In the field of machine learning, hyperbolic space demonstrates superior representation capabilities for hierarchical data compared to conventional Euclidean space. This work focuses on the Coarse-To-Fine Few-Shot Class-Incremental Learning (C2FSCIL) task. Our study follows the Knowe approach, which contrastively learns coarse class labels and subsequently normalizes and freezes the classifier weights of learned fine classes in the embedding space. To better interpret the "coarse-to-fine" paradigm, we propose embedding the feature extractor into hyperbolic space. Specifically, we employ the Poincaré ball model of hyperbolic space, enabling the feature extractor to transform input images into feature vectors within the Poincaré ball instead of Euclidean space. We further introduce hyperbolic contrastive loss and hyperbolic fully-connected layers to facilitate model optimization and classification in hyperbolic space. Additionally, to enhance performance under few-shot conditions, we implement maximum entropy distribution in hyperbolic space to estimate the probability distribution of fine-class feature vectors. This allows generation of augmented features from the distribution to mitigate overfitting during training with limited samples. Experiments on C2FSCIL benchmarks show that our method effectively improves both coarse and fine class accuracies.

Magnitude, obtained as a special case of Euler characteristic of enriched category, represents a sense of the size of metric spaces and is related to classical notions such as cardinality, dimension, and volume. While the studies have explained the meaning of magnitude from various perspectives, continuity also gives a valuable view of magnitude. Based on established results about continuity of magnitude and maximum diversity, this article focuses on continuity of weighting, a distribution whose totality is magnitude, and its variation corresponding to maximum diversity. Meanwhile, recent studies also illuminated the connection between magnitude and data analysis by applying magnitude theory to point clouds representing the data or the set of model parameters. This article will also provide an application for time series analysis by introducing a new kind of invariants of periodic time series, where the invariance follows directly from the continuity results. As a use-case, a simple machine learning experiment is conducted with real-world data, in which the suggested invariants improved the performance.

Timed regular expressions serve as a formalism for specifying real-time behaviors of Cyber-Physical Systems. In this paper, we consider the synthesis of timed regular expressions, focusing on generating a timed regular expression consistent with a given set of system behaviors including positive and negative examples, i.e., accepting all positive examples and rejecting all negative examples. We first prove the decidability of the synthesis problem through an exploration of simple timed regular expressions. Subsequently, we propose our method of generating a consistent timed regular expression with minimal length, which unfolds in two steps. The first step is to enumerate and prune candidate parametric timed regular expressions. In the second step, we encode the requirement that a candidate generated by the first step is consistent with the given set into a Satisfiability Modulo Theories (SMT) formula, which is consequently solved to determine a solution to parametric time constraints. Finally, we evaluate our approach on benchmarks, including randomly generated behaviors from target timed models and a case study.

Stein variational gradient descent (SVGD) is a kernel-based and non-parametric particle method for sampling from a target distribution, such as in Bayesian inference and other machine learning tasks. Different from other particle methods, SVGD does not require estimating the score, which is the gradient of the log-density. However, in practice, SVGD can be slow compared to score-estimation-based sampling algorithms. To design a fast and efficient high-dimensional sampling algorithm with the advantages of SVGD, we introduce accelerated SVGD (ASVGD), based on an accelerated gradient flow in a metric space of probability densities following Nesterov's method. We then derive a momentum-based discrete-time sampling algorithm, which evolves a set of particles deterministically. To stabilize the particles' position update, we also include a Wasserstein metric regularization. This paper extends the conference version \cite{SL2025}. For the bilinear kernel and Gaussian target distributions, we study the kernel parameter and damping parameters with an optimal convergence rate of the proposed dynamics. This is achieved by analyzing the linearized accelerated gradient flows at the equilibrium. Interestingly, the optimal parameter is a constant, which does not depend on the covariance of the target distribution. For the generalized kernel functions, such as the Gaussian kernel, numerical examples with varied target distributions demonstrate the effectiveness of ASVGD compared to SVGD and other popular sampling methods. Furthermore, we show that in the setting of Bayesian neural networks, ASVGD outperforms SVGD significantly in terms of log-likelihood and total iteration times.

Dimensionality reduction can distort vector space properties such as orthogonality and linear independence, which are critical for tasks including cross-modal retrieval, clustering, and classification. We propose a Relationship Preserving Loss (RPL), a loss function that preserves these properties by minimizing discrepancies between relationship matrices (e.g., Gram or cosine) of high-dimensional data and their low-dimensional embeddings. RPL trains neural networks for non-linear projections and is supported by error bounds derived from matrix perturbation theory. Initial experiments suggest that RPL reduces embedding dimensions while largely retaining performance on downstream tasks, likely due to its preservation of key vector space properties. While we describe here the use of RPL in dimensionality reduction, this loss can also be applied more broadly, for example to cross-domain alignment and transfer learning, knowledge distillation, fairness and invariance, dehubbing, graph and manifold learning, and federated learning, where distributed embeddings must remain geometrically consistent.

A fundamental aspect of the semantics of natural language is that novel meanings can be formed from the composition of previously known parts. Vision-language models (VLMs) have made significant progress in recent years, however, there is evidence that they are unable to perform this kind of composition. For example, given an image of a red cube and a blue cylinder, a VLM such as CLIP is likely to incorrectly label the image as a red cylinder or a blue cube, indicating it represents the image as a `bag-of-words' and fails to capture compositional semantics. Diffusion models have recently gained significant attention for their impressive generative abilities, and zero-shot classifiers based on diffusion models have been shown to perform competitively with CLIP in certain compositional tasks. In this work we explore whether the generative Diffusion Classifier has improved compositional generalisation abilities compared to discriminative models. We assess three models -- Diffusion Classifier, CLIP, and ViLT -- on their ability to bind objects with attributes and relations in both zero-shot learning (ZSL) and generalised zero-shot learning (GZSL) settings. Our results show that the Diffusion Classifier and ViLT perform well at concept binding tasks, but that all models struggle significantly with the relational GZSL task, underscoring the broader challenges VLMs face with relational reasoning. Analysis of CLIP embeddings suggests that the difficulty may stem from overly similar representations of relational concepts such as left and right. Code and dataset are available at: https://github.com/otmive/diffusion_classifier_clip

The paper introduces a novel framework based on category theory to enhance the explainability of artificial intelligence systems, particularly focusing on word embeddings. Furthermore, the paper defines the categories of configurations Conf and word embeddings Emb, accompanied by the concept of divergence as a decoration on Emb . It establishes a mathematically precise method for comparing word embeddings, demonstrating the equivalence between the GloVe and Word2Vec algorithms and the metric MDS algorithm, transitioning from neural network algorithms (black box) to a transparent framework. Finally, the paper presents a mathematical approach to computing biases before embedding and offers insights on mitigating biases at the semantic space level, advancing the field of explainable artificial intelligence. Introduction Word embeddings have emerged as a cornerstone in natural language processing (NLP) and machine learning (ML) applications, revolutionizing the representation of textual data (see [IUS23]). At the heart of word em-beddings lies the idea of capturing semantic relationships between words in a continuous vector space, enabling machines to understand and process human language more effectively (see [HAMJ16, LG14]). By mapping words to high-dimensional vectors, word embeddings encode semantic similarities and syntactic structures, thereby facilitating a wide array of downstream tasks such as sentiment analysis, named entity recognition, machine translation, and document classification. In addition to enhancing model performance and accuracy, word embeddings offer several practical advantages in ML applications. They provide a compact and dense representation of textual data, enabling efficient storage, retrieval, and computation. Moreover, word embeddings capture contextual nuances and semantic meanings that traditional bag-of-words or one-hot encoding schemes fail to capture, leading to more nuanced and context-aware language understanding. As such, word embeddings serve as foundational building blocks for a broad spectrum of ML tasks, empowering researchers and practitioners to unlock new capabilities in language understanding and processing. In recent years, word embeddings have become indispensable tools for natural language processing tasks, offering compact representations of textual data that capture semantic relationships between words. Biases embedded in the training data can be perpetuated in word embeddings, leading to unfair associations and stereotypes. Addressing these challenges requires interdisciplinary efforts from researchers in machine learning, natural language processing, and ethics. Strategies such as debiasing techniques, dimensionality reduction methods, and transparency-enhancing approaches are being actively explored to mitigate these challenges and improve the reliability and fairness of word embeddings in practical applications.

The suggested procedure involves integrating the proximity function null P as a mechanism to guide exploration on the space of walks. While the use of null P that we have explained has focused on classification and metric analysis, its geometric interpretation and ability to quantify similarity between walks suggest a broader applicability, particularly in settings where the reward landscape is sparse or the graph structure is too large for exhaustive exploration. We propose an improvement to the exploration strategy used in reinforcement learning algorithms that incrementally construct walks within graph-based environments. Traditionally, these algorithms alternate between exploitation (choosing the next node to maximize an estimated reward) and exploration (randomly selecting a new node). The novelty lies in replacing random exploration with a proximity-guided strategy using a function null P . Instead of sampling uniformly, the agent compares potential path extensions to a reference set of high-reward walks, prioritizing those that are most similar in structure. This approach introduces a more informed, data-driven method for exploration, focusing on areas of the graph that resemble previously successful trajectories.

To address this shortcoming, we introduce'Graph Structured Prediction Energy Networks,' for which we develop inference techniques that allow to both model explicit local and implicit

Structured prediction energy networks (SPENs) (Belanger & McCallum, 2016) are a type of energy-based model (LeCun et al., 2006) in which inference is done by gradient descent.