Data ultrametricity and clusterability

宽间隙与聚类公理

Wide Gaps and Clustering Axioms

针对k-means违反Kleinberg一致性公理的问题，本文提出变分与残差k-可分性两种可聚类性条件，证明在此条件下k-means满足一致性，并通过算法扩展近似满足丰富性公理，从而调和二者关系。

Computational Feasibility of Clustering under Clusterability Assumptions

It is well known that most of the common clustering objectives are NP-hard to optimize. In practice, however, clustering is being routinely carried out. One approach for providing theoretical understanding of this seeming discrepancy is to come up with notions of clusterability that distinguish realistically interesting input data from worst-case data sets. The hope is that there will be clustering algorithms that are provably efficient on such 'clusterable' instances. In other words, hope that "Clustering is difficult only when it does not matter" (CDNM thesis, for short). We believe that to …

Scaling Hierarchical Agglomerative Clustering to Billion-sized Datasets

Hierarchical Agglomerative Clustering (HAC) is one of the oldest but still most widely used clustering methods. However, HAC is notoriously hard to scale to large data sets as the underlying complexity is at least quadratic in the number of data points and many algorithms to solve HAC are inherently sequential. In this paper, we propose {Reciprocal Agglomerative Clustering (RAC)}, a distributed algorithm for HAC, that uses a novel strategy to efficiently merge clusters in parallel. We prove theoretically that RAC recovers the exact solution of HAC. Furthermore, under clusterability and balance…

Reliable Clustering of Bernoulli Mixture Models

A Bernoulli Mixture Model (BMM) is a finite mixture of random binary vectors with independent dimensions. The problem of clustering BMM data arises in a variety of real-world applications, ranging from population genetics to activity analysis in social networks. In this paper, we analyze the clusterability of BMMs from a theoretical perspective, when the number of clusters is unknown. In particular, we stipulate a set of conditions on the sample complexity and dimension of the model in order to guarantee the Probably Approximately Correct (PAC)-clusterability of a dataset. To the best of our k…

Distribution free optimality intervals for clustering

We address the problem of validating the ouput of clustering algorithms. Given data $\mathcal{D}$ and a partition $\mathcal{C}$ of these data into $K$ clusters, when can we say that the clusters obtained are correct or meaningful for the data? This paper introduces a paradigm in which a clustering $\mathcal{C}$ is considered meaningful if it is good with respect to a loss function such as the K-means distortion, and stable, i.e. the only good clustering up to small perturbations. Furthermore, we present a generic method to obtain post-inference guarantees of near-optimality and stability for a…

Clustering is Easy When ....What?

It is well known that most of the common clustering objectives are NP-hard to optimize. In practice, however, clustering is being routinely carried out. One approach for providing theoretical understanding of this seeming discrepancy is to come up with notions of clusterability that distinguish realistically interesting input data from worst-case data sets. The hope is that there will be clustering algorithms that are provably efficient on such "clusterable" instances. This paper addresses the thesis that the computational hardness of clustering tasks goes away for inputs that one really cares…

用于无监督学习的自编码UMAP增强聚类

Autoencoded UMAP-Enhanced Clustering for Unsupervised Learning

本研究提出三阶段无监督学习框架AUEC，结合自编码器与UMAP构建可聚类性嵌入，用复合损失训练，在MNIST数据集上聚类准确率超越现有前沿方法。

邻域稳定性作为最近邻搜索性的度量

Neighborhood Stability as a Measure of Nearest Neighbor Searchability

本文提出两种度量方法，用于评估数据集是否可通过基于聚类的近似最近邻搜索进行搜索，且不依赖特定距离函数。

Probabilistic Partitive Partitioning (PPP)

Clustering is a NP-hard problem. Thus, no optimal algorithm exists, heuristics are applied to cluster the data. Heuristics can be very resource-intensive, if not applied properly. For substantially large data sets computational efficiencies can be achieved by reducing the input space if a minimal loss of information can be achieved. Clustering algorithms, in general, face two common problems: 1) these converge to different settings with different initial conditions and; 2) the number of clusters has to be arbitrarily decided beforehand. This problem has become critical in the realm of big data…