Current exact hierarchical density-based clustering algorithms for high-dimensional data have asymptotically quadratic complexity. We present algorithms for approximate hierarchical density-based clustering, namely for single-linkage clustering and for HDBSCAN, with empirically near-linear time scalability. We explore both graph index-based incremental nearest neighbor search and an iterative exploration scheme on the graph index approximating the MST of the reachability graph similar to Kruskal. As graph index, we use both the bottom layer and a combination of all layers of an HNSW as a stand-in for connected search graphs. We provide experiments comparing the clusterings to baselines such as exact implementation and an algorithm using metric tree-based searchers. We explore the impact of the HNSW hyperparameters on the performance in terms of running time and clustering quality. For both single-linkage clustering and HDBSCAN, our algorithms yield highly accurate clusterings while being up to two orders of magnitude faster than industry-standard baselines such as scikit-learn’s hdbscan .

Abstract We perform an extensive experimental evaluation of clustering-based outlier detection methods. These methods offer benefits such as efficiency, the possibility to capitalize on more mature evaluation measures, more developed subspace analysis for high-dimensional data and better explainability, and yet they have so-far been neglected in literature. To our knowledge, our work is the first effort to analytically and empirically study their advantages and disadvantages. Our main goal is to evaluate whether or not clustering-based techniques can compete in efficiency and effectiveness against the most studied state-of-the-art algorithms in the literature. We consider the quality of the results, the resilience against different types of data and variations in parameter configuration, the scalability, and the ability to filter out inappropriate parameter values automatically based on internal measures of clustering quality. It has been recently shown that several classic, simple, unsupervised methods surpass many deep learning approaches and, hence, remain at the state-of-the-art of outlier detection. We therefore study 14 of the best classic unsupervised methods, in particular 11 clustering-based methods and 3 non-clustering-based ones, using a consistent parameterization heuristic to identify the pros and cons of each approach. We consider 46 real and synthetic datasets with up to 125k points and 1.5k dimensions aiming to achieve plausibility with the broadest possible diversity of real-world use cases. Our results indicate that the clustering-based methods are on par with (if not surpass) the non-clustering-based ones, and we argue that clustering-based methods like KMeans−− should be included as baselines in future benchmarking studies, as they often offer a competitive quality at a relatively low run time, besides several other benefits.

Searching for similar objects in intrinsically high-dimensional data sets is a challenging task. The use of compact sketches has been proposed for faster similarity search using linear scans. Binary sketches are one such approach to find a good mapping from the original data space to bit strings of a fixed length. These bit strings can be compared efficiently using only few XOR and bit count operations, replacing costly similarity computations with an inexpensive approximation. We propose a new scheme to initialize and improve binary sketches for similarity search in Euclidean spaces. Our optimization iteratively improves the quality of the sketches with a form of orthogonalization. We provide empirical evidence that the quality of the sketches has a peak beyond which it is not correlated to neither bit independence nor bit balance, which contradicts a previous hypothesis in the literature. Regularization in the form of noise added to the training data can turn the peak into a plateau and applying the optimization in a stochastic fashion, i.e., training on smaller subsets of the data, allows for rapid initialization. We provide a loss function that allows to approximate the same objective using neural network frameworks such as PyTorch, elevating the approach to GPU-based training.

Abstract The graph edit distance is an intuitive measure to quantify the dissimilarity of graphs, but its computation is $$\mathsf {NP}$$ NP -hard and challenging in practice. We introduce methods for answering nearest neighbor and range queries regarding this distance efficiently for large databases with up to millions of graphs. We build on the filter-verification paradigm, where lower and upper bounds are used to reduce the number of exact computations of the graph edit distance. Highly effective bounds for this involve solving a linear assignment problem for each graph in the database, which is prohibitive in massive datasets. Index-based approaches typically provide only weak bounds leading to high computational costs verification. In this work, we derive novel lower bounds for efficient filtering from restricted assignment problems, where the cost function is a tree metric. This special case allows embedding the costs of optimal assignments isometrically into $$\ell _1$$ ℓ 1 space, rendering efficient indexing possible. We propose several lower bounds of the graph edit distance obtained from tree metrics reflecting the edit costs, which are combined for effective filtering. Our method termed EmbAssi can be integrated into existing filter-verification pipelines as a fast and effective pre-filtering step. Empirically we show that for many real-world graphs our lower bounds are already close to the exact graph edit distance, while our index construction and search scales to very large databases.

BIRCH clustering is a widely known approach for clustering that has influenced much subsequent research and commercial products. The key contribution of BIRCH is the Clustering Feature tree (CF-Tree), which is a compressed representation of the input data. As new data arrives, the tree is eventually rebuilt to increase the compression. Afterward, the leaves of the tree are used for clustering. Because of the data compression, this method is very scalable. The idea has been adopted, for example, for k-means, data stream, and density-based clustering. Clustering features used by BIRCH are simple summary statistics that can easily be updated with new data: the number of points, the linear sums, and the sum of squared values. Unfortunately, how the sum of squares is then used in BIRCH is prone to catastrophic cancellation. We introduce a replacement cluster feature that does not have this numeric problem, that is not much more expensive to maintain, and which makes many computations simpler and, hence, more efficient. These cluster features can also easily be used in other work derived from BIRCH such as algorithms for streaming data. In the experiments, we demonstrate the numerical problem and compare the performance of the original algorithm compared to the improved cluster features. We furthermore explain how to improve clustering with different algorithms such as hierarchical clustering, k-means, k-means++ and Gaussian mixture modeling by using the variance information stored in the cluster features to obtain high-quality result approximations in a substantially reduced runtime (up to 500× faster in our experiments).

The intrinsic dimensionality refers to the “true” dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower than the number of variables. Local intrinsic dimensionality refers to the observation that this property can vary for different parts of the data set; and intrinsic dimensionality can serve as a proxy for the local data complexity of the data set. Most popular methods for estimating the local intrinsic dimensionality are based on distances, and the rate at which the distances to the nearest neighbors increase, a concept known as “expansion dimension”. In this paper we introduce an orthogonal concept, which does not use any distances: we use the distribution of angles among the neighbors with respect to the query point. We derive the theoretical distribution of angles and use this to construct an estimator for intrinsic dimensionality. We derive a regularized version that obeys an upper bound of the true intrinsic dimensionality for idealized data. We generalize the estimator to arbitrary moments and discuss the relation of the regularized second moment to approaches based on principal component analysis. Experimentally, we verify that this measure behaves similarly but complementarily to existing measures of intrinsic dimensionality. By introducing a new idea of intrinsic dimensionality to the research community we hope to contribute to a better understanding of intrinsic dimensionality and to spur new research in this direction.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids clustering. In Euclidean geometry the mean – as used in k-means – is a good estimator for the cluster center, but this does not exist for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains and applications. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm that achieve an O(k)-fold speedup in the second (“SWAP”) phase of the algorithm, but will still find the same results as the original PAM algorithm. If we relax the choice of swaps performed (while retaining comparable quality), we can further accelerate the algorithm by eagerly performing additional swaps in each iteration. With the substantially faster SWAP, we can now explore faster initialization strategies, because (i) the classic (“BUILD”) initialization now becomes the bottleneck, and (ii) our swap is fast enough to compensate for worse starting conditions. We also show how the CLARA and CLARANS algorithms benefit from the proposed modifications. While we do not study the parallelization of our approach in this work, it can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100,200, we observed a 458× respectively 1191× speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets, and in particular to higher k.

With the advent of big data, we see an increasing interest in computing correlations in huge data sets with both many instances and many variables. Essential descriptive statistics such as the variance, standard deviation, covariance, and correlation can suffer from a numerical instability known as "catastrophic cancellation" that can lead to problems when naively computing these statistics with a popular textbook equation. While this instability has been discussed in the literature already 50 years ago, we found that even today, some high-profile tools still employ the instable version.

Given a query object q, reverse k -nearest neighbor (R k NN) search aims to locate those objects of the database that have q among their k -nearest neighbors. In this paper, we propose an approximation method for solving R k NN queries, where the pruning operations and termination tests are guided by a characterization of the intrinsic dimensionality of the data. The method can accommodate any index structure supporting incremental (forward) nearest-neighbor search for the generation and verification of candidates, while avoiding impractically-high preprocessing costs. We also provide experimental evidence that our method significantly outperforms its competitors in terms of the tradeoff between execution time and the quality of the approximation. Our approach thus addresses many of the scalability issues surrounding the use of previous methods in data mining.

At SIGMOD 2015, an article was presented with the title “DBSCAN Revisited: Mis-Claim, Un-Fixability, and Approximation” that won the conference’s best paper award. In this technical correspondence, we want to point out some inaccuracies in the way DBSCAN was represented, and why the criticism should have been directed at the assumption about the performance of spatial index structures such as R-trees and not at an algorithm that can use such indexes. We will also discuss the relationship of DBSCAN performance and the indexability of the dataset, and discuss some heuristics for choosing appropriate DBSCAN parameters. Some indicators of bad parameters will be proposed to help guide future users of this algorithm in choosing parameters such as to obtain both meaningful results and good performance. In new experiments, we show that the new SIGMOD 2015 methods do not appear to offer practical benefits if the DBSCAN parameters are well chosen and thus they are primarily of theoretical interest. In conclusion, the original DBSCAN algorithm with effective indexes and reasonably chosen parameter values performs competitively compared to the method proposed by Gan and Tao.

The analysis of social media data poses several challenges: first of all, the data sets are very large, secondly they change constantly, and third they are heterogeneous, consisting of text, images, geographic locations and social connections. In this article, we focus on detecting events consisting of text and location information, and introduce an analysis method that is scalable both with respect to volume and velocity. We also address the problems arising from differences in adoption of social media across cultures, languages, and countries in our event detection by efficient normalization.

The challenges associated with handling uncertain data, in particular with querying and mining, are finding increasing attention in the research community. Here we focus on clustering uncertain data and describe a general framework for this purpose that also allows to visualize and understand the impact of uncertainty---using different uncertainty models---on the data mining results. Our framework constitutes release 0.7 of ELKI (http://elki.dbs.ifi.lmu.de/) and thus comes along with a plethora of implementations of algorithms, distance measures, indexing techniques, evaluation measures and visualization components.

We consider the problem of outlier detection and interpretation. While most existing studies focus on the first problem, we simultaneously address the equally important challenge of outlier interpretation. We propose an algorithm that uncovers outliers in subspaces of reduced dimensionality in which they are well discriminated from regular objects while at the same time retaining the natural local structure of the original data to ensure the quality of outlier explanation. Our algorithm takes a mathematically appealing approach from the spectral graph embedding theory and we show that it achieves the globally optimal solution for the objective of subspace learning. By using a number of real-world datasets, we demonstrate its appealing performance not only w.r.t. the outlier detection rate but also w.r.t. the discriminative human-interpretable features. This is the first approach to exploit discriminative features for both outlier detection and interpretation, leading to better understanding of how and why the hidden outliers are exceptional.

Social media such as Twitter or weblogs are a popular source for live textual data. Much of this popularity is due to the fast rate at which this data arrives, and there are a number of global events - such as the Arab Spring - where Twitter is reported to have had a major influence. However, existing methods for emerging topic detection are often only able to detect events of a global magnitude such as natural disasters or celebrity deaths, and can monitor user-selected keywords or operate on a curated set of hashtags only. Interesting emerging topics may, however, be of much smaller magnitude and may involve the combination of two or more words that themselves are not unusually hot at that time. Our contributions to the detection of emerging trends are three-fold first of all, we propose a significance measure that can be used to detect emerging topics early, long before they become "hot tags", by drawing upon experience from outlier detection. Secondly, by using hash tables in a heavy-hitters type algorithm for establishing a noise baseline, we show how to track even all keyword pairs using only a fixed amount of memory. Finally, we aggregate the detected co-trends into larger topics using clustering approaches, as often as a single event will cause multiple word combinations to trend at the same time.

We analyse the interplay of density estimation and outlier detection in density-based outlier detection. By clear and principled decoupling of both steps, we formulate a generalization of density-based outlier detection methods based on kernel density estimation. Embedded in a broader framework for outlier detection, the resulting method can be easily adapted to detect novel types of outliers: while common outlier detection methods are designed for detecting objects in sparse areas of the data set, our method can be modified to also detect unusual local concentrations or trends in the data set if desired. It allows for the integration of domain knowledge and specific requirements. We demonstrate the flexible applicability and scalability of the method on large real world data sets.

Parallel coordinates are an established technique to visualize high-dimensional data, in particular for data mining purposes. A major challenge is the ordering of axes, as any axis can have at most two neighbors when placed in parallel on a 2D plane. By extending this concept to a 3D visualization space we can place several axes next to each other. However, finding a good arrangement often does not necessarily become easier, as still not all axes can be arranged pairwise adjacently to each other. Here, we provide a tool to explore complex data sets using 3D-parallel-coordinate-trees, along with a number of approaches to arrange the axes.

When comparing clustering results, any evaluation metric breaks down the available information to a single number. However, a lot of evaluation metrics are around, that are not always concordant nor easily interpretable in judging the agreement of a pair of clusterings. Here, we provide a tool to visually support the assessment of clustering results in comparing multiple clusterings. Along the way, the suitability of a couple of clustering comparison measures can be judged in different scenarios.

In this paper, we propose a novel outlier detection model to find outliers that deviate from the generating mechanisms of normal instances by considering combinations of different subsets of attributes, as they occur when there are local correlations in the data set. Our model enables to search for outliers in arbitrarily oriented subspaces of the original feature space. We show how in addition to an outlier score, our model also derives an explanation of the outlierness that is useful in investigating the results. Our experiments suggest that our novel method can find different outliers than existing work and can be seen as a complement of those approaches.

Outlier detection research is currently focusing on the development of new methods and on improving the computation time for these methods. Evaluation however is rather heuristic, often considering just precision in the top k results or using the area under the ROC curve. These evaluation procedures do not allow for assessment of similarity between methods. Judging the similarity of or correlation between two rankings of outlier scores is an important question in itself but it is also an essential step towards meaningfully building outlier detection ensembles, where this aspect has been completely ignored so far. In this study, our generalized view of evaluation methods allows both to evaluate the performance of existing methods as well as to compare different methods w.r.t. their detection performance. Our new evaluation framework takes into consideration the class imbalance problem and offers new insights on similarity and redundancy of existing outlier detection methods. As a result, the design of effective ensemble methods for outlier detection is considerably enhanced.

Outlier scores provided by different outlier models differ widely in their meaning, range, and contrast between different outlier models and, hence, are not easily comparable or interpretable. We propose a unification of outlier scores provided by various outlier models and a translation of the arbitrary “outlier factors” to values in the range [0, 1] interpretable as values describing the probability of a data object of being an outlier. As an application, we show that this unification facilitates enhanced ensembles for outlier detection.

Many outlier detection methods do not merely provide the decision for a single data object being or not being an outlier but give also an outlier score or "outlier factor" signaling "how much" the respective data object is an outlier. A major problem for any user not very acquainted with the outlier detection method in question is how to interpret this "factor" in order to decide for the numeric score again whether or not the data object indeed is an outlier. Here, we formulate a local density based outlier detection method providing an outlier "score" in the range of [0, 1] that is directly interpretable as a probability of a data object for being an outlier.