Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

Investigators have tried to increase the precision of similarity searches of images by using distance functions that reflect the similarity of features. When the quadratic-form distance is used, however, dissimilar images can be judged to be similar. We therefore propose that the similarity of images be evaluated using a measure of distance in a multi-vector feature space based on pseudo-Euclidean space and an oblique basis (MVPO). In this space an image is represented by a set of vectors each of which represents each feature. And we propose a distance (called D-distance) between two sets of vectors. Roughly speaking, it is the distance between solids.Another representative distance used in similarity searches is the Earth Mover's Distance (EMD). It can be formalized using MVPO, and that explains well why EMD outperforms quad-ratic-form distance. The main difference between EMD and D-distance is that EMD is based on partial matching and D-distance is based on total matching.We also discuss performance issues of MPVO and D-distance to address practical use of them.

In this paper we present a new data structure for estimating distances in a pseudo-metric space. Given are a database of objects and a distance function for the objects, which is a pseudo-metric. We map the objects to vectors in a pseudo-Euclidean space with a reasonably low dimension while preserving the distance between two objects approximately. Such a data structure can be used as an approximate oracle to process a broad class of pattern-matching based queries. Experimental results on both synthetic and real data show the good performance of the oracle in distance estimation.

In this paper, we present an embedding technique, called MetricMap, which is capable of estimating distances in a pseudometric space. Given a database of objects and a distance function for the objects, which is a pseudometric, we map the objects to vectors in a pseudo-Euclidean space with a reasonably low dimension while preserving the distance between two objects approximately. Such an embedding technique can be used as an approximate oracle to process a broad class of distance-based queries. It is also adaptable to data mining applications such as data clustering and classification. We present the theory underlying MetricMap and conduct experiments to compare MetricMap with other methods including MVP-tree and M-tree in processing the distance-based queries. Experimental results on both protein and RNA data show the good performance and the superiority of MetricMap over the other methods.

Suppose a dispersed-dot dither matrix is treated as a collection of numbers, each number having a position in space; when the numbers are visited in increasing order, what is the distance in space between pairs of consecutive numbers visited? In Bayer’s matrices, this distance is always large. We hypothesize that this large consecutive distance is important for good dispersed-dot threshold matrices. To study the hypothesis, matrices that have this quality were generated by solving a more general problem: given an arbitrary set of points on the plane, sort them into a list where consecutive points are far apart. Our solution colors the nearestneighbor graph, hierarchically. The method does reproduce Bayer’s dispersed-dot dither matrices under some settings and, furthermore, can produce matrices of arbitrary dimensions. Multiple similar matrices can be created to minimize repetitive artifacts that plague Bayer dither while retaining its parallelizability. The method can also be used for halftoning with points on a hexagonal grid, or even randomly placed points. It can also be applied to artistic dithering, which creates a dither matrix from a motif image. Unlike in the artistic dither method of Ostromoukhov and Hersch, the motif image can be arbitrary and need not be specially constructed.

This study aims to analyze the spatial effects triggered by dollar liquidity by constructing a multidimensional spatial matrix that modifies the traditional monetary spatial framework. We utilized a three-level spatial econometric model (Spatial Lag, Durbin, and Generalized Nested Space) to measure Gross Domestic Product (GDP), Consumer Price Index (CPI), and Asset Price Bubbles (BBL) through five spillover channels (geography, linguistics, politics, war, and economy). Our aim is to establish a systematic relationship between the conduction mechanism, means, economic indicators, and dollar externalities to examine liquidity spillover effects at varying distances in the global monetary space. We find that the spatial effects induced by the global circulation of the US dollar behave significantly differently in a single matrix space compared to in a multidimensional space. While the model verifies the existence of a positive correlation between the complexity of a single space and the spillover effect from a conduction mechanism perspective, the measure of the multidimensional matrix shows that the significance of the spillover effect weakens with an increase in abstraction level from a conduction means perspective. It suggests that spatial matrices of different dimensions reflect different economic realities. The former shows hierarchical multivariate details in independent matrices, while the variation in the level of abstraction of matrices of different dimensions in the latter enhances their interactivity and complexity.