A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification

$k$ Nearest Neighbors ($k$NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based $k$NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an $R$-level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new $k$NN algorithm and its improvements to other version of $k$NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional $k$NN algorithm, the proposed manifold version $k$NN shows promising potential for classifying manifold-distributed data.Comment: 32 pages, 12 figures, 7 table

Efficient query processing on spatial and textual data: beyond individual queries

With the increasing popularity of GPS enabled mobile devices, queries with locational intent are quickly becoming the most common type of search task on the web. This development has driven several research work on efficient processing of spatial and spatial-textual queries in the past few decades. While most of the existing work focus on answering queries independently, e.g., one query at a time, many real-life applications require the processing of multiple queries in a short period of time, and can benefit from sharing computations. This thesis focuses on efficient processing of the queries on spatial and spatial-textual data for the applications where multiple queries are of interest. Specifically, the following queries are studied: (i) batch processing of top-k spatial-textual queries; (ii) optimal location and keyword selection queries; and (iii) top-m rank aggregation on streaming spatial queries. The batch processing of queries is motivated from different application scenarios that require computing the result of multiple queries efficiently, including (i) multiple-query optimization, where the overall efficiency and throughput can be improved by grouping or partitioning a large set of queries; and (ii) continuous processing of a query stream, where in each time slot, the queries that have arrived can be processed together. In this thesis, given a set of top-k spatial-textual queries, the problem of computing the results for all the queries concurrently and efficiently as a batch is addressed. Some applications require an aggregation over the results of multiple queries. An exam- ple application is to identify the optimal value of attributes (e.g., location, text) for a new facility/service, so that the facility will appear in the query result of the maximum number of potential customers. This problem is essentially an aggregation (maximization) over the results of queries issued by multiple potential customers, where each user can be treated as a top-k query. In this thesis, we address this problem for spatial and textual data where the computations for multiple users are shared to find the final result. Rank aggregation is the problem of combining multiple rank orderings to produce a single ordering of the objects. Thus, aggregating the ranks of spatial objects can provide key insights into the importance of the objects in many different scenarios. This translates into a natural extension of the problem that finds the top-m objects with the highest aggregate rank over multiple queries. As the users issue new queries, clearly the rank aggregations continuously change over time, and recency also play an important role when interpreting the final results. The top-m rank aggregation of spatial objects for streaming queries is studied in this thesis, where the problem is to report the updated top-m objects with the highest aggregate rank over a subset of the most recent queries from a stream