Order Restricted Statistical Inference.

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.

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Learning Mixtures of Submodular Functions for Image Collection Summarization

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

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Simple Unsupervised Keyphrase Extraction using Sentence Embeddings

In this paper we consider efficient construction of "composable core-sets" for basic diversity and coverage maximization problems. A core-set for a point-set in a metric space is a subset of the point-set with the property that an approximate solution to the whole point-set can be obtained given the core-set alone. A composable core-set has the property that for a collection of sets, the approximate solution to the union of the sets in the collection can be obtained given the union of the composable core-sets for the point sets in the collection. Using composable core-sets one can obtain efficient solutions to a wide variety of massive data processing applications, including nearest neighbor search, streaming algorithms and map-reduce computation.Our main results are algorithms for constructing composable core-sets for several notions of "diversity objective functions", a topic that attracted a significant amount of research over the last few years. The composable core-sets we construct are small and accurate: their approximation factor almost matches that of the best "off-line" algorithms for the relevant optimization problems (up to a constant factor). Moreover, we also show applications of our results to diverse nearest neighbor search, streaming algorithms and map-reduce computation. Finally, we show that for an alternative notion of diversity maximization based on the maximum coverage problem small composable core-sets do not exist.

https://dl.acm.org/doi/pdf/10.1145/2594538.2594560

Composable core-sets for diversity and coverage maximization

Graph pattern matching has been widely used in e.g., social data analysis. A number of matching algorithms have been developed that, given a graph pattern Q and a graph G, compute the set M(Q,G) of matches of Q in G. However, these algorithms often return an excessive number of matches, and are expensive on large real-life social graphs. Moreover, in practice many social queries are to find matches of a specific pattern node, rather than the entire M(Q,G).This paper studies top-k graph pattern matching. (1) We revise graph pattern matching defined in terms of simulation, by supporting a designated output node uo. Given G and Q, it is to find those nodes in M(Q,G) that match uo, instead of the large set M(Q,G). (2) We study two classes of functions for ranking the matches: relevance functions δr() based on, e.g., social impact, and distance functions δd() to cover diverse elements. (3) We develop two algorithms for computing top-k matches of uo based on δr(), with the early termination property, i.e., they find top-k matches without computing the entire M(Q,G). (4) We also study diversified top-k matching, a bi-criteria optimization problem based on both δr() and δd(). We show that its decision problem is NP-complete. Nonetheless, we provide an approximation algorithm with performance guarantees and a heuristic one with the early termination property. (5) Using real-life and synthetic data, we experimentally verify that our (diversified) top-k matching algorithms are effective, and outperform traditional matching algorithms in efficiency.

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Diversified top-k graph pattern matching

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

In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = (V, E) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v1, …, vn such that the property P holds on the induced subgraph G[N(vi)∩ Vi], where N(vi) is the neighborhood of vi and Vi = {vi, …, vn}. It turns out that if we take P as a graph with maximum independent set size no greater than k, then this definition gives a natural generalization of both chordal graphs and (k + 1)-claw-free graphs. We refer to such graphs as inductive k-independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3-independent graph; furthermore, any planar graph is an inductive 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for inductive k-independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.

Elimination graphs

A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v , the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [1] define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially k -independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially k -independent for small k . In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3-independent graph; furthermore, any planar graph is a sequentially 3-independent graph. For any fixed constant k , we develop simple, polynomial time approximation algorithms for sequentially k -independent graphs with respect to several well-studied NP-complete problems.

Elimination Graphs

Greedy algorithms are simple, but their relative power is not well understood. The priority framework (Borodin et al. in Algorithmica 37:295–326, 2003) captures a key notion of “greediness” in the sense that it processes (in some locally optimal manner) one data item at a time, depending on and only on the current knowledge of the input. This algorithmic model provides a tool to assess the computational power and limitations of greedy algorithms, especially in terms of their approximability. In this paper, we study priority algorithm approximation ratios for the Subset-Sum Problem, focusing on the power of revocable decisions, for which the accepted data items can be later rejected to maintain the feasibility of the solution. We first provide a tight bound of α≈0.657 for irrevocable priority algorithms. We then show that the approximation ratio of fixed order revocable priority algorithms is between β≈0.780 and γ≈0.852, and the ratio of adaptive order revocable priority algorithms is between 0.8 and δ≈0.893.

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Priority algorithms for the subset-sum problem

Greedy algorithms are simple, but their relative power is not well understood. The priority framework [5] captures a key notion of "greediness" in the sense that it processes (in some locally optimal manner) one data item at a time, depending on and only on the current knowledge of the input. This algorithmic model provides a tool to assess the computational power and limitations of greedy algorithms, especially in terms of their approximability. In this paper, we study priority algorithm approximation ratios for the Subset-Sum Problem, focusing on the power of revocable decisions. We first provide a tight bound of α ≈ 0.657 for irrevocable priority algorithms. We then show that the approximation ratio of fixed order revocable priority algorithms is between β; ≈ 0.780 and γ ≈ 0.852, and the ratio of adaptive order revocable priority algorithms is between 0.8 and δ ≈ 0.893.

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Yuli Ye

Papers

Max-Sum diversification, monotone submodular functions and dynamic updates

Elimination graphs

Elimination Graphs

Priority algorithms for the subset-sum problem

Priority algorithms for the subset-sum problem