Keyphrase extraction is the task of automatically selecting a small set of phrases that best describe a given free text document Supervised keyphrase extraction requires large amounts of labeled training data and generalizes very poorly outside the domain of the training data At the same time, unsupervised systems have poor accuracy, and often do not generalize well, as they require the input document to belong to a larger corpus also given as input Addressing these drawbacks, in this paper, we tackle keyphrase extraction from single documents with EmbedRank: a novel unsupervised method, that leverages sentence embeddings EmbedRank achieves higher F-scores than graph-based state of the art systems on standard datasets and is suitable for real-time processing of large amounts of Web data With EmbedRank, we also explicitly increase coverage and diversity among the selected keyphrases by introducing an embedding-based maximal marginal relevance (MMR) for new phrases A user study including over 200 votes showed that, although reducing the phrases’ semantic overlap leads to no gains in F-score, our high diversity selection is preferred by humans

/pdf/simple-unsupervised-keyphrase-extraction-using-sentence-e84euy7fjy.pdf

Simple Unsupervised Keyphrase Extraction using Sentence Embeddings

We address the problem of image collection summarization by learning mixtures of submodular functions. Submodularity is useful for this problem since it naturally represents characteristics such as fidelity and diversity, desirable for any summary. Several previously proposed image summarization scoring methodologies, in fact, instinctively arrived at submodularity. We provide classes of submodular component functions (including some which are instantiated via a deep neural network) over which mixtures may be learnt. We formulate the learning of such mixtures as a supervised problem via large-margin structured prediction. As a loss function, and for automatic summary scoring, we introduce a novel summary evaluation method called V-ROUGE, and test both submodular and non-submodular optimization (using the submodular-supermodular procedure) to learn a mixture of submodular functions. Interestingly, using non-submodular optimization to learn submodular functions provides the best results. We also provide a new data set consisting of 14 real-world image collections along with many human-generated ground truth summaries collected using Amazon Mechanical Turk. We compare our method with previous work on this problem and show that our learning approach outperforms all competitors on this new data set. This paper provides, to our knowledge, the first systematic approach for quantifying the problem of image collection summarization, along with a new data set of image collections and human summaries.

/pdf/learning-mixtures-of-submodular-functions-for-image-2c4qpoa7hc.pdf

Learning Mixtures of Submodular Functions for Image Collection Summarization

Result diversification has many important applications in databases, operations research, information retrieval, and finance. In this paper, we study and extend a particular version of result diversification, known as max-sum diversification. More specifically, we consider the setting where we are given a set of elements in a metric space and a set valuation function f defined on every subset. For any given subset S, the overall objective is a linear combination of f(S) and the sum of the distances induced by S. The goal is to find a subset S satisfying some constraints that maximizes the overall objective.This problem is first studied by Gollapudi and Sharma in [17] for modular set functions and for sets satisfying a cardinality constraint (uniform matroids). In their paper, they give a 2-approximation algorithm by reducing to an earlier result in [20]. The first part of this paper considers an extension of the modular case to the monotone submodular case, for which the algorithm in [17] no longer applies. Interestingly, we are able to maintain the same 2-approximation using a natural, but different greedy algorithm. We then further extend the problem by considering any matroid constraint and show that a natural single swap local search algorithm provides a 2-approximation in this more general setting. This extends the Nemhauser, Wolsey and Fisher approximation result [20] for the problem of submodular function maximization subject to a matroid constraint (without the distance function component).The second part of the paper focuses on dynamic updates for the modular case. Suppose we have a good initial approximate solution and then there is a single weight-perturbation either on the valuation of an element or on the distance between two elements. Given that users expect some stability in the results they see, we ask how easy is it to maintain a good approximation without significantly changing the initial set. We measure this by the number of updates, where each update is a swap of a single element in the current solution with a single element outside the current solution. We show that we can maintain an approximation ratio of 3 by just a single update if the perturbation is not too large.

Max-Sum diversification, monotone submodular functions and dynamic updates

We investigate the performance of the standard Greedy algorithm for cardinality constrained maximization of non-submodular nondecreasing set functions. While there are strong theoretical guarantees on the performance of Greedy for maximizing submodular functions, there are few guarantees for non-submodular ones. However, Greedy enjoys strong empirical performance for many important non-submodular functions, e.g., the Bayesian A-optimality objective in experimental design. We prove theoretical guarantees supporting the empirical performance. Our guarantees are characterized by a combination of the (generalized) curvature $\alpha$ and the submodularity ratio $\gamma$. In particular, we prove that Greedy enjoys a tight approximation guarantee of $\frac{1}{\alpha}(1- e^{-\gamma\alpha})$ for cardinality constrained maximization. In addition, we bound the submodularity ratio and curvature for several important real-world objectives, including the Bayesian A-optimality objective, the determinantal function of a square submatrix and certain linear programs with combinatorial constraints. We experimentally validate our theoretical findings for both synthetic and real-world applications.

Guarantees for Greedy Maximization of Non-submodular Functions with Applications

The classical K Shortest Paths (KSP) problem, which identifies the $k$ shortest paths in a directed graph, plays an important role in many application domains, such as providing alternative paths for vehicle routing services. However, the returned $k$ shortest paths may be highly similar , i.e., sharing significant amounts of edges, thus adversely affecting service qualities. In this paper, we formalize the K Shortest Paths with Diversity (KSPD) problem that identifies top- $k$ shortest paths such that the paths are dissimilar with each other and the total length of the paths is minimized. We first prove that the KSPD problem is NP-hard and then propose a generic greedy framework to solve the KSPD problem in the sense that (1) it supports a wide variety of path similarity metrics which are widely adopted in the literature and (2) it is also able to efficiently solve the traditional KSP problem if no path similarity metric is specified. The core of the framework includes the use of two judiciously designed lower bounds, where one is dependent on and the other one is independent on the chosen path similarity metric, which effectively reduces the search space and significantly improves efficiency. Empirical studies on five real-world and synthetic graphs and five different path similarity metrics offer insight into the design properties of the proposed general framework and offer evidence that the proposed lower bounds are effective.

Finding Top-k Shortest Paths with Diversity

Result diversification is an important aspect in web-based search, document summarization, facility location, portfolio management and other applications. Given a set of ranked results for a set of objects (e.g. web documents, facilities, etc.) with a distance between any pair, the goal is to select a subset $S$ satisfying the following three criteria: (a) the subset $S$ satisfies some constraint (e.g. bounded cardinality); (b) the subset contains results of high "quality"; and (c) the subset contains results that are "diverse" relative to the distance measure. The goal of result diversification is to produce a diversified subset while maintaining high quality as much as possible. We study a broad class of problems where the distances are a metric, where the constraint is given by independence in a matroid, where quality is determined by a monotone submodular function, and diversity is defined as the sum of distances between objects in $S$. Our problem is a generalization of the {\em max sum diversification} problem studied in \cite{GoSh09} which in turn is a generaliztion of the {\em max sum $p$-dispersion problem} studied extensively in location theory. It is NP-hard even with the triangle inequality. We propose two simple and natural algorithms: a greedy algorithm for a cardinality constraint and a local search algorithm for an arbitary matroid constraint. We prove that both algorithms achieve constant approximation ratios.

Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates

Result diversification is an important aspect in web-based search, document summarization, facility location, portfolio management, and other applications. Given a set of ranked results for a set of objects (e.g., web documents, facilities, etc.) with a distance between any pair, the goal is to select a subset S satisfying the following three criteria: (a) the subset S satisfies some constraint (e.g., bounded cardinality), (b) the subset contains results of high “quality,” and (c) the subset contains results that are “diverse” relative to the distance measure. The goal of result diversification is to produce a diversified subset while maintaining high quality as much as possible. We study a broad class of problems where the distances are a metric, where the constraint is given by independence in a matroid, where quality is determined by a monotone submodular function and diversity is defined as the sum of distances between objects in S. Our problem is a generalization of the max-sum diversification problem studied in Gollapudi and Sharma [2009], which in turn is a generalization of the max-sum p-dispersion problem studied extensively in location theory. It is NP-hard even with the triangle inequality. We propose two simple and natural algorithms: a greedy algorithm for a cardinality constraint and a local search algorithm for an arbitrary matroid constraint. We prove that both algorithms achieve constant approximation ratios.

pdf/max-sum-diversification-monotone-submodular-functions-and-1exom27gps.pdf

Max-Sum Diversification, Monotone Submodular Functions, and Dynamic Updates

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn-1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity nn-1 + n - 1, and two-sided ideals and factor-closed languages of syntactic complexity nn-2 + (n - 2)2n-2 + 1.

Syntactic complexity of ideal and closed languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, respectively, and exhibit languages with these syntactic complexities.

Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. 
We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases.

Weakly Submodular Functions.

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