Related papers: Final Topology for Preference Spaces
Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of Social Choice Theory, wherein…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
In a metric space, such as the real numbers with their standard metric, a set A is open if and only if no sequence with terms outside of A has a limit inside A. Moreover, a metric space is compact if and only if every sequence has a…
We consider a model of stable edge sets (``matchings'') in a bipartite graph $G=(V,E)$ in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistency, substitutability…
In this paper, we establish a mathematical duality between utility transforms and probability distortions. These transforms play a central role in decision under risk by forming the foundation for the classic theories of expected utility,…
Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly…
We first consider interval partitions whose complements are Lebesgue-null and introduce a complete metric that induces the same topology as the Hausdorff distance (between complements). This is done using correspondences between intervals.…
Many-to-many matching with contracts is studied in the framework of revealed preferences. All preferences are described by choice functions that satisfy natural conditions. Under a no-externality assumption individual preferences can be…
Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations)…
The article is devoted to a structure of topological spaces related with topological quasigroups. Regular and complete spaces over topological quasigroups are studied. Separations and embeddings are also investigated for them. Their…
We study tightness properties and selective versions of separability in bitopological function spaces endowed with set-open topologies.
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals. Unfortunately, continuous attractors suffer from severe structural instability…
Persistent Homology is a widely used topological data analysis tool that creates a concise description of the topological properties of a point cloud based on a specified filtration. Most filtrations used for persistent homology depend…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
Soft uniform structures provide a way to speak about uniform closeness in a parameterized setting. Working over a fixed parameter set, we treat entourages as soft relations and introduce a notion of \emph{soft uniformity} whose axioms…
The concept of typed topological space is introduced, for which open sets in a topology on a finite set will be assigned types (from lattice). The neighborhood system of a point, the closure and the connectedness can be defined according to…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
We show that probabilistic equivalence of a regret-based preference relationship over random variables is implied by a weak form of continuity and monotonicity.
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…