Edge integrity of nearest neighbor graphs and separator theorems

In this paper, a spectral clustering method based on relation-invariable persistent formation (RIPF) is proposed, which can character the similarity graph succinctly. RIPF is an optimal formation, which can remain the formation by using least edges in the comminution graph. After generating the RIPF, the algorithm builds a network graph weighted according to distance between neighborhoods, and then applies spectral clustering. Three methods to construct similarity graphs in standard spectral clustering are ε -neighborhood graph, k- nearest neighbor graph, or fully connected graph. However, ε -neighborhood graph or k -nearest neighbor graph is sensitive to the parameters ε or k, and fully connected similarity graph of which is redundant. The similarity graph is constructed by generating RIPF, which can model the concise local neighborhood relationships by using the least edges in similarity graph. In this paper, a new algorithm is proposed to weaken the parameter dependence while maintaining the solution quality. Experimental results show that out algorithm performs as well as or even better than the state-of-the-art standard spectral methods.

The nearest neighbor graph (NNG), defined for a set of points in Euclidean space, has found many uses in computational geometry and clustering analysis. Yet there seems to be surprisingly little knowledge about some basic properties of this graph. In this talk, we ask some natural questions that are motivated by geometric embedding problems. For example, in the simulation of many-body systems, it is desirable to map a set of particles to a regular data array so as to preserve neighborhood relations, i.e., to minimize the dilation of NNG. We will derive bounds on the dilation by studying the diameter of NNG. Other properties and applications of NNG will also be discussed. (Joint work with Mike Paterson)

The logic engine and the realization problem for nearest neighbor graphs

Spectral clustering has become a typical and efficient clustering method used in a variety of applications. The critical step of spectral clustering is the similarity measurement, which largely determines the performance of the spectral clustering method. In this paper, we propose a novel spectral clustering algorithm based on the local similarity measure of shared neighbors. This similarity measurement exploits the local density information between data points based on the weight of the shared neighbors in a directed k $$ k $$ -nearest neighbor graph with only one parameter k $$ k $$ , that is, the number of nearest neighbors. Numerical experiments on synthetic and real-world datasets demonstrate that our proposed algorithm outperforms other existing spectral clustering algorithms in terms of the clustering performance measured via the normalized mutual information, clustering accuracy, and F $$ F $$ -measure. As an example, the proposed method can provide an improvement of 15.82% in the clustering performance for the Soybean dataset.

We consider the problem of finding latency optimal storage codes in a geographically distributed storage network of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> nodes and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k \leq n$</tex> files, where inter-node communication involves certain round-trip times. The optimality is with respect to two metrics: the worst-case latency among files at each node, and the system-average latency across files and nodes. Storage can be uncoded where raw message files are stored on the nodes or coded where linear combinations of files are placed on some of the nodes. In our previous work, it was shown that a latency optimal uncoded scheme exists if and only if a certain extended graph associated with the storage network is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex>-colorable. In this paper, we explore the networks where this condition fails and thus require coded storage schemes for latency optimality. We first construct a family of worst-case latency optimal codes called MDS-chromatic codes that are built upon the vertex coloring of extended graph. Further, we derive necessary and sufficient conditions for binary-chromatic codes to exist, along with explicit construction of the codes. In specific networks that have unit-link (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-1$</tex>) -nearest neighbor graph, we show that there exists a MDSchromatic code that is also system-average latency optimal.

A k-ratio edge separator is a set of edges which separates a weighted graph into two disconnected sets of components neither of which contains more than k-1/k of the original graph's weight. An optimal quotient separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the weight of the smaller set of components is minimized. An optimal quotient k-ratio separator is an edge separator where the size of the separator (i.e., the number of edges) divided by the smaller of either 1/k of the total weight or the weight of the smaller set of components is minimized. In this paper we present an algorithm that finds the optimal quotient k-ratio separator for any k ≥ 3. We use the optimal quotient algorithm to obtain approximation algorithms for finding optimal k-ratio edge separators for any k ≥ 3. Given a planar graph with a size OPT k-ratio separator, we describe an algorithm which a finds k-ratio separator which costs less than O(OPT log n). More importantly the algorithm finds ck-ratio separators (for any c ≫ 1) which cost less than C(c)OPT, where C(c) depends only on c.

Manifold-ranking based retrieval using k-regular nearest neighbor graph

Hyperspectral band selection (BS) is an important technique to reduce data dimensionality for the classification applications of hyperspectral remote sensing images (HSIs). Recently, searching-based BS methods have received increasing attention for their ability to select the best subset of bands while preserving the essential information of the original data. However, existing searching-based BS methods neglect effective exploitation of the spatial and spectral prior information inherent in the data, thus limiting their performance. To address this problem, in this study, a novel unsupervised BS method called Spectral–Spatial Iterative Greedy Algorithm (SSIGA) is proposed. Specifically, to facilitate efficient local search using spectral information, SSIGA conducts clustering on all the bands by employing a K-means clustering method with balanced cluster size constraints and constructs a K-nearest neighbor graph for each cluster. Based on the nearest neighbor graphs, SSIGA can effectively explore the neighborhood solutions in local search. In addition, to efficiently evaluate the discriminability and information redundancy of the solution given by SSIGA using the spatial and spectral information of HSIs, we designed an effective objective function for SSIGA. The value of the objective function is derived by calculating the Fisher score for each band in the solution based on the results of the superpixel segmentation performed on the target HSI, as well as by computing the average information entropy and mutual information of the bands in the solution. Experimental results on three publicly available real HSI datasets demonstrate that the SSIG algorithm achieves superior performance compared to several state-of-the-art methods.

C.L.Monma and V.K.Wei [1986, J. Comb. Theory, Ser-B, 41, 141-181] proposed a unified approach to characterize several subclasses of chordal graphs using clique separator. The characterizations so obtained are called separator theorems. Separator theorems play an important role in designing algorithms in subclasses of chordal graphs. In this paper, we obtain separator theorems for interval graphs and proper interval graphs, which are subclasses of chordal graphs, following the framework of Monma and Wei.

In this paper, we study the problem of designing robust algorithms for computing the minimum spanning tree, the nearest neighbor graph, and the relative neighborhood graph of a set of points in the plane, under the Euclidean metric. We use the term “robust” to denote an algorithm that can properly handle degenerate configurations of the input (such as co-circularities and collinearities) and that is not affected by errors in the flow of control due to round-off approximations. Existing asymptotically optimal algorithms that compute such graphs are either suboptimal in terms of the arithmetic precision required for the implementation, or cannot handle degeneracies, or are based on complex data structures. We present a unified approach to the robust computation of the above graphs. The approach is a variant of the general region approach for the computation of proximity graphs based on Yao graphs, first introduced in Ref. 43 (A. C.-C. Yao, On constructing minimum spanning trees in [Formula: see text]-dimensional spaces and related problems, SIAM J. Comput. 11(4) (1982) 721–736). We show that a sparse supergraph of these geometric graphs can be computed in asymptotically optimal time and space, and requiring only double precision arithmetic, which is proved to be optimal. The arithmetic complexity of the approach is measured by using the notion of degree, introduced in Ref. 31 (G. Liotta, F. P. Preparata and R. Tamassia, Robust proximity queries: An illustration of degree-driven algorithm design, SIAM J. Comput. 28(3) (1998) 864–889) and Ref. 3 (J. D. Boissonnat and F. P. Preparata, Robust plane sweep for intersecting segments, SIAM J. Comput. 29(5) (2000) 1401–1421). As a side effect of our results, we solve a question left open by Katajainen27 (J. Katajainen, The region approach for computing relative neighborhood graphs in the [Formula: see text] metric, Computing 40 (1987) 147–161) about the existence of a subquadratic algorithm, based on the region approach, that computes the relative neighborhood graph of a set of points [Formula: see text] in the plane under the [Formula: see text] metric.

The expected size of some graphs in computational geometry

Affine invariant triangulations

In this paper, a novel infrared maritime small target detection method, called local dissimilarity measure with anti interference based on global graph clustering (LDMGGC), is proposed. The Wasserstein distance is introduced to calculate the dissimilarity of gray level distribution between a central region and its neighborhoods. These dissimilarities construct the feature of a region. With this feature, detection for recalling all suspected targets is achieved. As the maritime interferences among suspected targets are able to be clustered, relaxing mutual k nearest neighbor graph is introduced in global graph clustering for filtering interferences. With this method, real targets are detected and maritime interferences are filtered out. Experiments are conducted on three maritime datasets and a non-maritime dataset for comparison. On three datasets, the proposed method achieves the best Receiver Operating Characteristic curves and Area Under Curve (0.99529, 0.99945, 0.99573, and 0.9906) values, meaning that the proposed method has high detection probability and low false-alarm ratio. Target Hit Rate (98.04%, 97.96%, 100%, and 99.24%) and Intersection of Union (0.8170, 0.7542, 0.5824, 0.7707) on four datasets of the proposed method show it has a strong ability to suppress the interferences

From one graph to many: Ensemble transduction for content-based database retrieval

We propose a novel non-parametric adaptive outlier detection algorithm, called LPE, for high dimensional data based on score functions derived from nearest neighbor graphs on n-point nominal data. Outliers are predicted whenever the score of a test sample falls below α, which is supposed to be the desired false alarm level. The resulting outlier detector is shown to be asymptotically optimal in that it is uniformly most powerful for the specified false alarm level, α, for the case when the density associated with the outliers is a mixture of the nominal and a known density. Our algorithm is computationally efficient, being linear in dimension and quadratic in data size. The whole empirical Receiving Operating Characteristics (ROC) curve can be derived with almost no additional cost based on the estimated score function. It does not require choosing complicated tuning parameters or function approximation classes and it can adapt to local structure such as local change in dimensionality by incorporating the technique of manifold learning. We demonstrate the algorithm on both artificial and real data sets in high dimensional feature spaces.