Related papers: From Quantitative Spatial Operator to Qualitative …
Neural networks encode inputs as high-dimensional vectors, known as representations, that capture how models process data by encoding task-relevant structure and semantics. Representation alignment refers to the degree to which different…
In order to gain a better understanding of the state space of programs, with the aim of making their verification more tractable, models based on directed topological spaces have been introduced, allowing to take in account equivalence…
The prevailing approach to improving large language model (LLM) reasoning has centered on expanding context windows, implicitly assuming that more tokens yield better performance. However, empirical evidence - including the "lost in the…
Compilers use control flow graph (CFG) representations of low-level programs because they are suited to program analysis and optimizations. However, formalizing the behavior and metatheory of CFG programs is non-trivial: CFG programs don't…
The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological…
A proper scene representation is central to the pursuit of spatial intelligence where agents can robustly reconstruct and efficiently understand 3D scenes. A scene representation is either metric, such as landmark maps in 3D reconstruction,…
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a…
Understanding dataset complexity is fundamental to evaluating and comparing link prediction models on knowledge graphs (KGs). While the Cumulative Spectral Gradient (CSG) metric, derived from probabilistic divergence between classes within…
We apply a recent classification of topological action terms to Composite Higgs models based on a variety of coset spaces $G/H$ and discuss their phenomenology. The topological terms, which can all be obtained by integrating (possibly only…
Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. We introduce a hypergraph-based formalism, Blocks & Bonds (B&B), that generalizes…
We investigate topology change in 3D. Using Morse theory and handle decomposition we find the set of elementary cobordisms for 3-manifolds. These are: (i) \O <-> S^2; (ii) \Sigma_g <-> \Sigma_{g+1}; (iii) \Sigma_{g_1} \sqcup \Sigma_{g_2}…
The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators,…
Knowledge Graphs (KGs) are composed of structured information about a particular domain in the form of entities and relations. In addition to the structured information KGs help in facilitating interconnectivity and interoperability between…
In the present paper, we propose the model of {\it structural information learning machines} (SiLeM for short), leading to a mathematical definition of learning by merging the theories of computation and information. Our model shows that…
Generative modeling typically seeks the path of least action via deterministic flows (ODE). While effective for in-distribution tasks, we argue that these deterministic paths become brittle under causal interventions, which often require…
The configuration space network (CSN) of a dynamical system is an effective approach to represent the ensemble of configurations sampled during a simulation and their dynamic connectivity. To elucidate the connection between the CSN…
The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches…
Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism $\mbox{$\bR \rightarrow \bGamma$}$ between two relational…
Complex systems are difficult to study not only because they are nonlinear, multiscale, and often nonstationary, but because their scientifically relevant organization is often invisible at the level of individual components, pairwise…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…