{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/representational-renyi-heterogeneity","title":"Representational Rényi heterogeneity","arxiv_id":"1912.05031","date":"2019-12-10","proceeding":null,"authors":["Abraham Nunes","Martin Alda","Timothy Bardouille","Thomas Trappenberg"],"abstract":"A discrete system's heterogeneity is measured by the R\\'enyi heterogeneity family of indices (also known as Hill numbers or Hannah--Kay indices), whose units are {the numbers equivalent}. Unfortunately, numbers equivalent heterogeneity measures for non-categorical data require {a priori} (A) categorical partitioning and (B) pairwise distance measurement on the observable data space, thereby precluding application to problems with ill-defined categories or where semantically relevant features must be learned as abstractions from some data. We thus introduce representational R\\'enyi heterogeneity (RRH), which transforms an observable domain onto a latent space upon which the R\\'enyi heterogeneity is both tractable and semantically relevant. This method requires neither {a priori} binning nor definition of a distance function on the observable space. We show that RRH can generalize existing biodiversity and economic equality indices. Compared with existing indices on a beta-mixture distribution, we show that RRH responds more appropriately to changes in mixture component separation and weighting. Finally, we demonstrate the measurement of RRH in a set of natural images, with respect to abstract representations learned by a deep neural network. The RRH approach will further enable heterogeneity measurement in disciplines whose data do not easily conform to the assumptions of existing indices.","url_abs":"https://arxiv.org/abs/1912.05031v3","url_pdf":"https://arxiv.org/pdf/1912.05031v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"representational-renyi-heterogeneity","repo_url":"https://github.com/abrahamnunes/RRH","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}