Abstract $${{\textbf{Latent}}\varvec{Out}}$$ Latent Out is a recently introduced algorithm for unsupervised anomaly detection which enhances latent space-based neural methods, namely (Variational) Autoencoders, GANomaly and ANOGan architectures. The main idea behind it is to exploit both the latent space and the baseline score of these architectures in order to provide a refined anomaly score performing density estimation in the augmented latent-space/baseline-score feature space. In this paper we investigate the performance of $${{\textbf{Latent}}\varvec{Out}}$$ Latent Out acting as a one-class classifier and we experiment the combination of $${{\textbf{Latent}}\varvec{Out}}$$ Latent Out with GAAL architectures, a novel type of Generative Adversarial Networks for unsupervised anomaly detection. Moreover, we show that the feature space induced by $${{\textbf{Latent}}\varvec{Out}}$$ Latent Out has the characteristic to enhance the separation between normal and anomalous data. Indeed, we prove that standard data mining outlier detection methods perform better when applied on this novel augmented latent space rather than on the original data space.

Abstract Enabling information systems to face anomalies in the presence of uncertainty is a compelling and challenging task. In this work the problem of unsupervised outlier detection in large collections of data objects modeled by means of arbitrary multidimensional probability density functions is considered. We present a novel definition ofuncertain distance-based outlierunder the attribute level uncertainty model, according to which an uncertain object is an object that always exists but its actual value is modeled by a multivariate pdf. According to this definition an uncertain object is declared to be an outlier on the basis of the expected number of its neighbors in the dataset. To the best of our knowledge this is the first work that considers the unsupervised outlier detection problem on data objects modeled by means of arbitrarily shaped multidimensional distribution functions. We present the UDBOD algorithm which efficiently detects the outliers in an input uncertain dataset by taking advantages of three optimized phases, that are parameter estimation, candidate selection, and the candidate filtering. An experimental campaign is presented, including a sensitivity analysis, a study of the effectiveness of the technique, a comparison with related algorithms, also in presence of high dimensional data, and a discussion about the behavior of our technique in real case scenarios.

We present a novel notion of outlier, called the Concentration Free Outlier Factor, or CFOF. As a main contribution, we formalize the notion of concentration of outlier scores and theoretically prove that CFOF does not concentrate in the Euclidean space for any arbitrary large dimensionality. To the best of our knowledge, there are no other proposals of data analysis measures related to the Euclidean distance for which it has been provided theoretical evidence that they are immune to the concentration effect. We determine the closed form of the distribution of CFOF scores in arbitrarily large dimensionalities and show that the CFOF score of a point depends on its squared norm standard score and on the kurtosis of the data distribution, thus providing a clear and statistically founded characterization of this notion. Moreover, we leverage this closed form to provide evidence that the definition does not suffer of the hubness problem affecting other measures in high dimensions. We prove that the number of CFOF outliers coming from each cluster is proportional to cluster size and kurtosis, a property that we call semi-locality. We leverage theoretical findings to shed lights on properties of well-known outlier scores. Indeed, we determine that semi-locality characterizes existing reverse nearest neighbor-based outlier definitions, thus clarifying the exact nature of their observed local behavior. We also formally prove that classical distance-based and density-based outliers concentrate both for bounded and unbounded sample sizes and for fixed and variable values of the neighborhood parameter. We introduce the fast-CFOF algorithm for detecting outliers in large high-dimensional dataset. The algorithm has linear cost, supports multi-resolution analysis, and is embarrassingly parallel. Experiments highlight that the technique is able to efficiently process huge datasets and to deal even with large values of the neighborhood parameter, to avoid concentration, and to obtain excellent accuracy.

In this work, we introduce a novel definition of outlier, namely the Gradient Outlier Factor (or GOF), with the aim to provide a definition that unifies with the statistical one on some standard distributions but has a different behavior in the presence of mixture distributions. Intuitively, the GOF score measures the probability to stay in the neighborhood of a certain object. It is directly proportional to the density and inversely proportional to the variation of the density. We derive formal properties under which the GOF definition unifies the statistical outlier definition and show that the unification holds for some standard distributions, while the GOF is able to capture tails in the presence of different distributions even if their densities sensibly differ. Moreover, we provide a probabilistic interpretation of the GOF score, by means of the notion of density of the data density. Experimental results confirm that there are scenarios in which the novel definition can be profitably employed. To the best of our knowledge, except for distance-based outlier, no other data mining outlier definition has a so clearly established relationship with statistical outliers.

This work deals with the problem of classifying uncertain data. With this aim we introduce the Uncertain Nearest Neighbor (UNN) rule, which represents the generalization of the deterministic nearest neighbor rule to the case in which uncertain objects are available. The UNN rule relies on the concept of nearest neighbor class, rather than on that of nearest neighbor object. The nearest neighbor class of a test object is the class that maximizes the probability of providing its nearest neighbor. The evidence is that the former concept is much more powerful than the latter in the presence of uncertainty, in that it correctly models the right semantics of the nearest neighbor decision rule when applied to the uncertain scenario. An effective and efficient algorithm to perform uncertain nearest neighbor classification of a generic (un)certain test object is designed, based on properties that greatly reduce the temporal cost associated with nearest neighbor class probability computation. Experimental results are presented, showing that the UNN rule is effective and efficient in classifying uncertain data.

We introduce a distributed method for detecting distance-based outliers in very large data sets. Our approach is based on the concept of outlier detection solving set [2], which is a small subset of the data set that can be also employed for predicting novel outliers. The method exploits parallel computation in order to obtain vast time savings. Indeed, beyond preserving the correctness of the result, the proposed schema exhibits excellent performances. From the theoretical point of view, for common settings, the temporal cost of our algorithm is expected to be at least three orders of magnitude faster than the classical nested-loop like approach to detect outliers. Experimental results show that the algorithm is efficient and that its running time scales quite well for an increasing number of nodes. We discuss also a variant of the basic strategy which reduces the amount of data to be transferred in order to improve both the communication cost and the overall runtime. Importantly, the solving set computed by our approach in a distributed environment has the same quality as that produced by the corresponding centralized method.

We consider the problem of discovering attributes, or properties, accounting for the a priori stated abnormality of a group of anomalous individuals (the outliers) with respect to an overall given population (the inliers). To this aim, we introduce the notion of exceptional property and define the concept of exceptionality score, which measures the significance of a property. In particular, in order to single out exceptional properties, we resort to a form of minimum distance estimation for evaluating the badness of fit of the values assumed by the outliers compared to the probability distribution associated with the values assumed by the inliers. Suitable exceptionality scores are introduced for both numeric and categorical attributes. These scores are, both from the analytical and the empirical point of view, designed to be effective for small samples, as it is the case for outliers. We present an algorithm, called EXPREX, for efficiently discovering exceptional properties. The algorithm is able to reduce the needed computational effort by not exploring many irrelevant numerical intervals and by exploiting suitable pruning rules. The experimental results confirm that our technique is able to provide knowledge characterizing outliers in a natural manner.

In this study, we deal with the problem of efficiently answering range queries over uncertain objects in a general metric space. In this study, an uncertain object is an object that always exists but its actual value is uncertain and modeled by a multivariate probability density function. As a major contribution, this is the first work providing an effective technique for indexing uncertain objects coming from general metric spaces. We generalize the reverse triangle inequality to the probabilistic setting in order to exploit it as a discard condition. Then, we introduce a novel pivot-based indexing technique, called UP-index, and show how it can be employed to speed up range query computation. Importantly, the candidate selection phase of our technique is able to noticeably reduce the set of candidates with little time requirements. Finally, we provide a criterion to measure the quality of a set of pivots and study the problem of selecting a good set of pivots according to the introduced criterion. We report some intractability results and then design an approximate algorithm with statistical guarantees for selecting pivots. Experimental results validate the effectiveness of the proposed approach and reveal that the introduced technique may be even preferable to indexing techniques specifically designed for the euclidean space.

We present a novel definition of outlier in the context of inductive logic programming. Given a set of positive and negative examples, the definition aims at singling out the examples showing anomalous behavior. We note that the task here pursued is different from noise removal, and, in fact, the anomalous observations we discover are different in nature from noisy ones. We discuss pecularities of the novel approach, present an algorithm for detecting outliers, discuss some examples of knowledge mined, and compare it with alternative approaches.

In this work a novel distance-based outlier detection algorithm, named DOLPHIN, working on disk-resident datasets and whose I/O cost corresponds to the cost of sequentially reading the input dataset file twice, is presented. It is both theoretically and empirically shown that the main memory usage of DOLPHIN amounts to a small fraction of the dataset and that DOLPHIN has linear time performance with respect to the dataset size. DOLPHIN gains efficiency by naturally merging together in a unified schema three strategies, namely the selection policy of objects to be maintained in main memory, usage of pruning rules, and similarity search techniques. Importantly, similarity search is accomplished by the algorithm without the need of preliminarily indexing the whole dataset, as other methods do. The algorithm is simple to implement and it can be used with any type of data, belonging to either metric or nonmetric spaces. Moreover, a modification to the basic method allows DOLPHIN to deal with the scenario in which the available buffer of main memory is smaller than its standard requirements. DOLPHIN has been compared with state-of-the-art distance-based outlier detection algorithms, showing that it is much more efficient.

Assume you are given a data population characterized by a certain number of attributes. Assume, moreover, you are provided with the information that one of the individuals in this data population is abnormal, but no reason whatsoever is given to you as to why this particular individual is to be considered abnormal. In several cases, you will be indeed interested in discovering such reasons. This article is precisely concerned with this problem of discovering sets of attributes that account for the (a priori stated) abnormality of an individual within a given dataset. A criterion is presented to measure the abnormality of combinations of attribute values featured by the given abnormal individual with respect to the reference population. In this respect, each subset of attributes is intended to somehow represent a “property” of individuals. We distinguish between global and local properties. Global properties are subsets of attributes explaining the given abnormality with respect to the entire data population. With local ones, instead, two subsets of attributes are singled out, where the former one justifies the abnormality within the data subpopulation selected using the values taken by the exceptional individual on those attributes included in the latter one. The problem of individuating abnormal properties with associated explanations is formally stated and analyzed. Such a formal characterization is then exploited in order to devise efficient algorithms for detecting both global and local forms of most abnormal properties. The experimental evidence, which is accounted for in the article, shows that the algorithms are both able to mine meaningful information and to accomplish the computational task by examining a negligible fraction of the search space.

In this work a novel algorithm, named DOLPHIN, for detecting distance-based outliers is presented.

In this work a method for detecting distance-based outliers in data streams is presented. We deal with the sliding window model, where outlier queries are performed in order to detect anomalies in the current window. Two algorithms are presented. The first one exactly answers outlier queries, but has larger space requirements. The second algorithm is directly derived from the exact one, has limited memory requirements and returns an approximate answer based on accurate estimations with a statistical guarantee. Several experiments have been accomplished, confirming the effectiveness of the proposed approach and the high quality of approximate solutions.

This work has two main objectives, namely, to introduce a novel algorithm, called the fast condensed nearest neighbor (FCNN) rule, for computing a training-set-consistent subset for the nearest neighbor decision rule and to show that condensation algorithms for the nearest neighbor rule can be applied to huge collections of data. The FCNN rule has some interesting properties: it is order independent, its worst-case time complexity is quadratic but often with a small constant prefactor, and it is likely to select points very close to the decision boundary. Furthermore, its structure allows for the triangle inequality to be effectively exploited to reduce the computational effort. The FCNN rule outperformed even here-enhanced variants of existing competence preservation methods both in terms of learning speed and learning scaling behavior and, often, in terms of the size of the model while it guaranteed the same prediction accuracy. Furthermore, it was three orders of magnitude faster than hybrid instance-based learning algorithms on the MNIST and Massachusetts Institute of Technology (MIT) Face databases and computed a model of accuracy comparable to that of methods incorporating a noise-filtering pass.

In this work, the parallel fast condensed nearest neighbor (PFCNN) rule, a distributed method for computing a consistent subset of a very large data set for the nearest neighbor classification rule is presented. In order to cope with the communication overhead typical of distributed environments and to reduce memory requirements, different variants of the basic PFCNN method are introduced. An analysis of spatial cost, CPU cost, and communication overhead is accomplished for all the algorithms. Experimental results, performed on both synthetic and real very large data sets, revealed that these methods can be profitably applied to enormous collections of data. Indeed, they scale up well and are efficient in memory consumption, confirming the theoretical analysis, and achieve noticeable data reduction and good classification accuracy. To the best of our knowledge, this is the first distributed algorithm for computing a training set consistent subset for the nearest neighbor rule.

A distance-based outlier detection method that finds the top outliers in an unlabeled data set and provides a subset of it, called outlier detection solving set, that can be used to predict the outlierness of new unseen objects, is proposed. The solving set includes a sufficient number of points that permits the detection of the top outliers by considering only a subset of all the pairwise distances from the data set. The properties of the solving set are investigated, and algorithms for computing it, with subquadratic time requirements, are proposed. Experiments on synthetic and real data sets to evaluate the effectiveness of the approach are presented. A scaling analysis of the solving set size is performed, and the false positive rate, that is, the fraction of new objects misclassified as outliers using the solving set instead of the overall data set, is shown to be negligible. Finally, to investigate the accuracy in separating outliers from inliers, ROC analysis of the method is accomplished. Results obtained show that using the solving set instead of the data set guarantees a comparable quality of the prediction, but at a lower computational cost.

A new definition of distance-based outlier and an algorithm, called HilOut, designed to efficiently detect the top n outliers of a large and high-dimensional data set are proposed. Given an integer k, the weight of a point is defined as the sum of the distances separating it from its k nearest-neighbors. Outlier are those points scoring the largest values of weight. The algorithm HilOut makes use of the notion of space-filling curve to linearize the data set, and it consists of two phases. The first phase provides an approximate solution, within a rough factor, after the execution of at most d + 1 sorts and scans of the data set, with temporal cost quadratic in d and linear in N and in k, where d is the number of dimensions of the data set and N is the number of points in the data set. During this phase, the algorithm isolates points candidate to be outliers and reduces this set at each iteration. If the size of this set becomes n, then the algorithm stops reporting the exact solution. The second phase calculates the exact solution with a final scan examining further the candidate outliers that remained after the first phase. Experimental results show that the algorithm always stops, reporting the exact solution, during the first phase after much less than d + 1 steps. We present both an in-memory and disk-based implementation of the HilOut algorithm and a thorough scaling analysis for real and synthetic data sets showing that the algorithm scales well in both cases.

Metaquerying is a datamining technology by which hidden dependencies among several database relations can be discovered. This tool has already been successfully applied to several real-world applications. Recent papers provide only very preliminary results about the complexity of metaquerying. In this paper we define several variants of metaquerying that encompass, as far as we know, all variants defined in the literature. We study both the combined complexity and the data complexity of these variants. We show that, under the combined complexity measure, metaquerying is generally intractable (unless P=NP), but we are able to single out some tractable interesting metaquerying cases (whose combined complexity is LOGCFL-complete). As for the data complexity of metaquerying, we prove that, in general, this is in P, but lies within AC0 in some interesting cases. Finally, we discuss the issue of equivalence between metaqueries, which is useful for optimization purposes.