Knowledge Refinement via Rule Selection

In several different applications, including data transformation and entity resolution, rules are used to capture aspects of knowledge about the application at hand. Often, a large set of such rules is generated automatically or semi-automatically, and the challenge is to refine the encapsulated knowledge by selecting a subset of rules based on the expected operational behavior of the rules on available data. In this paper, we carry out a systematic complexity-theoretic investigation of the following rule selection problem: given a set of rules specified by Horn formulas, and a pair of an input database and an output database, find a subset of the rules that minimizes the total error, that is, the number of false positive and false negative errors arising from the selected rules. We first establish computational hardness results for the decision problems underlying this minimization problem, as well as upper and lower bounds for its approximability. We then investigate a bi-objective optimization version of the rule selection problem in which both the total error and the size of the selected rules are taken into account. We show that testing for membership in the Pareto front of this bi-objective optimization problem is DP-complete. Finally, we show that a similar DP-completeness result holds for a bi-level optimization version of the rule selection problem, where one minimizes first the total error and then the size

Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Abstract. Matrix factorizations are a popular tool to mine regularities from data. There are many ways to interpret the factorizations, but one particularly suited for data mining utilizes the fact that a matrix product can be interpreted as a sum of rank-1 matrices. Then the factorization of a matrix becomes the task of finding a small number of rank-1 matrices, sum of which is a good representation of the original matrix. Seen this way, it becomes obvious that many problems in data mining can be expressed as matrix factorizations with correct definitions of what a rank-1 matrix and a sum of rank-1 matrices mean. This paper develops a unified theory, based on generalized outer product operators, that encompasses many pattern set mining tasks. The focus is on the computational aspects of the theory and studying the computational complexity and approximability of many problems related to generalized matrix factorizations. The results immediately apply to a large number of data mining problems, and hopefully allow generalizing future results and algorithms, as well.

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

An Improved Algorithm for Learning to Perform Exception-Tolerant Abduction

Abstract Inference from an observed or hypothesized condition to a plausible cause or explanation for this condition is known as abduction. For many tasks, the acquisition of the necessary knowledge by machine learning has been widely found to be highly effective. However, the semantics of learned knowledge are weaker than the usual classical semantics, and this necessitates new formulations of many tasks. We focus on a recently introduced formulation of the abductive inference task that is thus adapted to the semantics of machine learning. A key problem is that we cannot expect that our causes or explanations will be perfect, and they must tolerate some error due to the world being more complicated than our formalization allows. This is a version of the qualification problem, and in machine learning, this is known as agnostic learning. In the work by Juba that introduced the task of learning to make abductive inferences, an algorithm is given for producing k-DNF explanations that tolerates such exceptions: if the best possible k-DNF explanation fails to justify the condition with probability , then the algorithm is promised to find a k-DNF explanation that fails to justify the condition with probability at most , where n is the number of propositional attributes used to describe the domain. Here, we present an improved algorithm for this task. When the best k-DNF fails with probability , our algorithm finds a k-DNF that fails with probability at most (i.e., suppressing logarithmic factors in n and ).We examine the empirical advantage of this new algorithm over the previous algorithm in two test domains, one of explaining conditions generated by a “noisy k-DNF rule, and another of explaining conditions that are actually generated by a linear threshold rule. We also apply the algorithm on the real world application Anomaly explanation. In this work, as opposed to anomaly detection, we are interested in finding possible descriptions of what may be causing anomalies in visual data. We use PCA to perform anomaly detection. The task is attaching semantics drawn from the image meta-data to a portion of the anomalous images from some source such as web-came. Such a partial description of the anomalous images in terms of the meta-data is useful both because it may help to explain what causes the identified anomalies, and also because it may help to identify the truly unusual images that defy such simple categorization. We find that it is a good match to apply our approximation algorithm on this task. Our algorithm successfully finds plausible explanations of the anomalies. It yields low error rate when the data set is large(\u3e80,000 inputs) and also works well when the data set is not very large(\u3c 50,000 examples). It finds small 2-DNFs that are easy to interpret and capture a non-negligible

CSP-Completeness And Its Applications

We build off of previous ideas used to study both reductions between CSPrefutation problems and improper learning and between CSP-refutation problems themselves to expand some hardness results that depend on the assumption that refuting random CSP instances are hard for certain choices of predicates (like k-SAT). First, we are able argue the hardness of the fundamental problem of learning conjunctions in a one-sided PAC-esque learning model that has appeared in several forms over the years. In this model we focus on producing a hypothesis that foremost guarantees a small false-positive rate while minimizing the false-negative rate for such hypotheses. Further, we formalize a notion of CSP-refutation reductions and CSP-refutation completeness that and use these, along with candidate CSP-refutatation complete predicates, to provide further evidence for the hardness of several problems

Review Selection Using Micro-Reviews

Singapore National Research Foundation under International Research Centre @ Singapore Funding Initiativ