## Rough Notes - Kleinberg (2022): There is no ***consistent*** method of clustering with partitions - Rich (all partitions realizable) - Scale-Invariant - Intra/inter-cluster distance transformation invariant - [mathematical statistics - Clustering -- Intuition behind Kleinberg's Impossibility Theorem - Cross Validated (stackexchange.com)](https://stats.stackexchange.com/questions/173313/clustering-intuition-behind-kleinbergs-impossibility-theorem) - [nips15.pdf (cornell.edu)](https://www.cs.cornell.edu/home/kleinber/nips15.pdf) - Carlsson Memoli (2008) There's a unique [[Functor|functorial]] method (single-linkage clustering) for assigning a hierarchical clustering to a metric space - Hierarchical Clustering - Category of [[Metric Space|metric spaces]], morphisms are the non-expansive maps $d(f(x), f(y)) \leq d(x,y)$ - ie you can decrease the distances as part of data processing but you shouldn't be able to increase distances - dendrograms form a hierarchical partition (category of ultrametric spaces). Single Linkage (SL) is a unique functor from Metrics spaces to ultra-metric spaces - Extend hierarchy of partitions to hierarchy of covers (Sieves) (Culbertson, Guralnik, Stiller 2016) ## Resources - [Jared Culbertson: "Applying Categorical Thinking to Practical Domains" - YouTube](https://www.youtube.com/watch?v=FukyaV27_S0) --- - Links: [[Principal component analysis|PCA]] [[t-Distributed Stochastic Neighbor Embeddings|t-SNE]] [[UMAP]] [[Variational Auto Encoders]] - Created at: 2023-06-22