Current transfer learning algorithm designs mainly focus on the similarities between source and target tasks, while the impacts of the sample sizes of these tasks are often not sufficiently addressed. This paper proposes a mathematical framework for quantifying the transferability in multi-source transfer learning problems, with both the task similarities and the sample complexity of learning models taken into account. In particular, we consider the setup where the models learned from different tasks are linearly combined for learning the target task, and use the optimal combining coefficients to measure the transferability. Then, we demonstrate the analytical expression of this transferability measure, characterized by the sample sizes, model complexity, and the similarities between source and target tasks, which provides fundamental insights of the knowledge transferring mechanism and the guidance for algorithm designs. Furthermore, we apply our analyses for practical learning tasks, and establish a quantifiable transferability measure by exploiting a parameterized model. In addition, we develop an alternating iterative algorithm to implement our theoretical results for training deep neural networks in multi-source transfer learning tasks. Finally, experiments on image classification tasks show that our approach outperforms existing transfer learning algorithms in multi-source and few-shot scenarios.

AI/ML models have rapidly gained prominence as innovations for solving previously unsolved problems and their unintended consequences from amplifying human biases. Advocates for responsible AI/ML have sought ways to draw on the richer causal models of system dynamics to better inform the development of responsible AI/ML. However, a major barrier to advancing this work is the difficulty of bringing together methods rooted in different underlying assumptions (i.e., Dana Meadow's "the unavoidable a priori"). This paper brings system dynamics and structural equation modeling together into a common mathematical framework that can be used to generate systems from distributions, develop methods, and compare results to inform the underlying epistemology of system dynamics for data science and AI/ML applications.

The rise of multi-paradigm languages challenges traditional classification methods, leading to practical software engineering issues like interoperability defects. This systematic literature review (SLR) maps the formal foundations of programming paradigms. Our objective is twofold: (1) to assess the state of the art of classification formalisms and their limitations, and (2) to identify the conceptual primitives and mathematical frameworks for a more powerful, reconstructive approach. Based on a synthesis of 74 primary studies, we find that existing taxonomies lack conceptual granularity, a unified formal basis, and struggle with hybrid languages. In response, our analysis reveals a strong convergence toward a compositional reconstruction of paradigms. This approach identifies a minimal set of orthogonal, atomic primitives and leverages mathematical frameworks, predominantly Type theory, Category theory and Unifying Theories of Programming (UTP), to formally guarantee their compositional properties. We conclude that the literature reflects a significant intellectual shift away from classification towards these promising formal, reconstructive frameworks. This review provides a map of this evolution and proposes a research agenda for their unification.

This paper presents the Coordinate Heart System (CHS), a geometric framework for emotion representation in artificial intelligence applications. We position eight core emotions as coordinates on a unit circle, enabling mathematical computation of complex emotional states through coordinate mixing and vector operations. Our initial five-emotion model revealed significant coverage gaps in the emotion space, leading to the development of an eight-emotion system that provides complete geometric coverage with mathematical guarantees. The framework converts natural language input to emotion coordinates and supports real-time emotion interpolation through computational algorithms. The system introduces a re-calibrated stability parameter S in [0,1], which dynamically integrates emotional load, conflict resolution, and contextual drain factors. This stability model leverages advanced Large Language Model interpretation of textual cues and incorporates hybrid temporal tracking mechanisms to provide nuanced assessment of psychological well-being states. Our key contributions include: (i) mathematical proof demonstrating why five emotions are insufficient for complete geometric coverage, (ii) an eight-coordinate system that eliminates representational blind spots, (iii) novel algorithms for emotion mixing, conflict resolution, and distance calculation in emotion space, and (iv) a comprehensive computational framework for AI emotion recognition with enhanced multi-dimensional stability modeling. Experimental validation through case studies demonstrates the system's capability to handle emotionally conflicted states, contextual distress factors, and complex psychological scenarios that traditional categorical emotion models cannot adequately represent. This work establishes a new mathematical foundation for emotion modeling in artificial intelligence systems.

In class-incremental learning (class-IL), models must classify all previously seen classes at test time without task-IDs, leading to task confusion. Despite being a key challenge, task confusion lacks a theoretical understanding. We present a novel mathematical framework for class-IL and prove the Infeasibility Theorem, showing optimal class-IL is impossible with discriminative modeling due to task confusion. However, we establish the Feasibility Theorem, demonstrating that generative modeling can achieve optimal class-IL by overcoming task confusion. We then assess popular class-IL strategies, including regularization, bias-correction, replay, and generative classifier, using our framework.

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive Operators (GENEOs) transformations. The central concept involves quantifying the distance between GEOs by measuring the non-commutativity of specific diagrams. Additionally, the paper proposes a definition of interpretability of GEOs according to a complexity measure that can be defined according to each user preferences. Moreover, we explore the formal properties of this framework and show how it can be applied in classical machine learning scenarios, like image classification with convolutional neural networks.

Current transfer learning algorithm designs mainly focus on the similarities between source and target tasks, while the impacts of the sample sizes of these tasks are often not sufficiently addressed. This paper proposes a mathematical framework for quantifying the transferability in multi-source transfer learning problems, with both the task similarities and the sample complexity of learning models taken into account. In particular, we consider the setup where the models learned from different tasks are linearly combined for learning the target task, and use the optimal combining coefficients to measure the transferability. Then, we demonstrate the analytical expression of this transferability measure, characterized by the sample sizes, model complexity, and the similarities between source and target tasks, which provides fundamental insights of the knowledge transferring mechanism and the guidance for algorithm designs. Furthermore, we apply our analyses for practical learning tasks, and establish a quantifiable transferability measure by exploiting a parameterized model.

We propose that inherent stochasticity enables synaptic plasticity to carry out probabilistic inference by sampling from a posterior distribution of synaptic parameters. This view provides a viable alternative to existing models that propose convergence of synaptic weights to maximum likelihood parameters. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience. In simulations we show that our model for synaptic plasticity allows spiking neural networks to compensate continuously for unforeseen disturbances. Furthermore it provides a normative mathematical framework to better understand the permanent variability and rewiring observed in brain networks.

This paper introduces a neuro-symbolic approach for relational exploration in cultural heritage knowledge graphs, leveraging Large Language Models (LLMs) for explanation generation and a novel mathematical framework to quantify the interestingness of relationships. We demonstrate the importance of interestingness measure using a quantitative analysis, by highlighting its impact on the overall performance of our proposed system, particularly in terms of precision, recall, and F1-score. Using the Wikidata Cultural Heritage Linked Open Data (WCH-LOD) dataset, our approach yields a precision of 0.70, recall of 0.68, and an F1-score of 0.69, representing an improvement compared to graph-based (precision: 0.28, recall: 0.25, F1-score: 0.26) and knowledge-based baselines (precision: 0.45, recall: 0.42, F1-score: 0.43). Furthermore, our LLM-powered explanations exhibit better quality, reflected in BLEU (0.52), ROUGE-L (0.58), and METEOR (0.63) scores, all higher than the baseline approaches. We show a strong correlation (0.65) between interestingness measure and the quality of generated explanations, validating its effectiveness. The findings highlight the importance of LLMs and a mathematical formalization for interestingness in enhancing the effectiveness of relational exploration in cultural heritage knowledge graphs, with results that are measurable and testable. We further show that the system enables more effective exploration compared to purely knowledge-based and graph-based methods. Keywords Knowledge Graphs, Large Language Models (LLMs), Explainable AI (XAI), Cultural Heritage, Neuro-Symbolic AI, Interestingness Score, Contextual Relevance, Relational Search 1. Introduction The digitization of cultural heritage artifacts and historical records has generated a vast amount of knowledge encoded in the form of interconnected knowledge graphs (KGs) [1, 2]. Unlocking meaningful insights from these KGs requires more than simple keyword searches [3].

Ranking entities such as algorithms, devices, methods, or models based on their performances, while accounting for application-specific preferences, is a challenge. To address this challenge, we establish the foundations of a universal theory for performance-based ranking. First, we introduce a rigorous framework built on top of both the probability and order theories. Our new framework encompasses the elements necessary to (1) manipulate performances as mathematical objects, (2) express which performances are worse than or equivalent to others, (3) model tasks through a variable called satisfaction, (4) consider properties of the evaluation, (5) define scores, and (6) specify application-specific preferences through a variable called importance. On top of this framework, we propose the first axiomatic definition of performance orderings and performance-based rankings. Then, we introduce a universal parametric family of scores, called ranking scores, that can be used to establish rankings satisfying our axioms, while considering application-specific preferences. Finally, we show, in the case of two-class classification, that the family of ranking scores encompasses well-known performance scores, including the accuracy, the true positive rate (recall, sensitivity), the true negative rate (specificity), the positive predictive value (precision), and F1. However, we also show that some other scores commonly used to compare classifiers are unsuitable to derive performance orderings satisfying the axioms. Therefore, this paper provides the computer vision and machine learning communities with a rigorous framework for evaluating and ranking entities.