Semi-definite rank minimization problems model a wide range of applications in both signal processing and machine learning fields. This class of problem is NP-hard in general. In this paper, we propose a proximal Alternating Direction Method (ADM) for the well-known semi-definite rank regularized minimization problem. Specifically, we first reformulate this NP-hard problem as an equivalent biconvex MPEC (Mathematical Program with Equilibrium Constraints), and then solve it using proximal ADM, which involves solving a sequence of structured convex semi-definite subproblems to find a desirable solution to the original rank regularized optimization problem. Moreover, based on the Kurdyka-Lojasiewicz inequality, we prove that the proposed method always converges to a KKT stationary point under mild conditions. We apply the proposed method to the widely studied and popular sensor network localization problem. Our extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art low-rank semi-definite minimization algorithms in terms of solution quality.

The Maximum Inner Product Search (MIPS) problem, prevalent in matrix factorization-based recommender systems, scales linearly with the number of objects to score. Recent work has shown that clever post-processing steps can turn the MIPS problem into a nearest neighbour one, allowing sublinear retrieval time either through Locality Sensitive Hashing or various tree structures that partition the Euclidian space. This work shows that instead of employing post-processing steps, substantially faster retrieval times can be achieved for the same accuracy when inference is not decoupled from the indexing process. By framing matrix factorization to be natively indexable, so that any solution is immediately sublinearly searchable, we use the machinery of Machine Learning to best learn such a solution. We introduce Indexable Probabilistic Matrix Factorization (IPMF) to shift the traditional post-processing complexity into the training phase of the model. Its inference procedure is based on Geodesic Monte Carlo, and adds minimal additional computational cost to standard Monte Carlo methods for matrix factorization. By coupling inference and indexing in this way, we achieve more than a 50% improvement in retrieval time against two state of the art methods, for a given level of accuracy in the recommendations of two large-scale recommender systems.

Large-scale Nuclear Norm penalized Least Square problem (NNLS) is frequently encountered in estimation of low rank structures. In this paper we accelerate the solution procedure by combining non-smooth convex optimization with smooth Riemannian method. Our methods comprise of two phases. In the first phase, we use Alternating Direction Method of Multipliers (ADMM) both to identify the fix rank manifold where an optimum resides and to provide an initializer for the subsequent refinement. In the second phase, two superlinearly convergent Riemannian methods: Riemannian NewTon (NT) and Riemannian Conjugate Gradient descent (CG) are adopted to improve the approximation over a fix rank manifold. We prove that our Hybrid method of ADMM and NT (HADMNT) converges to an optimum of NNLS at least quadratically. The experiments on large-scale collaborative filtering datasets demonstrate very competitive performance of these fast hybrid methods compared to the state-of-the-arts.

In this paper, we investigate the usage of autoencoders in modeling textual data. Traditional autoencoders suffer from at least two aspects: scalability with the high dimensionality of vocabulary size and dealing with task-irrelevant words. We address this problem by introducing supervision via the loss function of autoencoders. In particular, we first train a linear classifier on the labeled data, then define a loss for the autoencoder with the weights learned from the linear classifier. To reduce the bias brought by one single classifier, we define a posterior probability distribution on the weights of the classifier, and derive the marginalized loss of the autoencoder with Laplace approximation. We show that our choice of loss function can be rationalized from the perspective of Bregman Divergence, which justifies the soundness of our model. We evaluate the effectiveness of our model on six sentiment analysis datasets, and show that our model significantly outperforms all the competing methods with respect to classification accuracy. We also show that our model is able to take advantage of unlabeled dataset and get improved performance. We further show that our model successfully learns highly discriminative feature maps, which explains its superior performance.

Graph-based image segmentation organizes the image elements into graphs and partitions an image based on the graph. It has been widely used and many promising results are obtained. Since the segmentation performance highly depends on the graph, most of existing methods focus on obtaining a precise similarity graph or on designing efficient cutting/merging strategies. However, these two components are often conducted in two separated steps, and thus the obtained graph similarity may not be the optimal one for segmentation and this may lead to suboptimal results. In this paper, we propose a novel framework, Graph-Without-Cut (GWC), for learning the similarity graph and image segmentations simultaneously. GWC learns the similarity graph by assigning adaptive and optimal neighbors to each vertex based on the spatial and visual information. Meanwhile, the new rank constraint is imposed to the Laplacian matrix of the similarity graph, such that the connected components in the resulted similarity graph are exactly equal to the region number. Extensive empirical results on three public data sets (i.e, BSDS300, BSDS500 and MSRC) show that our unsupervised GWC achieves state-of-the-art performance compared with supervised and unsupervised image segmentation approaches.

Classical inconsistency-tolerant query answering relies on selecting maximal components of an ABox/database which are consistent with the ontology. However, some rules in ontologies might be unreliable if they are extracted from ontology learning or written by unskillful knowledge engineers. In this paper we present a framework of handling inconsistent existential rules under stable model semantics, which is defined by a notion called rule repairs to select maximal components of the existential rules. Surprisingly, for R-acyclic existential rules with R-stratified or guarded existential rules with stratified negations, both the data complexity and combined complexity of query answering under the rule repair semantics remain the same as that under the conventional query answering semantics. This leads us to propose several approaches to handle the rule repair semantics by calling answer set programming solvers. An experimental evaluation shows that these approaches have good scalability of query answering under rule repairs on realistic cases.

We consider a new formulation of abduction in which degrees of "plausibility" of explanations, along with the rules of the domain, are learned from concrete examples (settings of attributes). Our version of abduction thus falls in the " learning to reason " framework of Khardon and Roth. Such approaches enable us to capture a natural notion of "plausibility" in a domain while avoiding the extremely difficult problem of specifying an explicit representation of what is "plausible." We specifically consider the question of which syntactic classes of formulas have efficient algorithms for abduction. We find that the class of k -DNF explanations can be found in polynomial time for any fixed k ; but, we also find evidence that even weak versions of our abduction task are intractable for the usual class of conjunctions . This evidence is provided by a connection to the usual, inductive PAC-learning model proposed by Valiant. We also consider an exception-tolerant variant of abduction. We observe that it is possible for polynomial-time algorithms to tolerate a few adversarially chosen exceptions, again for the class of k -DNF explanations. All of the algorithms we study are particularly simple, and indeed are variants of a rule proposed by Mill.

The widespread adoption of Linked Data has been driven by the increasing demand for information exchange between organisations, as well as by data publishing regulations in domains such as health care and governance. In this setting, sensitive information is at risk of disclosure since published data can be linked with arbitrary external data sources. In this paper we lay the foundations of privacy-preserving data publishing (PPDP) in the context of Linked Data. We consider anonymisations of RDF graphs (and, more generally, relational datasets with labelled nulls) and define notions of safe and optimal anonymisations. Safety ensures that the anonymised data can be published with provable protection guarantees against linking attacks, whereas optimality ensures that it preserves as much information from the original data as possible, while satisfying the safety requirement. We establish the complexity of the underpinning decision problems both under open-world semantics inherent to RDF and a closed-world semantics, where we assume that an attacker has complete knowledge over some part of the original data.

In recent years, terrorist organizations (e.g., ISIS or al-Qaeda) are increasingly directing terrorists to launch coordinated attacks in their home countries. One example is the Paris shootings on January 7, 2015.By monitoring potential terrorists, security agencies are able to detect and stop terrorist plots at their planning stage.Although security agencies may have knowledge about potential terrorists (e.g., who they are, how they interact), they usually have limited resources and cannot monitor all terrorists.Moreover, a terrorist planner may strategically choose to arouse terrorists considering the security agency's monitoring strategy. This paper makes five key contributions toward the challenging problem of computing optimal monitoring strategies: 1) A new Stackelberg game model for terrorist plot detection;2) A modified double oracle framework for computing the optimal strategy effectively;3) Complexity results for both defender and attacker oracle problems;4) Novel mixed-integer linear programming (MILP) formulations for best response problems of both players;and 5) Effective approximation algorithms for generating suboptimal responses for both players.Experimental evaluation shows that our approach can obtain a robust enough solution outperforming widely-used centrality based heuristics significantly and scale up to realistic-sized problems.

Evolutionary game theory provides the principal tools to model the dynamics of multi-agent learning algorithms. While there is a long-standing literature on evolutionary game theory in strategic-form games, in the case of extensive-form games few results are known and the exponential size of the representations currently adopted makes the evolutionary analysis of such games unaffordable. In this paper, we focus on dynamics for the sequence form of extensive-form games, providing three dynamics: one realization equivalent to the normal-form logit dynamic, one realization equivalent to the agent-form replicator dynamic, and one realization equivalent to the agent-form logit dynamic. All the considered dynamics require polynomial time and space, providing an exponential compression w.r.t. the dynamics currently known and providing thus tools that can be effectively employed in practice. Moreover, we use our tools to compare the agent-form and normal-form dynamics and to provide new "hybrid" dynamics.