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Given samples from an unknown discrete distribution p, is it possible to distinguish whether p belongs to some class of distributions C versus p being far from every distribution in C? This fundamental question has received tremendous attention in statistics, focusing primarily on asymptotic analysis, as well as in information theory and theoretical computer science, where the emphasis has been on small sample size and computational complexity. Nevertheless, even for basic properties of discrete distributions such as monotonicity, independence, log-concavity, unimodality, and monotone-hazard rate, the optimal sample complexity is unknown. We provide a general approach via which we obtain sample-optimal and computationally efficient testers for all these distribution families.

Learning the number of clusters is a key problem in data clustering. We present dip-means, a novel robust incremental method to learn the number of data clusters that can be used as a wrapper around any iterative clustering algorithm of k-means family. In contrast to many popular methods which make assumptions about the underlying cluster distributions, dip-means only assumes a fundamental cluster property: each cluster to admit a unimodal distribution. The proposed algorithm considers each cluster member as an individual'viewer' and applies a univariate statistic hypothesis test for unimodality (dip-test) on the distribution of distances between the viewer and the cluster members. Important advantages are: i) the unimodality test is applied on univariate distance vectors, ii) it can be directly applied with kernel-based methods, since only the pairwise distances are involved in the computations. Experimental results on artificial and real datasets indicate the effectiveness of our method and its superiority over analogous approaches.

Given samples from an unknown discrete distribution p, is it possible to distinguish whether p belongs to some class of distributions C versus p being far from every distribution in C? This fundamental question has received tremendous attention in statistics, focusing primarily on asymptotic analysis, as well as in information theory and theoretical computer science, where the emphasis has been on small sample size and computational complexity. Nevertheless, even for basic properties of discrete distributions such as monotonicity, independence, logconcavity, unimodality, and monotone-hazard rate, the optimal sample complexity is unknown. We provide a general approach via which we obtain sample-optimal and computationally efficient testers for all these distribution families.

Over the last decade, the Dip-test of unimodality has gained increasing interest in the data mining community as it is a parameter-free statistical test that reliably rates the modality in one-dimensional samples. It returns a so called Dip-value and a corresponding probability for the sample's unimodality (Dip-p-value). These two values share a sigmoidal relationship. However, the specific transformation is dependent on the sample size. Many Dip-based clustering algorithms use bootstrapped look-up tables translating Dip- to Dip-p-values for a certain limited amount of sample sizes. We propose a specifically designed sigmoid function as a substitute for these state-of-the-art look-up tables. This accelerates computation and provides an approximation of the Dip- to Dip-p-value transformation for every single sample size. Further, it is differentiable and can therefore easily be integrated in learning schemes using gradient descent. We showcase this by exploiting our function in a novel subspace clustering algorithm called Dip'n'Sub. We highlight in extensive experiments the various benefits of our proposal.

Unimodality, pivotal in statistical analysis, offers insights into dataset structures and drives sophisticated analytical procedures. While unimodality's confirmation is straightforward for one-dimensional data using methods like Silverman's approach and Hartigans' dip statistic, its generalization to higher dimensions remains challenging. By extrapolating one-dimensional unimodality principles to multi-dimensional spaces through linear random projections and leveraging point-to-point distancing, our method, rooted in $\alpha$-unimodality assumptions, presents a novel multivariate unimodality test named mud-pod. Both theoretical and empirical studies confirm the efficacy of our method in unimodality assessment of multidimensional datasets as well as in estimating the number of clusters.

Clustering algorithms are useful tools for data mining, compression, probability density es- timation, and many other important tasks. However, most clustering algorithms require the user to specify the number of clusters (called k), and it is not always clear what is the best value for k. Figure 1 shows examples where k has been improperly chosen. Choosing k is often an ad hoc decision based on prior knowledge, assumptions, and practical experience. Choosing k is made more difficult when the data has many dimensions, even when clusters are well-separated. Center-based clustering algorithms (in particular k-means and Gaussian expectation- maximization) usually assume that each cluster adheres to a unimodal distribution, such as Gaussian. With these methods, only one center should be used to model each subset of data that follows a unimodal distribution. If multiple centers are used to describe data drawn from one mode, the centers are a needlessly complex description of the data, and in fact the multiple centers capture the truth about the subset less well than one center. In this paper we present a simple algorithm called G-means that discovers an appropriate k using a statistical test for deciding whether to split a k-means center into two centers.