Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, while a computed perturbation moves a barcode by more than $δ$, so the factor cannot be dropped. We apply this to quantitative equational theories, whose free algebras are metric spaces built from syntax: an inclusion of theories induces a morphism of the presenting monads and a comparison of barcodes with an explicit bound, so the invariant measures axiomatic strength. Four examples are computed, one in every degree.
Denis Mayr Lima Martins, Gottfried Vossencs.DB cs.LG
Self-Organizing Maps (SOMs) have long been used as exploratory tools for high-dimensional data: they organize objects into a two-dimensional topology that reveals clusters, gradients, sparse regions, dense regions, and boundaries. Yet, in modern data systems, SOMs are typically trained and visualized outside the DBMS, disconnected from the relational data they summarize. We introduce the abstraction of a queryable data map: a learned topological artifact consisting of representatives, neighborhood relations, object assignments, and derived summaries. We instantiate this idea with MapDB, a lightweight prototype that makes SOM artifacts queryable so users can explore data topology without leaving the database. Experimental study shows that SOM training is feasible at moderate analytical scale, that map queries are interactive after materialization, and that SOM regions provide meaningful targets for exploratory SQL.
Adam Wesołowski, Dimitrios Thanos, Daniel Leykam +1quant-ph cs.LG
Many datasets encountered across a wide range of domains possess rich geometric and topological structure that is difficult to capture using conventional vector-based representations. Quantum machine learning offers the possibility of processing high-dimensional data in Hilbert spaces, but its practical success depends critically on how classical data is encoded into quantum states. We introduce \emph{quantum topological data encoding} (QTDE), a general framework for encoding topological information into quantum states via topology-driven quantum evolution. Our method generalises an existing topology-driven quantum encoding framework to higher-dimensional data. We test the proposed method on clique-complexes classification tasks, and provide preliminary evidence that topology-driven quantum representations can capture discriminative information beyond that available through direct comparisons of classical topological descriptors. The proposed quantum representations consistently outperform a baseline based on direct comparisons of the combinatorial Laplacians describing the underlying topological structure. We indicate several areas of application where the framework can be used to provide a more efficient and reliable data representation.