Nearest neighbors revisited

The logic engine and the realization problem for nearest neighbor graphs

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.

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.

Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive isolines generation

Two strongly NP-hard problems of clustering a finite set of points in Euclidean space are considered. In the first problem, given an input set, we need to find a cluster (i.e., a subset) of given size that minimizes the sum of the squared distances between the elements of this cluster and its centroid (geometric center). Every point outside this cluster is considered a singleton cluster. In the second problem, we need to partition a finite set into two clusters minimizing the sum, over both clusters, of weighted intracluster sums of the squared distances between the elements of the clusters and their centers. The center of one of the clusters is unknown and is determined as its centroid, while the center of the other cluster is set at some point of space (without loss of generality, at the origin). The weighting factors for both intracluster sums are the given cluster sizes. Parameterized randomized algorithms are presented for both problems. For given upper bounds on the relative error and the failure probability, the parameter value is defined for which both algorithms find approximation solutions in polynomial time. This running time is linear in the space dimension and the size of the input set. The conditions are found under which the algorithms are asymptotically exact and their time complexity is linear in the space dimension and quadratic in the input set size.

The expected size of some graphs in computational geometry

Edge integrity of nearest neighbor graphs and separator theorems

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.

$$K$$ -nearest neighbor (K-NN) graphs are an essential component of many established methods for content-based image retrieval and automated image annotation. The performance of such methods relies heavily on the semantic quality of the graphs, which can be measured as the proportion of neighbors sharing the same class label as their query images. Due to the noise in image features, the K-NN graphs produced by existing methods may suffer from low semantic quality. In this article, we propose NNF-Descent for the efficient construction of K-NN graphs based on nearest-neighbor and feature descent, in which selective sparsification of feature vectors is interleaved with neighborhood refinement operations in an effort to improve the semantic quality of the result. A variant of the Laplacian Score is proposed for the identification of noisy features local to individual images, whose values are then set to $$0$$ (the global mean value after standardization). We show through extensive experiments on several datasets that NNF-Descent is able to increase the proportion of semantically-related images over unrelated images within the neighbor sets, and that the proposed method generalizes well for other types of data which are represented by high-dimensional feature vectors.

Basic program for a significance test for two clusters in euclidean space as measured by their overlap

Distributed computation of the knn graph for large high-dimensional point sets

Quantum computing introduces a revolutionary approach to solving problems that are computationally beyond the reach of classical systems. Among its most promising applications is the simulation of many-body systems, where strong particle interactions lead to complex quantum behaviors. Traditional computational techniques often fail due to exponential resource requirements, but quantum simulators can efficiently model these interactions through entanglement and superposition. Recent advances in superconducting qubits, trapped ions, and photonic quantum devices have enabled small- scale experimental demonstrations of spin models, Hubbard systems, and correlated electron dynamics. These developments offer new insights into condensed matter physics and quantum phase transitions. As quantum technologies continue to evolve, they hold the potential to transform the study of materials, superconductivity, and other many-body phenomena with remarkable precision and scalability.

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

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.

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