In this paper, we have derived a set of sufficient conditions for reliable clustering of data produced by Bernoulli Mixture Models (BMM), when the number of clusters is unknown. A BMM refers to a random binary vector whose components are independent Bernoulli trials with cluster-specific frequencies. The problem of clustering BMM data arises in many real-world applications, most notably in population genetics where researchers aim at inferring the population structure from multilocus genotype data. Our findings stipulate a minimum dataset size and a minimum number of Bernoulli trials (or genotyped loci) per sample, such that the existence of a clustering algorithm with a sufficient accuracy is guaranteed. Moreover, the mathematical intuitions and tools behind our work can help researchers in designing more effective and theoretically-plausible heuristic methods for similar problems.

The methodology of community detection can be divided into two principles: imposing a network model on a given graph, or optimizing a designed objective function. The former provides guarantees on theoretical detectability but falls short when the graph is inconsistent with the underlying model. The latter is model-free but fails to provide quality assurance for the detected communities. In this paper, we propose a novel unified framework to combine the advantages of these two principles. The presented method, SGC-GEN, not only considers the detection error caused by the corresponding model mismatch to a given graph, but also yields a theoretical guarantee on community detectability by analyzing Spectral Graph Clustering (SGC) under GENerative community models (GCMs). SGC-GEN incorporates the predictability on correct community detection with a measure of community fitness to GCMs. It resembles the formulation of supervised learning problems by enabling various community detection loss functions and model mismatch metrics. We further establish a theoretical condition for correct community detection using the normalized graph Laplacian matrix under a GCM, which provides a novel data-driven loss function for SGC-GEN. In addition, we present an effective algorithm to implement SGC-GEN, and show that the computational complexity of SGC-GEN is comparable to the baseline methods. Our experiments on 18 real-world datasets demonstrate that SGC-GEN possesses superior and robust performance compared to 6 baseline methods under 7 representative clustering metrics.

Plain vanilla K-means clustering is prone to produce unbalanced clusters and suffers from outlier sensitivity. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, which assigns a prescribed number of datapoints to an auxiliary outlier cluster and performs cardinality-constrained K-means clustering on the residual dataset. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes that transform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions are optimal in the MILP if a cluster separation condition holds.

Inferring predictive maps between multiple input and multiple output variables or tasks has innumerable applications in data science. Multi-task learning attempts to learn the maps to several output tasks simultaneously with information sharing between them. We propose a novel multi-task learning framework for sparse linear regression, where a full task hierarchy is automatically inferred from the data, with the assumption that the task parameters follow a hierarchical tree structure. The leaves of the tree are the parameters for individual tasks, and the root is the global model that approximates all the tasks. We apply the proposed approach to develop and evaluate: (a) predictive models of plant traits using large-scale and automated remote sensing data, and (b) GWAS methodologies mapping such derived phenotypes in lieu of hand-measured traits. We demonstrate the superior performance of our approach compared to other methods, as well as the usefulness of discovering hierarchical groupings between tasks. Our results suggest that richer genetic mapping can indeed be obtained from the remote sensing data. In addition, our discovered groupings reveal interesting insights from a plant science perspective.

Clustering is a type of Unsupervised learning. This is very often used when you don't have labeled data. K-Means Clustering is one of the popular clustering algorithm. The goal of this algorithm is to find groups(clusters) in the given data. In this post we will implement K-Means algorithm using Python from scratch.

Imagine you have some data-related problem that you want to solve. You have heard of all the amazing things that machine learning algorithms can achieve and want to try it for yourself -- but you have no prior experience or knowledge in this area. You start googling some terms like "machine learning models" and "machine learning methodologies," but after some time, you find yourself ready to give up, completely lost somewhere between the different algorithms. Hold on, you brave guy! Lucky for you, I'm going to walk you through three major categories of machine learning algorithms that will allow you to solve confidently a large range of data science problems. In the following post, we are going to talk about decision trees, clustering algorithms, and regression, point out their differences, and figure out how to choose the most suitable model for your case.

Efficient algorithms for $k$-means clustering frequently converge to suboptimal partitions, and given a partition, it is difficult to detect $k$-means optimality. In this paper, we develop an a posteriori certifier of approximate optimality for $k$-means clustering. The certifier is a sub-linear Monte Carlo algorithm based on Peng and Wei's semidefinite relaxation of $k$-means. In particular, solving the relaxation for small random samples of the dataset produces a high-confidence lower bound on the $k$-means objective, and being sub-linear, our algorithm is faster than $k$-means++ when the number of data points is large. We illustrate the performance of our algorithm with both numerical experiments and a performance guarantee: If the data points are drawn independently from any mixture of two Gaussians over $\mathbb{R}^m$ with identity covariance, then with probability $1-O(1/m)$, our $\operatorname{poly}(m)$-time algorithm produces a 3-approximation certificate with 99% confidence.

Clustering is a central approach for unsupervised learning. After clustering is applied, the most fundamental analysis is to quantitatively compare clusterings. Such comparisons are crucial for the evaluation of clustering methods as well as other tasks such as consensus clustering. It is often argued that, in order to establish a baseline, clustering similarity should be assessed in the context of a random ensemble of clusterings. The prevailing assumption for the random clustering ensemble is the permutation model in which the number and sizes of clusters are fixed. However, this assumption does not necessarily hold in practice; for example, multiple runs of K-means clustering returns clusterings with a fixed number of clusters, while the cluster size distribution varies greatly. Here, we derive corrected variants of two clustering similarity measures (the Rand index and Mutual Information) in the context of two random clustering ensembles in which the number and sizes of clusters vary. In addition, we study the impact of one-sided comparisons in the scenario with a reference clustering. The consequences of different random models are illustrated using synthetic examples, handwriting recognition, and gene expression data. We demonstrate that the choice of random model can have a drastic impact on the ranking of similar clustering pairs, and the evaluation of a clustering method with respect to a random baseline; thus, the choice of random clustering model should be carefully justified.

In an effort to overcome the data deluge in computational biology and bioinformatics and to facilitate bioinformatics research in the era of big data, we identify some of the most influential algorithms that have been widely used in the bioinformatics community. These top data mining and machine learning algorithms cover classification, clustering, regression, graphical model-based learning, and dimensionality reduction. The goal of this study is to guide the focus of scalable computing experts in the endeavor of applying new storage and scalable computation designs to bioinformatics algorithms that merit their attention most, following the engineering maxim of "optimize the common case".

A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal multiplicative update rules to solve the subspace clustering problem. In nonlinear orthogonal NMF framework, we propose two subspace clustering algorithms, named kernel-based non-negative subspace clustering KNSC-Ncut and KNSC-Rcut and establish their connection with spectral normalized cut and ratio cut clustering. We further extend the nonlinear orthogonal NMF framework and introduce a graph regularization to obtain a factorization that respects a local geometric structure of the data after the nonlinear mapping. The proposed NMF-based approach to subspace clustering takes into account the nonlinear nature of the manifold, as well as its intrinsic local geometry, which considerably improves the clustering performance when compared to the several recently proposed state-of-the-art methods.