Related papers: NIP and Distal Metric Structures

For a commutative quantale $\mathcal{V}$, the category $\mathcal{V}-cat$ can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor $T$ (formalised as an endofunctor on sets) can be…

The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete…

We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure…

This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a…

We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain…

In many high-impact applications, it is important to ensure the quality of output of a machine learning algorithm as well as its reliability in comparison with the complexity of the algorithm used. In this paper, we have initiated a…

A visceral structure on M is given by a definable base for a uniform topology on its universe in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. This context includes o-minimal ordered…

Modeling relations between components of 3D objects is essential for many geometry editing tasks. Existing techniques commonly rely on labeled components, which requires substantial annotation effort and limits components to a dictionary of…

While the universal approximation property holds both for hierarchical and shallow networks, we prove that deep (hierarchical) networks can approximate the class of compositional functions with the same accuracy as shallow networks but with…

In the field of Machine Learning Interatomic Potentials (MLIPs), understanding the intricate relationship between data biases, specifically conformational and structural diversity, and model generalization is critical in improving the…

We investigate the VC-dimension of the perceptron and simple two-layer networks like the committee- and the parity-machine with weights restricted to values $\pm1$. For binary inputs, the VC-dimension is determined by atypical pattern sets,…

Using thermodynamic relations and dimensional analysis we derive a general formula for the thermodynamical trace $2\mathcal{E}-DP$ for non-relativistic systems and $\mathcal{E-DP}$ for relativistic systems, where $D$ is the number of…

We propose a general framework called Network Dissection for quantifying the interpretability of latent representations of CNNs by evaluating the alignment between individual hidden units and a set of semantic concepts. Given any CNN model,…

Neural network architectures have been extensively employed in the fair representation learning setting, where the objective is to learn a new representation for a given vector which is independent of sensitive information. Various…

We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a…

We prove an existence and uniqueness theorem for fixed points of contraction maps in the framework of quantum metric spaces, where distinguishability is defined by the $L^2$ norm: $d_Q(\psi_1,\psi_2) = \|\psi_1 - \psi_2\|$. The result…

Neeman shows that the completion of a triangulated category with respect to a good metric yields a triangulated category. We compute completions of discrete cluster categories with respect to metrics induced by internal t-structures. In…

In this paper we combine the notions of partial metric spaces with negative distances, $G_p$-metric spaces and n-metric spaces together into one structure called the partial n-metric spaces. These are generalizations of all the said…

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of…

Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…