arXiv subjects

Nikolaj Tatti

Publications and source records attributed to Nikolaj Tatti.

At least 37 records · Page 2Linked to original sources

arXiv2022

Coresets remembered and items forgotten: submodular maximization with deletions

In recent years we have witnessed an increase on the development of methods for submodular optimization, which have been motivated by the wide applicability of submodular functions in real-world data-science problems. In this paper, we contribute to this line of work by considering the problem of robust submodular maximization against unexpected deletions, which may occur due to privacy issues or user preferences. Specifically, we consider the minimum number of items an algorithm has to remember, in order to achieve a non-trivial approximation guarantee against adversarial deletion of up to $d$ items. We refer to the set of items that an algorithm has to keep before adversarial deletions as a deletion-robust coreset. Our theoretical contributions are two-fold. First, we propose a single-pass streaming algorithm that yields a $(1-2ε)/(4p)$-approximation for maximizing a non-decreasing submodular function under a general p-matroid constraint and requires a coreset of size $k + d/ε$, where $k$ is the maximum size of a feasible solution. To the best of our knowledge, this is the first work to achieve an (asymptotically) optimal coreset, as no constant-factor approximation is possible with a coreset of size sublinear in $d$. Second, we devise an effective offline algorithm that guarantees stronger approximation ratios with a coreset of size $O(d \log(k)/ε)$. We also demonstrate the superior empirical performance of the proposed algorithms in real-life applications.

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arXiv2022

Ranking with submodular functions on a budget

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (MSR). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the MSR problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.

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arXiv2021

Maintaining AUC and $H$-measure over time

Measuring the performance of a classifier is a vital task in machine learning. The running time of an algorithm that computes the measure plays a very small role in an offline setting, for example, when the classifier is being developed by a researcher. However, the running time becomes more crucial if our goal is to monitor the performance of a classifier over time. In this paper we study three algorithms for maintaining two measures. The first algorithm maintains area under the ROC curve (AUC) under addition and deletion of data points in $O(\log n)$ time. This is done by maintaining the data points sorted in a self-balanced search tree. In addition, we augment the search tree that allows us to query the ROC coordinates of a data point in $O(\log n)$ time. In doing so we are able to maintain AUC in $O(\log n)$ time. Our next two algorithms involve in maintaining $H$-measure, an alternative measure based on the ROC curve. Computing the measure is a two-step process: first we need to compute a convex hull of the ROC curve, followed by a sum over the convex hull. We demonstrate that we can maintain the convex hull using a minor modification of the classic convex hull maintenance algorithm. We then show that under certain conditions, we can compute the $H$-measure exactly in $O(\log^2 n)$ time, and if the conditions are not met, then we can estimate the $H$-measure in $O((\log n + ε^{-1})\log n)$ time. We show empirically that our methods are significantly faster than the baselines.

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arXiv2021

Approximation algorithms for confidence bands for time series

Confidence intervals are a standard technique for analyzing data. When applied to time series, confidence intervals are computed for each time point separately. Alternatively, we can compute confidence bands, where we are required to find the smallest area enveloping $k$ time series, where $k$ is a user parameter. Confidence bands can be then used to detect abnormal time series, not just individual observations within the time series. We will show that despite being an NP-hard problem it is possible to find optimal confidence band for some $k$. We do this by considering a different problem: discovering regularized bands, where we minimize the envelope area minus the number of included time series weighted by a parameter $α$. Unlike normal confidence bands we can solve the problem exactly by using a minimum cut. By varying $α$ we can obtain solutions for various $k$. If we have a constraint $k$ for which we cannot find appropriate $α$, we demonstrate a simple algorithm that yields $O(\sqrt{n})$ approximation guarantee by connecting the problem to a minimum $k$-union problem. This connection also implies that we cannot approximate the problem better than $O(n^{1/4})$ under some (mild) assumptions. Finally, we consider a variant where instead of minimizing the area we minimize the maximum width. Here, we demonstrate a simple 2-approximation algorithm and show that we cannot achieve better approximation guarantee.

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arXiv2020

Decomposable Families of Itemsets

The problem of selecting a small, yet high quality subset of patterns from a larger collection of itemsets has recently attracted lot of research. Here we discuss an approach to this problem using the notion of decomposable families of itemsets. Such itemset families define a probabilistic model for the data from which the original collection of itemsets has been derived from. Furthermore, they induce a special tree structure, called a junction tree, familiar from the theory of Markov Random Fields. The method has several advantages. The junction trees provide an intuitive representation of the mining results. From the computational point of view, the model provides leverage for problems that could be intractable using the entire collection of itemsets. We provide an efficient algorithm to build decomposable itemset families, and give an application example with frequency bound querying using the model. Empirical results show that our algorithm yields high quality results.

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arXiv2020

Tell Me Something I Don't Know: Randomization Strategies for Iterative Data Mining

There is a wide variety of data mining methods available, and it is generally useful in exploratory data analysis to use many different methods for the same dataset. This, however, leads to the problem of whether the results found by one method are a reflection of the phenomenon shown by the results of another method, or whether the results depict in some sense unrelated properties of the data. For example, using clustering can give indication of a clear cluster structure, and computing correlations between variables can show that there are many significant correlations in the data. However, it can be the case that the correlations are actually determined by the cluster structure. In this paper, we consider the problem of randomizing data so that previously discovered patterns or models are taken into account. The randomization methods can be used in iterative data mining. At each step in the data mining process, the randomization produces random samples from the set of data matrices satisfying the already discovered patterns or models. That is, given a data set and some statistics (e.g., cluster centers or co-occurrence counts) of the data, the randomization methods sample data sets having similar values of the given statistics as the original data set. We use Metropolis sampling based on local swaps to achieve this. We describe experiments on real data that demonstrate the usefulness of our approach. Our results indicate that in many cases, the results of, e.g., clustering actually imply the results of, say, frequent pattern discovery.

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arXiv2020

Density-friendly Graph Decomposition

Decomposing a graph into a hierarchical structure via $k$-core analysis is a standard operation in any modern graph-mining toolkit. $k$-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph. Despite the fact that $k$-core analysis relies on vertex degrees, $k$-cores do not satisfy a certain, rather natural, density property. Simply put, the most central $k$-core is not necessarily the densest subgraph. This inconsistency between $k$-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by $k$-cores, but in this case the components are arranged in order of increasing density. We show that such a locally-dense decomposition for a graph $G=(V,E)$ can be computed in polynomial time. The running time of the exact decomposition algorithm is $O(|V|^2|E|)$ but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the $k$-core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice $k$-cores have different structure than locally-dense subgraphs, and as predicted by the theory, $k$-cores are not always well-aligned with graph density.

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arXiv2019

Maximum Entropy Based Significance of Itemsets

We consider the problem of defining the significance of an itemset. We say that the itemset is significant if we are surprised by its frequency when compared to the frequencies of its sub-itemsets. In other words, we estimate the frequency of the itemset from the frequencies of its sub-itemsets and compute the deviation between the real value and the estimate. For the estimation we use Maximum Entropy and for measuring the deviation we use Kullback-Leibler divergence. A major advantage compared to the previous methods is that we are able to use richer models whereas the previous approaches only measure the deviation from the independence model. We show that our measure of significance goes to zero for derivable itemsets and that we can use the rank as a statistical test. Our empirical results demonstrate that for our real datasets the independence assumption is too strong but applying more flexible models leads to good results.

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arXiv2019

Summarizing Data Succinctly with the Most Informative Itemsets

Knowledge discovery from data is an inherently iterative process. That is, what we know about the data greatly determines our expectations, and therefore, what results we would find interesting and/or surprising. Given new knowledge about the data, our expectations will change. Hence, in order to avoid redundant results, knowledge discovery algorithms ideally should follow such an iterative updating procedure. With this in mind, we introduce a well-founded approach for succinctly summarizing data with the most informative itemsets; using a probabilistic maximum entropy model, we iteratively find the itemset that provides us the most novel information--that is, for which the frequency in the data surprises us the most---and in turn we update our model accordingly. As we use the Maximum Entropy principle to obtain unbiased probabilistic models, and only include those itemsets that are most informative with regard to the current model, the summaries we construct are guaranteed to be both descriptive and non-redundant. The algorithm that we present, called MTV, can either discover the top-$k$ most informative itemsets, or we can employ either the Bayesian Information Criterion (BIC) or the Minimum Description Length (MDL) principle to automatically identify the set of itemsets that together summarize the data well. In other words, our method will `tell you what you need to know' about the data. Importantly, it is a one-phase algorithm: rather than picking itemsets from a user-provided candidate set, itemsets and their supports are mined on-the-fly. To further its applicability, we provide an efficient method to compute the maximum entropy distribution using Quick Inclusion-Exclusion. Experiments on our method, using synthetic, benchmark, and real data, show that the discovered summaries are succinct, and correctly identify the key patterns in the data.

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arXiv2019

Mining Closed Strict Episodes

Discovering patterns in a sequence is an important aspect of data mining. One popular choice of such patterns are episodes, patterns in sequential data describing events that often occur in the vicinity of each other. Episodes also enforce in which order the events are allowed to occur. In this work we introduce a technique for discovering closed episodes. Adopting existing approaches for discovering traditional patterns, such as closed itemsets, to episodes is not straightforward. First of all, we cannot define a unique closure based on frequency because an episode may have several closed superepisodes. Moreover, to define a closedness concept for episodes we need a subset relationship between episodes, which is not trivial to define. We approach these problems by introducing strict episodes. We argue that this class is general enough, and at the same time we are able to define a natural subset relationship within it and use it efficiently. In order to mine closed episodes we define an auxiliary closure operator. We show that this closure satisfies the needed properties so that we can use the existing framework for mining closed patterns. Discovering the true closed episodes can be done as a post-processing step. We combine these observations into an efficient mining algorithm and demonstrate empirically its performance in practice.

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arXiv2019

Comparing Apples and Oranges: Measuring Differences between Exploratory Data Mining Results

Deciding whether the results of two different mining algorithms provide significantly different information is an important, yet understudied, open problem in exploratory data mining. Whether the goal is to select the most informative result for analysis, or to decide which mining approach will most likely provide the most novel insight, it is essential that we can tell how different the information is that different results by possibly different methods provide. In this paper we take a first step towards comparing exploratory data mining results on binary data. We propose to meaningfully convert results into sets of noisy tiles, and compare between these sets by Maximum Entropy modelling and Kullback-Leibler divergence, well-founded notions from Information Theory. We so construct a measure that is highly flexible, and allows us to naturally include background knowledge, such that differences in results can be measured from the perspective of what a user already knows. Furthermore, adding to its interpretability, it coincides with Jaccard dissimilarity when we only consider exact tiles. Our approach provides a means to study and tell differences between results of different exploratory data mining methods. As an application, we show that our measure can also be used to identify which parts of results best redescribe other results. Furthermore, we study its use for iterative data mining, where one iteratively wants to find that result that will provide maximal novel information. Experimental evaluation shows our measure gives meaningful results, correctly identifies methods that are similar in nature, automatically provides sound redescriptions of results, and is highly applicable for iterative data mining.

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arXiv2019

Finding Robust Itemsets Under Subsampling

Mining frequent patterns is plagued by the problem of pattern explosion making pattern reduction techniques a key challenge in pattern mining. In this paper we propose a novel theoretical framework for pattern reduction. We do this by measuring the robustness of a property of an itemset such as closedness or non-derivability. The robustness of a property is the probability that this property holds on random subsets of the original data. We study four properties: if an itemset is closed, free, non-derivable or totally shattered, and demonstrate how to compute the robustness analytically without actually sampling the data. Our concept of robustness has many advantages: Unlike statistical approaches for reducing patterns, we do not assume a null hypothesis or any noise model and in contrast to noise tolerant or approximate patterns, the robust patterns for a given property are always a subset of the patterns with this property. If the underlying property is monotonic, then the measure is also monotonic, allowing us to efficiently mine robust itemsets. We further derive a parameter-free technique for ranking itemsets that can be used for top-$k$ approaches. Our experiments demonstrate that we can successfully use the robustness measure to reduce the number of patterns and that ranking yields interesting itemsets.

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arXiv2019

Tiers for peers: a practical algorithm for discovering hierarchy in weighted networks

Interactions in many real-world phenomena can be explained by a strong hierarchical structure. Typically, this structure or ranking is not known; instead we only have observed outcomes of the interactions, and the goal is to infer the hierarchy from these observations. Discovering a hierarchy in the context of directed networks can be formulated as follows: given a graph, partition vertices into levels such that, ideally, there are only edges from upper levels to lower levels. The ideal case can only happen if the graph is acyclic. Consequently, in practice we have to introduce a penalty function that penalizes edges violating the hierarchy. A practical variant for such penalty is agony, where each violating edge is penalized based on the severity of the violation. Hierarchy minimizing agony can be discovered in $O(m^2)$ time, and much faster in practice. In this paper we introduce several extensions to agony. We extend the definition for weighted graphs and allow a cardinality constraint that limits the number of levels. While, these are conceptually trivial extensions, current algorithms cannot handle them, nor they can be easily extended. We solve the problem by showing the connection to the capacitated circulation problem, and we demonstrate that we can compute the exact solution fast in practice for large datasets. We also introduce a provably fast heuristic algorithm that produces rankings with competitive scores. In addition, we show that we can compute agony in polynomial time for any convex penalty, and, to complete the picture, we show that minimizing hierarchy with any concave penalty is an NP-hard problem.

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arXiv2019

Mining Closed Episodes with Simultaneous Events

Sequential pattern discovery is a well-studied field in data mining. Episodes are sequential patterns describing events that often occur in the vicinity of each other. Episodes can impose restrictions to the order of the events, which makes them a versatile technique for describing complex patterns in the sequence. Most of the research on episodes deals with special cases such as serial, parallel, and injective episodes, while discovering general episodes is understudied. In this paper we extend the definition of an episode in order to be able to represent cases where events often occur simultaneously. We present an efficient and novel miner for discovering frequent and closed general episodes. Such a task presents unique challenges. Firstly, we cannot define closure based on frequency. We solve this by computing a more conservative closure that we use to reduce the search space and discover the closed episodes as a postprocessing step. Secondly, episodes are traditionally presented as directed acyclic graphs. We argue that this representation has drawbacks leading to redundancy in the output. We solve these drawbacks by defining a subset relationship in such a way that allows us to remove the redundant episodes. We demonstrate the efficiency of our algorithm and the need for using closed episodes empirically on synthetic and real-world datasets.

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arXiv2019

Discovering Episodes with Compact Minimal Windows

Discovering the most interesting patterns is the key problem in the field of pattern mining. While ranking or selecting patterns is well-studied for itemsets it is surprisingly under-researched for other, more complex, pattern types. In this paper we propose a new quality measure for episodes. An episode is essentially a set of events with possible restrictions on the order of events. We say that an episode is significant if its occurrence is abnormally compact, that is, only few gap events occur between the actual episode events, when compared to the expected length according to the independence model. We can apply this measure as a post-pruning step by first discovering frequent episodes and then rank them according to this measure. In order to compute the score we will need to compute the mean and the variance according to the independence model. As a main technical contribution we introduce a technique that allows us to compute these values. Such a task is surprisingly complex and in order to solve it we develop intricate finite state machines that allow us to compute the needed statistics. We also show that asymptotically our score can be interpreted as a P-value. In our experiments we demonstrate that despite its intricacy our ranking is fast: we can rank tens of thousands episodes in seconds. Our experiments with text data demonstrate that our measure ranks interpretable episodes high.

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arXiv2019

Discovering Bands from Graphs

Discovering the underlying structure of a given graph is one of the fundamental goals in graph mining. Given a graph, we can often order vertices in a way that neighboring vertices have a higher probability of being connected to each other. This implies that the edges form a band around the diagonal in the adjacency matrix. Such structure may rise for example if the graph was created over time: each vertex had an active time interval during which the vertex was connected with other active vertices. The goal of this paper is to model this phenomenon. To this end, we formulate an optimization problem: given a graph and an integer $K$, we want to order graph vertices and partition the ordered adjacency matrix into $K$ bands such that bands closer to the diagonal are more dense. We measure the goodness of a segmentation using the log-likelihood of a log-linear model, a flexible family of distributions containing many standard distributions. We divide the problem into two subproblems: finding the order and finding the bands. We show that discovering bands can be done in polynomial time with isotonic regression, and we also introduce a heuristic iterative approach. For discovering the order we use Fiedler order accompanied with a simple combinatorial refinement. We demonstrate empirically that our heuristic works well in practice.

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arXiv2019

Using Background Knowledge to Rank Itemsets

Assessing the quality of discovered results is an important open problem in data mining. Such assessment is particularly vital when mining itemsets, since commonly many of the discovered patterns can be easily explained by background knowledge. The simplest approach to screen uninteresting patterns is to compare the observed frequency against the independence model. Since the parameters for the independence model are the column margins, we can view such screening as a way of using the column margins as background knowledge. In this paper we study techniques for more flexible approaches for infusing background knowledge. Namely, we show that we can efficiently use additional knowledge such as row margins, lazarus counts, and bounds of ones. We demonstrate that these statistics describe forms of data that occur in practice and have been studied in data mining. To infuse the information efficiently we use a maximum entropy approach. In its general setting, solving a maximum entropy model is infeasible, but we demonstrate that for our setting it can be solved in polynomial time. Experiments show that more sophisticated models fit the data better and that using more information improves the frequency prediction of itemsets.

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arXiv2019

Fast Sequence Segmentation using Log-Linear Models

Sequence segmentation is a well-studied problem, where given a sequence of elements, an integer K, and some measure of homogeneity, the task is to split the sequence into K contiguous segments that are maximally homogeneous. A classic approach to find the optimal solution is by using a dynamic program. Unfortunately, the execution time of this program is quadratic with respect to the length of the input sequence. This makes the algorithm slow for a sequence of non-trivial length. In this paper we study segmentations whose measure of goodness is based on log-linear models, a rich family that contains many of the standard distributions. We present a theoretical result allowing us to prune many suboptimal segmentations. Using this result, we modify the standard dynamic program for one-dimensional log-linear models, and by doing so reduce the computational time. We demonstrate empirically, that this approach can significantly reduce the computational burden of finding the optimal segmentation.

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