as well as the interactions among notes, chords, and key signatures. Objective and subjective experiments validate the effectiveness of our framework in both emotional valence and arousal modeling. We further leverage our framework in a novel application of emotional controls, showing a broad potential in emotion-driven music generation.

Emotion-driven melody harmonization aims to generate diverse harmonies for a single melody to convey desired emotions. Previous research found it hard to alter the perceived emotional valence of lead sheets only by harmonizing the same melody with different chords, which may be attributed to the constraints imposed by the melody itself and the limitation of existing music representation. In this paper, we propose a novel functional representation for symbolic music. This new method takes musical keys into account, recognizing their significant role in shaping music's emotional character through major-minor tonality. It also allows for melodic variation with respect to keys and addresses the problem of data scarcity for better emotion modeling. A Transformer is employed to harmonize key-adaptable melodies, allowing for keys determined in rule-based or model-based manner. Experimental results confirm the effectiveness of our new representation in generating key-aware harmonies, with objective and subjective evaluations affirming the potential of our approach to convey specific valence for versatile melody.

Neural models learn data representations that lie on low-dimensional manifolds, yet modeling the relation between these representational spaces is an ongoing challenge. By integrating spectral geometry principles into neural modeling, we show that this problem can be better addressed in the functional domain, mitigating complexity, while enhancing interpretability and performances on downstream tasks. To this end, we introduce a multi-purpose framework to the representation learning community, which allows to: (i) compare different spaces in an interpretable way and measure their intrinsic similarity; (ii) find correspondences between them, both in unsupervised and weakly supervised settings, and (iii) to effectively transfer representations between distinct spaces. We validate our framework on various applications, ranging from stitching to retrieval tasks, demonstrating that latent functional maps can serve as a swiss-army knife for representation alignment.

Data assimilation is a vital component in modern global medium-range weather forecasting systems to obtain the best estimation of the atmospheric state by combining the short-term forecast and observations. Recently, AI-based data assimilation approaches have attracted increasing attention for their significant advantages over traditional techniques in terms of computational consumption. However, existing AI-based data assimilation methods can only handle observations with a specific resolution, lacking the compatibility and generalization ability to assimilate observations with other resolutions. Considering that complex real-world observations often have different resolutions, we propose the \textit{\textbf{Fourier Neural Processes}} (FNP) for \textit{arbitrary-resolution data assimilation} in this paper. Leveraging the efficiency of the designed modules and flexible structure of neural processes, FNP achieves state-of-the-art results in assimilating observations with varying resolutions, and also exhibits increasing advantages over the counterparts as the resolution and the amount of observations increase. Moreover, our FNP trained on a fixed resolution can directly handle the assimilation of observations with out-of-distribution resolutions and the observational information reconstruction task without additional fine-tuning, demonstrating its excellent generalization ability across data resolutions as well as across tasks.

Many conventional statistical and machine learning methods face challenges when applied directly to high dimensional temporal observations. In recent decades, Functional Data Analysis (FDA) has gained widespread popularity as a framework for modeling and analyzing data that are, by their nature, functions in the domain of time. Although supervised classification has been extensively explored in recent decades within the FDA literature, ensemble learning of functional classifiers has only recently emerged as a topic of significant interest. Thus, the latter subject presents unexplored facets and challenges from various statistical perspectives. The focal point of this paper lies in the realm of ensemble learning for functional data and aims to show how different functional data representations can be used to train ensemble members and how base model predictions can be combined through majority voting. The so-called Functional Voting Classifier (FVC) is proposed to demonstrate how different functional representations leading to augmented diversity can increase predictive accuracy. Many real-world datasets from several domains are used to display that the FVC can significantly enhance performance compared to individual models. The framework presented provides a foundation for voting ensembles with functional data and can stimulate a highly encouraging line of research in the FDA context.