Optimization in large language models (LLMs) unfolds over high-dimensional parameter spaces with non-Euclidean structure. Information geometry frames this landscape using the Fisher information metric, enabling more principled learning via natural gradient descent. Though often impractical, this geometric lens clarifies phenomena such as sharp minima, generalization, and observed scaling laws. We argue that curvature-based approaches deepen our understanding of LLM training. Finally, we speculate on quantum analogies based on the Fubini-Study metric and Quantum Fisher Information, hinting at efficient optimization in quantum-enhanced systems.

Robot localization is a fundamental component of autonomous navigation in unknown environments. Among various sensing modalities, visual input from cameras plays a central role, enabling robots to estimate their position by tracking point features across image frames. However, image frames often contain a large number of features, many of which are redundant or uninformative for localization. Processing all features can introduce significant computational latency and inefficiency. This motivates the need for intelligent feature selection, identifying a subset of features that are most informative for localization over a prediction horizon. In this work, we propose two fast and memory-efficient feature selection algorithms that enable robots to actively evaluate the utility of visual features in real time. Unlike existing approaches with high computational and memory demands, the proposed methods are explicitly designed to reduce both time and memory complexity while achieving a favorable trade-off between computational efficiency and localization accuracy.

The inconsistency issue in the Visual-Inertial Navigation System (VINS) is a long-standing and fundamental challenge. While existing studies primarily attribute the inconsistency to observability mismatch, these analyses are often based on simplified theoretical formulations that consider only prediction and SLAM correction. Such formulations fail to cover the non-standard estimation steps, such as MSCKF correction and delayed initialization, which are critical for practical VINS estimators. Furthermore, the lack of a comprehensive understanding of how inconsistency dynamically emerges across estimation steps has hindered the development of precise and efficient solutions. As a result, current approaches often face a trade-off between estimator accuracy, consistency, and implementation complexity. To address these limitations, this paper proposes a novel analysis framework termed Unobservable Subspace Evolution (USE), which systematically characterizes how the unobservable subspace evolves throughout the entire estimation pipeline by explicitly tracking changes in its evaluation points. This perspective sheds new light on how individual estimation steps contribute to inconsistency. Our analysis reveals that observability misalignment induced by certain steps is the antecedent of observability mismatch. Guided by this insight, we propose a simple yet effective solution paradigm, Unobservable Subspace Alignment (USA), which eliminates inconsistency by selectively intervening only in those estimation steps that induce misalignment. We design two USA methods: transformation-based and re-evaluation-based, both offering accurate and computationally lightweight solutions. Extensive simulations and real-world experiments validate the effectiveness of the proposed methods.

In many areas of systems biology, including virology, pharmacokinetics, and population biology, dynamical systems are commonly used to describe biological processes. These systems can be characterized by estimating their parameters from sampled data. The key problem is how to optimally select sampling points to achieve accurate parameter estimation. Classical approaches often rely on Fisher information matrix-based criteria such as A-, D-, and E-optimality, which require an initial parameter estimate and may yield suboptimal results when the estimate is inaccurate. This study proposes two simulation-based methods for optimal sampling design that do not depend on initial parameter estimates. The first method, E-optimal-ranking (EOR), employs the E-optimal criterion, while the second utilizes a Long Short-Term Memory (LSTM) neural network. Simulation studies based on the Lotka-Volterra and three-compartment models demonstrate that the proposed methods outperform both random selection and classical E-optimal design.

The rapid spread of fake news threatens social stability and public trust, rendering its detection an imperative research priority. Although large language models (LLMs) excel at numerous natural language processing tasks with their remarkable contextual understanding and extensive prior knowledge, the time-bounded knowledge coverage and tendency for generating hallucination content reduce their reliability when handling fast-evolving news streams. Furthermore, models trained on existing static datasets also often lack the generalization needed for emerging news topics. To address these challenges, we propose ZoFia, a novel two-stage zero-shot fake news detection framework. First, we introduce Hierarchical Salience to quantify the importance of entities in the news content, and propose the SC-MMR algorithm to effectively select an informative and diverse set of keywords that serve as queries for retrieving up-to-date external evidence. Subsequently, a multi LLM interactive system, in which each agent assumes a distinct role, performs multi-view collaborative analysis and adversarial debate over the news text and its related information, and finally produces an interpretable and robust judgment. Comprehensive experiments on two public datasets demonstrate that ZoFia obviously outperforms existing zero-shot baselines and most of few-shot methods. Our codes will be open-sourced to facilitate related communities.

Analysing how information flows along the layers of a multilayer perceptron is a topic of paramount importance in the field of artificial neural networks. After framing the problem from the point of view of information theory, in this position article a specific investigation is conducted on the way information is processed, with particular reference to the requirements imposed by supervised learning. To this end, the concept of information matrix is devised and then used as formal framework for understanding the aetiology of optimisation strategies and for studying the information flow. The underlying research for this article has also produced several key outcomes: i) the definition of a parametric optimisation strategy, ii) the finding that the optimisation strategy proposed in the information bottleneck framework shares strong similarities with the one derived from the information matrix, and iii) the insight that a multilayer perceptron serves as a kind of "adaptor", meant to process the input according to the given objective.

Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean, $α$-$z$, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter $α$, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter $α$ even though the original quantities do not. Additionally, I derive and establish basic properties of $α$-$z$ information matrices resulting from the $α$-$z$ Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their $α$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

We consider the problem of minimizing a functional over a parametric family of probability measures, where the parameterization is characterized via a push-forward structure. An important application of this problem is in training generative adversarial networks.