A geometrical pattern is a set of points with all pairwise distances (or, more generally, relative distances) specified. Finding matches to such patterns has applications to spatial data in seismic, astronomical, and transportation contexts. For example, a particularly interesting geometric pattern in astronomy is the Einstein cross, which is an astronomical phenomenon in which a single quasar is observed as four distinct sky objects (due to gravitational lensing) when captured by earth telescopes. Finding such crosses, as well as other geometric patterns, is a challenging problem as the potential number of sets of elements that compose shapes is exponentially large in the size of the dataset and the pattern. In this paper, we denote geometric patterns as constellation queries and propose algorithms to find them in large data applications. Our methods combine quadtrees, matrix multiplication, and unindexed join processing to discover sets of points that match a geometric pattern within some additive factor on the pairwise distances. Our distributed experiments show that the choice of composition algorithm (matrix multiplication or nested loops) depends on the freedom introduced in the query geometry through the distance additive factor. Three clearly identified blocks of threshold values guide the choice of the best composition algorithm. Finally, solving the problem for relative distances requires a novel continuous-to-discrete transformation. To the best of our knowledge this paper is the first to investigate constellation queries at scale.
Finding collections of objects having some metric relationship of interest is an area with many applications. The problem has different names depending on the discipline, including @cite_0 , @cite_13 , and @cite_18 .
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Metric learning aims at finding a distance that approximates a task-specific notion of semantic similarity. Typically, a Mahalanobis distance is learned from pairs of data labeled as being semantically similar or not. In this paper, we learn such metrics in a weakly supervised setting where "bags" of instances are labeled with "bags" of labels. We formulate the problem as a multiple instance learning (MIL) problem over pairs of bags. If two bags share at least one label, we label the pair positive, and negative otherwise. We propose to learn a metric using those labeled pairs of bags, leading to MildML, for multiple instance logistic discriminant metric learning. MildML iterates between updates of the metric and selection of putative positive pairs of examples from positive pairs of bags. To evaluate our approach, we introduce a large and challenging data set, Labeled Yahoo! News, which we have manually annotated and contains 31147 detected faces of 5873 different people in 20071 images. We group the faces detected in an image into a bag, and group the names detected in the caption into a corresponding set of labels. When the labels come from manual annotation, we find that MildML using the bag-level annotation performs as well as fully supervised metric learning using instance-level annotation. We also consider performance in the case of automatically extracted labels for the bags, where some of the bag labels do not correspond to any example in the bag. In this case MildML works substantially better than relying on noisy instance-level annotations derived from the bag-level annotation by resolving face-name associations in images with their captions. The problem of searching the elements of a set that are close to a given query element under some similarity criterion has a vast number of applications in many branches of computer science, from pattern recognition to textual and multimedia information retrieval. We are interested in the rather general case where the similarity criterion defines a metric space, instead of the more restricted case of a vector space. Many solutions have been proposed in different areas, in many cases without cross-knowledge. Because of this, the same ideas have been reconceived several times, and very different presentations have been given for the same approaches. We present some basic results that explain the intrinsic difficulty of the search problem. This includes a quantitative definition of the elusive concept of "intrinsic dimensionality." We also present a unified view of all the known proposals to organize metric spaces, so as to be able to understand them under a common framework. Most approaches turn out to be variations on a few different concepts. We organize those works in a taxonomy that allows us to devise new algorithms from combinations of concepts not noticed before because of the lack of communication between different communities. We present experiments validating our results and comparing the existing approaches. We finish with recommendations for practitioners and open questions for future development. Similarity search in metric spaces represents an important paradigm for content-based retrieval in many applications. Existing centralized search structures can speed-up retrieval, but they do not scale up to large volume of data because the response time is linearly increasing with the size of the searched file. In this article, we study the problem of executing the nearest neighbor(s) queries in a distributed metric structure, which is based on the P2P communication paradigm and the generalized hyperplane partitioning. By exploiting parallelism in a dynamic network of computers, the query execution scales up very well considering both the number of distance computations and the hop count between the peers. Results are verified by experiments on real-life data sets. The metric 2-clustering problem is defined as follows: given a metric (or weighted graph) (X,d), partition X into two sets S(1) and S(2) in order to minimize the value of spl Sigma sub i spl Sigma sub u,v spl sub S(i) d(u,v). In this paper, we show an approximation scheme for this problem.