Limits of Dense Simplicial Complexes
Authors: T. Mitchell Roddenberry, Santiago Segarra
JMLR 2023 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | We develop a theory of limits for sequences of dense abstract simplicial complexes, where a sequence is considered convergent if its homomorphism densities converge. The limiting objects are represented by stacks of measurable [0, 1]-valued functions on unit cubes of increasing dimension, each corresponding to a dimension of the abstract simplicial complex. We show that convergence in homomorphism density implies convergence in a cut-metric, and vice versa, as well as showing that simplicial complexes sampled from the limit objects closely resemble its structure. Applying this framework, we also partially characterize the convergence of nonuniform hypergraphs. |
| Researcher Affiliation | Academia | T. Mitchell Roddenberry EMAIL Department of Electrical and Computer Engineering Rice University Houston, TX 77005-1827, USA Santiago Segarra EMAIL Department of Electrical and Computer Engineering Rice University Houston, TX 77005-1827, USA |
| Pseudocode | No | The paper does not contain any structured pseudocode or algorithm blocks. It focuses on theoretical derivations, proofs, and definitions related to the theory of limits for sequences of dense abstract simplicial complexes. |
| Open Source Code | No | The paper does not provide any concrete access to source code, explicit code release statements, or links to repositories for the methodology described. |
| Open Datasets | No | The paper does not provide concrete access information for any publicly available or open datasets. It uses conceptual examples like 'ˇCech complexes from a bouquet of circles in R2' and 'a finite set of n points Xn = {X1, . . . , Xn} in X sampled i.i.d. according to the probability measure µ' for theoretical illustration, not for experimental data analysis. |
| Dataset Splits | No | The paper does not describe any experimental setup involving datasets, and therefore no information about training/test/validation dataset splits is provided. |
| Hardware Specification | No | The paper does not describe any experimental procedures or results that would require specific hardware. It is a theoretical work. |
| Software Dependencies | No | The paper does not describe any computational implementations or experiments, and therefore no specific software dependencies with version numbers are mentioned. |
| Experiment Setup | No | The paper does not describe any experiments, training processes, or models that would require details on experimental setup, hyperparameters, or system-level training settings. The work is purely theoretical. |