Online learning algorithms have become a ubiquitous tool in the machine learning toolbox and are frequently used in small, resource-constraint environments. Among the most successful online learning methods are Decision Tree (DT) ensembles. DT ensembles provide excellent performance while adapting to changes in the data, but they are not resource efficient. Incremental tree learners keep adding new nodes to the tree but never remove old ones increasing the memory consumption over time. Gradient-based tree learning, on the other hand, requires the computation of gradients over the entire tree which is costly for even moderately sized trees. In this paper, we propose a novel memory-efficient online classification ensemble called shrub ensembles for resource-constraint systems. Our algorithm trains small to medium-sized decision trees on small windows and uses stochastic proximal gradient descent to learn the ensemble weights of these `shrubs'. We provide a theoretical analysis of our algorithm and include an extensive discussion on the behavior of our approach in the online setting. In a series of 2~959 experiments on 12 different datasets, we compare our method against 8 state-of-the-art methods. Our Shrub Ensembles retain an excellent performance even when only little memory is available. We show that SE offers a better accuracy-memory trade-off in 7 of 12 cases, while having a statistically significant better performance than most other methods. Our implementation is available under https://github.com/sbuschjaeger/se-online .

Abstract Matrix tri-factorization subject to binary constraints is a versatile and powerful framework for the simultaneous clustering of observations and features, also known as biclustering. Applications for biclustering encompass the clustering of high-dimensional data and explorative data mining, where the selection of the most important features is relevant. Unfortunately, due to the lack of suitable methods for the optimization subject to binary constraints, the powerful framework of biclustering is typically constrained to clusterings which partition the set of observations or features. As a result, overlap between clusters cannot be modelled and every item, even outliers in the data, have to be assigned to exactly one cluster. In this paper we propose Broccoli , an optimization scheme for matrix factorization subject to binary constraints, which is based on the theoretically well-founded optimization scheme of proximal stochastic gradient descent. Thereby, we do not impose any restrictions on the obtained clusters. Our experimental evaluation, performed on both synthetic and real-world data, and against 6 competitor algorithms, show reliable and competitive performance, even in presence of a high amount of noise in the data. Moreover, a qualitative analysis of the identified clusters shows that Broccoli may provide meaningful and interpretable clustering structures.

When it comes to clustering nonconvex shapes, two paradigms are used to find the most suitable clustering: minimum cut and maximum density. The most popular algorithms incorporating these paradigms are Spectral Clustering and DBSCAN. Both paradigms have their pros and cons. While minimum cut clusterings are sensitive to noise, density-based clusterings have trouble handling clusters with varying densities. In this paper, we propose SPECTACL: a method combining the advantages of both approaches, while solving the two mentioned drawbacks. Our method is easy to implement, such as Spectral Clustering, and theoretically founded to optimize a proposed density criterion of clusterings. Through experiments on synthetic and real-world data, we demonstrate that our approach provides robust and reliable clusterings.

Boolean matrix factorization (BMF) is a popular and powerful technique for inferring knowledge from data. The mining result is the Boolean product of two matrices, approximating the input dataset. The Boolean product is a disjunction of rank-1 binary matrices, each describing a feature-relation, called pattern, for a group of samples. Yet, there are no guarantees that any of the returned patterns do not actually arise from noise, i.e., are false discoveries. In this paper, we propose and discuss the usage of the false discovery rate in the unsupervised BMF setting. We prove two bounds on the probability that a found pattern is constituted of random Bernoulli-distributed noise. Each bound exploits a specific property of the factorization which minimizes the approximation error—yielding new insights on the minimizers of Boolean matrix factorization. This leads to improved BMF algorithms by replacing heuristic rank selection techniques with a theoretically well-based approach. Our empirical demonstration shows that both bounds deliver excellent results in various practical settings.