In this paper, we propose PCKID, a novel, robust, kernel function for spectral clustering, specifically designed to handle incomplete data. By combining posterior distributions of Gaussian Mixture Models for incomplete data on different scales, we are able to learn a kernel for incomplete data that does not depend on any critical hyperparameters, unlike the commonly used RBF kernel. To evaluate our method, we perform experiments on two real datasets. PCKID outperforms the baseline methods for all fractions of missing values and in some cases outperforms the baseline methods with up to 25 percentage points.

Clustering is the grouping of objects together so that objects belonging in the same group (cluster) are more similar to each other than those in other groups (clusters). In this intro cluster analysis tutorial, we'll check out a few algorithms in Python so you can get a basic understanding of the fundamentals of clustering on a real dataset. For the clustering problem, we will use the famous Zachary's Karate Club dataset. The story behind the data set is quite simple: There was a Karate Club that had an administrator "John A" and an instructor "Mr. Then a conflict arose between them, causing the students (Nodes) to split into two groups.

Being an unsupervised machine learning and data mining technique, biclustering and its multimodal extensions are becoming popular tools for analysing object-attribute data in different domains. Apart from conventional clustering techniques, biclustering is searching for homogeneous groups of objects while keeping their common description, e.g., in binary setting, their shared attributes. In bioinformatics, biclustering is used to find genes, which are active in a subset of situations, thus being candidates for biomarkers. However, the authors of those biclustering techniques that are popular in gene expression analysis, may overlook the existing methods. For instance, BiMax algorithm is aimed at finding biclusters, which are well-known for decades as formal concepts. Moreover, even if bioinformatics classify the biclustering methods according to reasonable domain-driven criteria, their classification taxonomies may be different from survey to survey and not full as well. So, in this paper we propose to use concept lattices as a tool for taxonomy building (in the biclustering domain) and attribute exploration as means for cross-domain taxonomy completion.

We introduce a semi-supervised discrete choice model to calibrate discrete choice models when relatively few requests have both choice sets and stated preferences but the majority only have the choice sets. Two classic semi-supervised learning algorithms, the expectation maximization algorithm and the cluster-and-label algorithm, have been adapted to our choice modeling problem setting. We also develop two new algorithms based on the cluster-and-label algorithm. The new algorithms use the Bayesian Information Criterion to evaluate a clustering setting to automatically adjust the number of clusters. Two computational studies including a hotel booking case and a large-scale airline itinerary shopping case are presented to evaluate the prediction accuracy and computational effort of the proposed algorithms. Algorithmic recommendations are rendered under various scenarios.

Subspace clustering has been widely applied to detect meaningful clusters in high-dimensional data spaces. A main challenge in subspace clustering is to quickly calculate a "good" affinity matrix. ℓ 0 , ℓ 1 , ℓ 2 or nuclear norm regularization is used to construct the affinity matrix in many subspace clustering methods because of their theoretical guarantees and empirical success. However, they suffer from the following problems: (1) ℓ 2 and nuclear norm regularization require very strong assumptions to guarantee a subspace-preserving affinity; (2) although ℓ 1 regularization can be guaranteed to give a subspace-preserving affinity under certain conditions, it needs more time to solve a large-scale convex optimization problem; (3) ℓ 0 regularization can yield a tradeoff between computationally efficient and subspace-preserving affinity by using the orthogonal matching pursuit (OMP) algorithm, but this still takes more time to search the solution in OMP when the number of data points is large. In order to overcome these problems, we first propose a learned OMP (LOMP) algorithm to learn a single hidden neural network (SHNN) to fast approximate the ℓ 0 code. We then exploit a sparse subspace clustering method based on ℓ 0 code which is fast computed by SHNN. Two sufficient conditions are presented to guarantee that our method can give a subspace-preserving affinity. Experiments on handwritten digit and face clustering show that our method not only quickly computes the ℓ 0 code, but also outperforms the relevant subspace clustering methods in clustering results. In particular, our method achieves the state-of-the-art clustering accuracy (94.32%) on MNIST.

Feature selection is an important technique in machine learning research. An effective and robust feature selection method is desired to simultaneously identify the informative features and eliminate the noisy ones of data. In this paper, we consider the unsupervised feature selection problem which is particularly difficult as there is not any class labels that would guide the search for relevant features. To solve this, we propose a novel algorithmic framework which performs unsupervised feature selection. Firstly, the proposed framework implements structure learning, where the data structures (including intrinsic distribution structure and the data segment) are found via a combination of the alternative optimization and clustering. Then, both the intrinsic data structure and data segmentation are formulated as regularization terms for discriminant feature selection. The results of the feature selection also affect the structure learning step in the following iterations. By leveraging the interactions between structure learning and feature selection, we are able to capture more accurate structure of data and select more informative features. Clustering and classification experiments on real world image data sets demonstrate the effectiveness of our method.

Multiple kernel $k$-means (MKKM) aims to improve clustering performance by learning an optimal kernel, which is usually assumed to be a linear combination of a group of pre-specified base kernels. However, we observe that this assumption could: i) cause limited kernel representation capability; and ii) not sufficiently consider the negotiation between the process of learning the optimal kernel and that of clustering, leading to unsatisfying clustering performance. To address these issues, we propose an optimal neighborhood kernel clustering (ONKC) algorithm to enhance the representability of the optimal kernel and strengthen the negotiation between kernel learning and clustering. We theoretically justify this ONKC by revealing its connection with existing MKKM algorithms. Furthermore, this justification shows that existing MKKM algorithms can be viewed as a special case of our approach and indicates the extendability of the proposed ONKC for designing better clustering algorithms. An efficient algorithm with proved convergence is designed to solve the resultant optimization problem. Extensive experiments have been conducted to evaluate the clustering performance of the proposed algorithm. As demonstrated, our algorithm significantly outperforms the state-of-the-art ones in the literature, verifying the effectiveness and advantages of ONKC.

Recommender systems have achieved great success in recent years, and matrix approximation (MA) is one of the most popular techniques for collaborative filtering (CF) based recommendation. However, a major issue is that MA methods perform poorly at detecting strong localized associations among closely related users and items. Recently, some MA-based CF methods adopt clustering methods to discover meaningful user-item subgroups and perform ensemble on different clusterings to improve the recommendation accuracy. However, ensemble learning suffers from lower efficiency due to the increased overall computation overhead. In this paper, we propose GLOMA, a new clustering-based matrix approximation method, which can embed global information in local matrix approximation models to improve recommendation accuracy. In GLOMA, a MA model is first trained on the entire data to capture global information. The global MA model is then utilized to guide the training of cluster-based local MA models, such that the local models can detect strong localized associations shared within clusters and at the same time preserve global associations shared among all users/items. Evaluation results using MovieLens and Netflix datasets demonstrate that, by integrating global information in local models, GLOMA can outperform five state-of-the-art MA-based CF methods in recommendation accuracy while achieving descent efficiency.

Non-negative Matrix Factorization (NMF) has attracted much attention and been widely used in real-world applications. As a clustering method, it fails to handle the case where data points lie in a complicated geometry structure. Existing methods adopt single global centroid for each cluster, failing to capture the manifold structure. In this paper, we propose a novel local centroids structured NMF to address this drawback. Instead of using single centroid for each cluster, we introduce multiple local centroids for individual cluster such that the manifold structure can be captured by the local centroids. Such a novel NMF method can improve the clustering performance effectively. Furthermore, a novel bipartite graph is incorporated to obtain the clustering indicator directly without any post process. Experiments on both toy datasets and real-world datasets have verified the effectiveness of the proposed method.

Multi-View Clustering (MVC) has garnered more attention recently since many real-world data are comprised of different representations or views. The key is to explore complementary information to benefit the clustering problem. In this paper, we present a deep matrix factorization framework for MVC, where semi-nonnegative matrix factorization is adopted to learn the hierarchical semantics of multi-view data in a layer-wise fashion. To maximize the mutual information from each view, we enforce the non-negative representation of each view in the final layer to be the same. Furthermore, to respect the intrinsic geometric structure in each view data, graph regularizers are introduced to couple the output representation of deep structures. As a non-trivial contribution, we provide the solution based on alternating minimization strategy, followed by a theoretical proof of convergence. The superior experimental results on three face benchmarks show the effectiveness of the proposed deep matrix factorization model.