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Originally, tangles were invented as an abstract tool in mathematical graph theory to prove the famous graph minor theorem. In this paper, we showcase the practical potential of tangles in machine learning applications. Given a collection of cuts of any dataset, tangles aggregate these cuts to point in the direction of a dense structure. As a result, a cluster is softly characterized by a set of consistent pointers. This highly flexible approach can solve clustering problems in various setups, ranging from questionnaires over community detection in graphs to clustering points in metric spaces. The output of our proposed framework is hierarchical and induces the notion of a soft dendrogram, which can help explore the cluster structure of a dataset. The computational complexity of aggregating the cuts is linear in the number of data points. Thus the bottleneck of the tangle approach is to generate the cuts, for which simple and fast algorithms form a sufficient basis. In our paper we construct the algorithmic framework for clustering with tangles, prove theoretical guarantees in various settings, and provide extensive simulations and use cases. Python code is available on github. [ABSTRACT FROM AUTHOR]

Many large-scale applications amount to finding relevant results from an enormous output space of potential candidates. For example, finding the best matching product from a large catalog or suggesting related search phrases on a search engine. The size of the output space for these problems can range from millions to billions, and can even be infinite in some applications. Moreover, training data is often limited for the long-tail items in the output space. Fortunately, items in the output space are often correlated thereby presenting an opportunity to alleviate the data sparsity issue. In this paper, we propose the Prediction for Enormous and Correlated Output Spaces (PECOS) framework, a versatile and modular machine learning framework for solving prediction problems for very large output spaces, and apply it to the eXtreme Multilabel Ranking (XMR) problem: given an input instance, find and rank the most relevant items from an enormous but fixed and finite output space. We propose a three phase framework for PECOS: (i) in the first phase, PECOS organizes the output space using a semantic indexing scheme, (ii) in the second phase, PECOS uses the indexing to narrow down the output space by orders of magnitude using a machine learned matching scheme, and (iii) in the third phase, PECOS ranks the matched items using a final ranking scheme. The versatility and modularity of PECOS allows for easy plug-and-play of various choices for the indexing, matching, and ranking phases. We also develop very fast inference procedures which allow us to perform XMR predictions in real time; for example, inference takes less than 1 millisecond per input on the dataset with 2.8 million labels. [ABSTRACT FROM AUTHOR]

Many important machine learning problems can be formulated as highly constrained convex optimization problems. One important example is metric constrained problems. In this paper, we show that standard optimization techniques can not be used to solve metric constrained problem. To solve such problems, we provide a general active set framework, called Project and Forget, and several variants thereof that use Bregman projections. Project and Forget is a general purpose method that can be used to solve highly constrained convex problems with many (possibly exponentially) constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithms converge to the global optimal solution and have a linear rate of convergence. We demonstrate that using our method, we can solve large problem instances of general weighted correlation clustering, metric nearness, information theoretic metric learning and quadratically regularized optimal transport; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes. [ABSTRACT FROM AUTHOR]