Spectral dimensionality reduction is frequently used to identify low-dimensional structure in high-dimensional data. However, learning manifolds, especially from the streaming data, is computationally and memory expensive. In this paper, we argue that a stable manifold can be learned using only a fraction of the stream, and the remaining stream can be mapped to the manifold in a significantly less costly manner. Identifying the transition point at which the manifold is stable is the key step. We present error metrics that allow us to identify the transition point for a given stream by quantitatively assessing the quality of a manifold learned using Isomap. We further propose an efficient mapping algorithm, called S-Isomap, that can be used to map new samples onto the stable manifold. We describe experiments on a variety of data sets that show that the proposed approach is computationally efficient without sacrificing accuracy.
Given the high computational complexity of Isomap, variants of Isomap, such as Landmark Isomap @cite_17 and out-of-sample extension techniques @cite_1 , have been proposed as a computationally viable alternative. Both of these methods either use a smaller set of landmark points or approximations to avoid performing the costly eigen decomposition on the @math geodesic distance matrix, where @math is the number of points in the entire data set. However, they still require computing the full geodesic distance matrix which is @math , where @math is the diameter of the embedded @math NN graph. The Incremental Isomap algorithm @cite_10 avoids both eigen decomposition and a recreation of the geodesic distance matrix. However, it requires updates to the geodesic distance matrix that incurs a significant cost, as discussed in . Consequently, the method is unsuitable for the streaming setting .
@cite_10 @cite_1 @cite_17
2143839989 2162114735 2121765205 2172107427
We propose a numerical method for finding packings of multiple-input and multiple-output (MIMO) semi-unitary precoding matrices in Grassmannian manifolds with different metrics. The proposed expansion-compression algorithm (ECA) is practical, simple and produces efficient packings without the need for a sophisticated initialization. With chordal distance metric, the algorithm tends to converge into a degenerated point constellation, where two points contain identical as well as orthogonal columns and distance between them cannot increase further along geodesic. Therefore, we alternate between max-min and min-max clustering parts of ECA algorithm, where the latter prevents degenerated constellations. With Fubini-Study distance metric, the algorithm converges to best known packings without extra min-max processing. We present a heuristic algorithm to compute approximate geodesic distances on a triangular manifold S containing n vertices with partially missing data. The proposed method computes an approximation of the geodesic distance between two vertices pi and pj on S and provides an upper bound of the geodesic distance that is shown to be optimal in the worst case. This yields a relative error bound of the estimate that is worst-case optimal. The algorithm approximates the geodesic distance without trying to reconstruct the missing data by embedding the surface in a low dimensional space via multi-dimensional scaling (MDS). We derive a new heuristic method to add an object to the embedding computed via least-squares MDS. In this paper, we present a new framework for geo-locating an image utilizing a novel multiple nearest neighbor feature matching method using Generalized Minimum Clique Graphs (GMCP). First, we extract local features (e.g., SIFT) from the query image and retrieve a number of nearest neighbors for each query feature from the reference data set. Next, we apply our GMCP-based feature matching to select a single nearest neighbor for each query feature such that all matches are globally consistent. Our approach to feature matching is based on the proposition that the first nearest neighbors are not necessarily the best choices for finding correspondences in image matching. Therefore, the proposed method considers multiple reference nearest neighbors as potential matches and selects the correct ones by enforcing consistency among their global features (e.g., GIST) using GMCP. In this context, we argue that using a robust distance function for finding the similarity between the global features is essential for the cases where the query matches multiple reference images with dissimilar global features. Towards this end, we propose a robust distance function based on the Gaussian Radial Basis Function (G-RBF). We evaluated the proposed framework on a new data set of 102k street view images; the experiments show it outperforms the state of the art by 10 percent. In this paper we study approximate landmark-based methods for point-to-point distance estimation in very large networks. These methods involve selecting a subset of nodes as landmarks and computing offline the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, it can be estimated quickly by combining the precomputed distances. We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. We therefore explore theoretical insights to devise a variety of simple methods that scale well in very large networks. The efficiency of the suggested techniques is tested experimentally using five real-world graphs having millions of edges. While theoretical bounds support the claim that random landmarks work well in practice, our extensive experimentation shows that smart landmark selection can yield dramatically more accurate results: for a given target accuracy, our methods require as much as 250 times less space than selecting landmarks at random. In addition, we demonstrate that at a very small accuracy loss our techniques are several orders of magnitude faster than the state-of-the-art exact methods. Finally, we study an application of our methods to the task of social search in large graphs.