{"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/pareto-optimal-data-compression-for-binary","title":"Pareto-optimal data compression for binary classification tasks","arxiv_id":"1908.08961","date":"2019-08-23","proceeding":null,"authors":["Max Tegmark","Tailin Wu"],"abstract":"The goal of lossy data compression is to reduce the storage cost of a data set $X$ while retaining as much information as possible about something ($Y$) that you care about. For example, what aspects of an image $X$ contain the most information about whether it depicts a cat? Mathematically, this corresponds to finding a mapping $X\\to Z\\equiv f(X)$ that maximizes the mutual information $I(Z,Y)$ while the entropy $H(Z)$ is kept below some fixed threshold. We present a method for mapping out the Pareto frontier for classification tasks, reflecting the tradeoff between retained entropy and class information. We first show how a random variable $X$ (an image, say) drawn from a class $Y\\in\\{1,...,n\\}$ can be distilled into a vector $W=f(X)\\in \\mathbb{R}^{n-1}$ losslessly, so that $I(W,Y)=I(X,Y)$; for example, for a binary classification task of cats and dogs, each image $X$ is mapped into a single real number $W$ retaining all information that helps distinguish cats from dogs. For the $n=2$ case of binary classification, we then show how $W$ can be further compressed into a discrete variable $Z=g_\\beta(W)\\in\\{1,...,m_\\beta\\}$ by binning $W$ into $m_\\beta$ bins, in such a way that varying the parameter $\\beta$ sweeps out the full Pareto frontier, solving a generalization of the Discrete Information Bottleneck (DIB) problem. We argue that the most interesting points on this frontier are \"corners\" maximizing $I(Z,Y)$ for a fixed number of bins $m=2,3...$ which can be conveniently be found without multiobjective optimization. We apply this method to the CIFAR-10, MNIST and Fashion-MNIST datasets, illustrating how it can be interpreted as an information-theoretically optimal image clustering algorithm.","url_abs":"https://arxiv.org/abs/1908.08961v2","url_pdf":"https://arxiv.org/pdf/1908.08961v2.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":"pareto-optimal-data-compression-for-binary","repo_url":"https://github.com/tailintalent/distillation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"binary-classification","task_name":"Binary Classification"},{"task_slug":"classification-1","task_name":"Classification"},{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"data-compression","task_name":"Data Compression"},{"task_slug":"classification","task_name":"General Classification"},{"task_slug":"image-clustering","task_name":"Image Clustering"},{"task_slug":"multiobjective-optimization","task_name":"Multiobjective Optimization"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1908.08961","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}