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1. Near-optimal Algorithms for Explainable k-Medians and k-Means

   Makarychev, Konstantin ; Shan, Liren ( July 2021 , International Conference on Machine Learning)

   null (Ed.)

   We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with k-medians with ℓ1 norm and O(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for k-medians with ℓ2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ2 norm. 

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2. Explainable k-means: don’t be greedy, plant bigger trees!

   https://doi.org/10.1145/3519935.3520056  

   Makarychev, Konstantin ; Shan, Liren ( June 2022 , STOC 2022)

   We provide a new bi-criteria O(log2k) competitive algorithm for explainable k-means clustering. Explainable k-means was recently introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). It is described by an easy to interpret and understand (threshold) decision tree or diagram. The cost of the explainable k-means clustering equals to the sum of costs of its clusters; and the cost of each cluster equals the sum of squared distances from the points in the cluster to the center of that cluster. The best non bi-criteria algorithm for explainable clustering O(k) competitive, and this bound is tight. Our randomized bi-criteria algorithm constructs a threshold decision tree that partitions the data set into (1+δ)k clusters (where δ∈(0,1) is a parameter of the algorithm). The cost of this clustering is at most O(1/δ⋅log2k) times the cost of the optimal unconstrained k-means clustering. We show that this bound is almost optimal. 

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3. Performance of Johnson-Lindenstrauss transform for k-means and k-medians clustering

   https://doi.org/10.1145/3313276.3316350  

   Makarychev, Konstantin ; Makarychev, Yury ; Razenshteyn, Ilya ( June 2019 , Performance of Johnson-Lindenstrauss transform for k-means and k-medians clustering)

   Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1+ε) under a projection onto a random O(log(k /ε) / ε2)-dimensional subspace. Further, the cost of every clustering is preserved within (1+ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p. For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan. 

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4. Using Event Log Timing Information to Assist Process Scenario Discoveries

   https://doi.org/10.1109/AIKE48582.2020.00017  

   Zhang, Zhenyu ; Guo, Chunhui ; Peng, Wenyu ; Ren, Shangping ( December 2020 , 2020 IEEE Third International Conference on Artificial Intelligence and Knowledge Engineering (AIKE))

   Event logs contain abundant information, such as activity names, time stamps, activity executors, etc. However, much of existing trace clustering research has been focused on applying activity names to assist process scenarios discovery. In addition, many existing trace clustering algorithms commonly used in the literature, such as k-means clustering approach, require prior knowledge about the number of process scenarios existed in the log, which sometimes are not known aprior. This paper presents a two-phase approach that obtains timing information from event logs and uses the information to assist process scenario discoveries without requiring any prior knowledge about process scenarios. We use five real-life event logs to compare the performance of the proposed two-phase approach for process scenario discoveries with the commonly used k-means clustering approach in terms of model’s harmonic mean of the weighted average fitness and precision, i.e., the F1 score. The experiment data shows that (1) the process scenario models obtained with the additional timing information have both higher fitness and precision scores than the models obtained without the timing information; (2) the two-phase approach not only removes the need for prior information related to k, but also results in a comparable F1 score compared to the optimal k-means approach with the optimal k obtained through exhaustive search. 

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5. A near-tight lower bound on the density of forward sampling schemes

   https://doi.org/10.1093/bioinformatics/btae736  

   Kille, Bryce ; Groot_Koerkamp, Ragnar ; McAdams, Drake ; Liu, Alan ; Treangen, Todd_J ; Ponty, ed., Yann ( December 2024 , Bioinformatics)

   Sampling k-mers is a ubiquitous task in sequence analysis algorithms. Sampling schemes such as the often-used random minimizer scheme are particularly appealing as they guarantee at least one k-mer is selected out of every w consecutive k-mers. Sampling fewer k-mers often leads to an increase in efficiency of downstream methods. Thus, developing schemes that have low density, i.e. have a small proportion of sampled k-mers, is an active area of research. After over a decade of consistent efforts in both decreasing the density of practical schemes and increasing the lower bound on the best possible density, there is still a large gap between the two.

   We prove a near-tight lower bound on the density of forward sampling schemes, a class of schemes that generalizes minimizer schemes. For small w and k, we observe that our bound is tight when k≡1(mod w). For large w and k, the bound can be approximated by 1w+k⌈w+kw⌉. Importantly, our lower bound implies that existing schemes are much closer to achieving optimal density than previously known. For example, with the current default minimap2 HiFi settings w = 19 and k = 19, we show that the best known scheme for these parameters, the double decycling-set-based minimizer of Pellow et al. is at most 3% denser than optimal, compared to the previous gap of at most 50%. Furthermore, when k≡1(mod w) and the alphabet size σ goes to ∞, we show that mod-minimizers introduced by Groot Koerkamp and Pibiri achieve optimal density matching our lower bound.

   Minimizer implementations: github.com/RagnarGrootKoerkamp/minimizers ILP and analysis: github.com/treangenlab/sampling-scheme-analysis.

    

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