Classical Distance Research Articles

A new shape-based clustering algorithm for time series

Time series clustering is a research hotspot in data mining. Most of the existing clustering algorithms combine with the classical distance measure which ignore the offset of sequence shape. As a result, shape-based clustering algorithms are becoming increasingly popular. On the majority of data sets, the most representative shape-based clustering algorithm, KShape, which defines a shape-based distance with shift invariance, has been shown to outperform other algorithms.In this paper, we propose a new shape-based clustering algorithm named Fractional Order Shape-based k-cluster(FrOKShape), which defines a multi-variable shape-based distance by normalized fractional order cross-correlation and uses the DTW Barycenter Averaging (DBA) as a center computation strategy. Our distance exhibits excellent shape shift deviating properties and good compatibility integrated with a variety of existing clustering center strategies so that it can provide more potential good results. Experiments show that combining our distance with a traditional clustering algorithm produces excellent clustering indicators. In a series of comparative experiments, FrOKShape also exhibits a comparable result to the existing better shape-based clustering algorithm KShape.

Learning quantum data with the quantum earth mover’s distance

Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such as poor local minima and exponentially decaying gradients. To overcome these obstacles, we consider here the recently proposed quantum earth mover’s (EM) or Wasserstein-1 distance as a quantum analog to the classical EM distance. We show that the quantum EM distance possesses unique properties, not found in other commonly used quantum distance metrics, that make quantum learning more stable and efficient. We propose a quantum Wasserstein generative adversarial network (qWGAN) which takes advantage of the quantum EM distance and provides an efficient means of performing learning on quantum data. We provide examples where our qWGAN is capable of learning a diverse set of quantum data with only resources polynomial in the number of qubits.

Using K-Means Clustering in Python with Periodic Boundary Conditions

Periodic boundary conditions are natural in many scientific problems, and often lead to particular symmetries. Working with datasets that express periodicity properties requires special approaches when analyzing these phenomena. Periodic boundary conditions often help to solve or describe the problem in a much simpler way. The angular rotational symmetry is an example of periodic boundary conditions. This symmetry implies angular momentum conservation. On the other hand, clustering is one of the first and most basic methods used in data analysis. It is often a starting point when new data are acquired and understood. K-means clustering is one of the most commonly used clustering methods. It can be applied to many different situations with reasonably good results. Unfortunately, the original k-means approach does not cope well with the periodic properties of the data. For example, the original k-means algorithm treats a zero angle as very far from an angle that is 359 degrees. Periodic boundary conditions often change the classical distance measure and introduce an error in k-means clustering. In the paper, we discuss the problem of periodicity in the dataset and present a periodic k-means algorithm that modifies the original approach. Considering that many data scientists prefer on-the-shelf solutions, such as libraries available in Python, we present how easily they can incorporate periodicity into existing k-means implementation in the PyClustering library. It allows anyone to integrate periodic conditions without significant additional costs. The paper evaluates the described method using three different datasets: the artificial dataset, wind direction measurement, and the New York taxi service dataset. The proposed periodic k-means provides better results when the dataset manifests some periodic properties.

Communicating With Extremely Large-Scale Array/Surface: Unified Modeling and Performance Analysis

Wireless communications with extremely large-scale array (XL-array) correspond to systems whose antenna sizes are so large that conventional modeling assumptions, such as uniform plane wave (UPW) impingement, are no longer valid. This paper studies the mathematical modeling and performance analysis of XL-array communications. By deviating from the conventional modeling approach that treats the array elements as sizeless points, we explicitly model their physical area/aperture, which enables a unified modeling for the classical discrete antenna arrays and the emerging active continuous surfaces. As such, a generic array/surface model that accurately takes into account the variations of signal phase, amplitude and projected aperture across array elements is proposed. Based on the proposed model, a closed-form expression of the resulting signal-to-noise ratio (SNR) with the optimal single-user maximum ratio combining/transmission (MRC/MRT) beamforming is derived. The expression reveals that instead of scaling linearly with the antenna number <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> as in conventional UPW modeling, the SNR with the more generic model increases with <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> with diminishing return, which is governed by the collective properties of the array, such as the <i>array occupation ratio</i> and the physical sizes of the array along each dimension, while irrespective of the properties of the individual array element. In addition, we have derived an alternative insightful expression for the optimal SNR in terms of the <i>vertical</i> and <i>horizontal angular spans</i>, which are fully determined by the geometric angles formed by the array/surface and user location. Furthermore, we also show that our derived results include the far-field UPW modeling as a special case. One important finding during the study of far-field approximation is the necessity to introduce a new distance criterion to complement the classical Rayleigh distance, termed <i>uniform-power distance</i> (UPD), which concerns the signal amplitude/power variations across array elements, instead of phase variations as for Rayleigh distance. Extensive numerical results are provided to demonstrate the necessity of proper modeling for XL-array communications by comparing the proposed model with various benchmark models.

A new distance between multivariate clusters of varying locations, elliptical shapes, and directions

Clustering methods are based on the computations of both the distances between every pair of the n observations in a multivariate dataset as well as the distances between every pair of clusters in the dataset. The clusters can have different locations and varying elliptical shapes and directions. Numerous methods have been proposed in the literature for computing both of these two types of distances. The contributions of this paper are two folds. First, we propose a new elliptical distance between pairs of clusters in a dataset with different cluster centers and elliptical shapes and directions, Second, we proved analytically that the Ward distance and the Euclidean distance are equivalent. We propose a new classical method for computing the distance between a pair of clusters in the dataset. It is the only distance that does not assume spherical clusters. The proposed classical distances could also be made robust by replacing estimates of location and scale by their respective robust estimators. The proposed distance has a number of advantages including simplicity, interpretability, computational efficiency as well as the ability to accurately capture both the variability of the cluster centers as well as the variability of shapes and directions of their respective covariance matrices. The method is also illustrated by several motivating examples that demonstrate the need of the new proposed distance. The superiority of the proposed method is also demonstrated by application to real-life as well as challenging synthetic data.

Human-supervised clustering of multidimensional data using crowdsourcing.

Clustering is a central task in many data analysis applications. However, there is no universally accepted metric to decide the occurrence of clusters. Ultimately, we have to resort to a consensus between experts. The problem is amplified with high-dimensional datasets where classical distances become uninformative and the ability of humans to fully apprehend the distribution of the data is challenged. In this paper, we design a mobile human-computing game as a tool to query human perception for the multidimensional data clustering problem. We propose two clustering algorithms that partially or entirely rely on aggregated human answers and report the results of two experiments conducted on synthetic and real-world datasets. We show that our methods perform on par or better than the most popular automated clustering algorithms. Our results suggest that hybrid systems leveraging annotations of partial datasets collected through crowdsourcing platforms can be an efficient strategy to capture the collective wisdom for solving abstract computational problems.

Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Persistence diagrams, an important summary in topological data analysis, consist of a set of ordered pairs, each with positive multiplicity. Persistence diagrams are obtained via Mobius inversion and may be compared using a one-parameter family of metrics called Wasserstein distances. In certain cases, Mobius inversion produces sets of ordered pairs which may have negative multiplicity. We call these virtual persistence diagrams. Divol and Lacombe recently showed that there is a Wasserstein distance for Radon measures on the half plane of ordered pairs that generalizes both the Wasserstein distance for persistence diagrams and the classical Wasserstein distance from optimal transport theory. Following this work, we define compatible Wasserstein distances for persistence diagrams and Radon measures on arbitrary metric spaces. We show that the 1-Wasserstein distance extends to virtual persistence diagrams and to signed measures. In addition, we characterize the Cauchy completion of persistence diagrams with respect to the Wasserstein distances. We also give a universal construction of a Banach space with a 1-Wasserstein norm. Persistence diagrams with the 1-Wasserstein distance isometrically embed into this Banach space.

A New Adaptive Filter to Remove Impulsive Noise in Color Images

In this paper, a new adaptive vector directional distance filter (AVDDF) is proposed. The AVDDF filter is developed to remove the ‘salt and papper’ impulsive noise in color images. This algorithm consists in first step to detect pixels that are likely to have been contaminated with noise by using a threshold value. In second step, after recognizing the corrupted pixels, the smallest angular‐magnitude distance is used to replace the noisy pixels. The proposed filter is tested with several standard color images which are contaminated with various levels of ‘salt and pepper’ impulsive noise (3%, 5%, 10%, 20% and 30%). The performance of the proposed filter is measured with peak signal‐to‐noise ratio (PSNR) and normalized color difference (NCD). The simulation results show that the proposed filter permits to ameliorate the performance of the classic vector directional distance filter (VDDF). Indeed, it provides an improvement by an average of 5% and 26% in the PSNR and NCD relative to the VDDF filter, respectively, with a small increase in the execution time by 6%. Besides, the AVDDF filter allows also an enhancement by an average of 2% and 14% in the PSNR and NCD relative to adaptive vector median filter (AVMF), respectively. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

3GOLD: optimized Levenshtein distance for clustering third-generation sequencing data

BackgroundThird-generation sequencing offers some advantages over next-generation sequencing predecessors, but with the caveat of harboring a much higher error rate. Clustering-related sequences is an essential task in modern biology. To accurately cluster sequences rich in errors, error type and frequency need to be accounted for. Levenshtein distance is a well-established mathematical algorithm for measuring the edit distance between words and can specifically weight insertions, deletions and substitutions. However, there are drawbacks to using Levenshtein distance in a biological context and hence has rarely been used for this purpose. We present novel modifications to the Levenshtein distance algorithm to optimize it for clustering error-rich biological sequencing data.ResultsWe successfully introduced a bidirectional frameshift allowance with end-user determined accommodation caps combined with weighted error discrimination. Furthermore, our modifications dramatically improved the computational speed of Levenstein distance. For simulated ONT MinION and PacBio Sequel datasets, the average clustering sensitivity for 3GOLD was 41.45% (S.D. 10.39) higher than Sequence-Levenstein distance, 52.14% (S.D. 9.43) higher than Levenshtein distance, 55.93% (S.D. 8.67) higher than Starcode, 42.68% (S.D. 8.09) higher than CD-HIT-EST and 61.49% (S.D. 7.81) higher than DNACLUST. For biological ONT MinION data, 3GOLD clustering sensitivity was 27.99% higher than Sequence-Levenstein distance, 52.76% higher than Levenshtein distance, 56.39% higher than Starcode, 48% higher than CD-HIT-EST and 70.4% higher than DNACLUST.ConclusionOur modifications to Levenshtein distance have improved its speed and accuracy compared to the classic Levenshtein distance, Sequence-Levenshtein distance and other commonly used clustering approaches on simulated and biological third-generation sequenced datasets. Our clustering approach is appropriate for datasets of unknown cluster centroids, such as those generated with unique molecular identifiers as well as known centroids such as barcoded datasets. A strength of our approach is high accuracy in resolving small clusters and mitigating the number of singletons.

A massively parallel accurate conservative level set algorithm for simulating turbulent atomization on adaptive unstructured grids

This article presents a massively parallel and robust strategy to perform the simulation of turbulent incompressible two-phase flows on unstructured grids in complex geometries. This strategy relies on a combination of a narrow-band accurate conservative level set (ACLS)/ghost-fluid framework with isotropic adaptive mesh refinement. This combination enables to accurately capture interface dynamics and topology, and the small physical scales at the liquid-gas interface are resolved at an affordable cost. The ACLS method, even if not strictly mass-conserving, ensures the exact conservation of a smoothed phase indicator, which minimizes the liquid mass conservation errors. In the accurate conservative level set framework, presented first in Desjardins et al. (2008) [25], the interface is defined as the iso-contour of a hyperbolic tangent function, which is advected by the fluid, and then reshaped using a reinitialization equation. Several forms of this reinitialization exist: the original ACLS form proposed by Desjardins et al. involves numerical estimation of the hyperbolic tangent gradient, which is difficult to compute accurately on unstructured meshes. It is thus susceptible to induce artificial deformation of the interface. A new form has been recently proposed in Chiodi and Desjardins (2017) [29], which takes advantage of a mapping onto a classical distance level set while much better preserving the interface shape. Nevertheless, the implementation of this new form on unstructured grids requires special care. In this work, a robust implementation of this new form on unstructured meshes is proposed and implemented in the YALES2 low-Mach flow solver. In order to compute interface normals and curvature, the signed-distance function is reconstructed in parallel at nodes in the narrow band around the interface using a Geometric-Projection Marker Method (GPMM). This method relies on the triangulation of the level set iso-contour and exact geometric projection to the closest surface elements. Spatial convergence, robustness and efficiency of the overall procedure are firstly demonstrated through classical interface transport test cases and two-phase flow examples. Eventually, to emphasize the significant computational gain using adaptive mesh refinement and the ability to compute complex turbulent flows with large density ratios, two Large-Eddy Simulations (LES) of atomizing liquid jets in air are presented, each one at various resolutions. The first one is a low-pressure water jet in quiescent air from a compound nozzle with full computation of the internal injector flow, while the latter is a high-pressure kerosene jet in crossflow. Both simulations are validated against experiments, demonstrating the potential of the method to access a deep numerical insight into jet instabilities and internal flow dynamics with 3D unstructured meshes.

Multi-view distance metric learning via independent and shared feature subspace with applications to face and forest fire recognition, and remote sensing classification

Distance Metric Learning for Large Margin Nearest Neighbor (LMNN), as a classic distance metric learning (DML) method, has attracted much attention among researchers. However, it, like most of the existing DML methods, cannot be guaranteed to achieve independent and shared feature subspaces from multiple sources or different feature subsets, such that much statistical feature information is ignored in model learning. In this paper, we propose a novel DML model, called Multi-view DML Based on Independent and Shared Feature Subspace (MVML-ISFS), which learns multiple distance metrics to unify the information from multiple views. The proposed method finds a distance metric for each view in an independent feature space to preserve its specific property as well as a sparse representation related to the distance metrics from distinct views in a shared feature space to remain their common properties. The objective problem of MVML-ISFS is formulated based on LMNN, thus encouraging a large margin for each view that makes the distance between each of the same class pairs of samples be smaller than that between each of the different class pairs of samples. The proposed model in MVML-ISFS involves multivariate variables, which are optimized by a gradient descent strategy. The experimental results show the effectiveness of our MVML-ISFS on remote sensing, face, forest fire, and UCI datasets.

Optimal transport improves cell-cell similarity inference in single-cell omics data.

MotivationHigh‐throughput single-cell molecular profiling is revolutionizing biology and medicine by unveiling the diversity of cell types and states contributing to development and disease. The identification and characterization of cellular heterogeneity are typically achieved through unsupervised clustering, which crucially relies on a similarity metric.ResultsWe here propose the use of Optimal Transport (OT) as a cell–cell similarity metric for single-cell omics data. OT defines distances to compare high-dimensional data represented as probability distributions. To speed up computations and cope with the high dimensionality of single-cell data, we consider the entropic regularization of the classical OT distance. We then extensively benchmark OT against state-of-the-art metrics over 13 independent datasets, including simulated, scRNA-seq, scATAC-seq and single-cell DNA methylation data. First, we test the ability of the metrics to detect the similarity between cells belonging to the same groups (e.g. cell types, cell lines of origin). Then, we apply unsupervised clustering and test the quality of the resulting clusters. OT is found to improve cell–cell similarity inference and cell clustering in all simulated and real scRNA-seq data, as well as in scATAC-seq and single-cell DNA methylation data.Availability and implementationAll our analyses are reproducible through the OT-scOmics Jupyter notebook available at https://github.com/ComputationalSystemsBiology/OT-scOmics.Supplementary information Supplementary data are available at Bioinformatics online.

Deep Discriminative Feature Models (DDFMs) for Set Based Face Recognition and Distance Metric Learning.

This article introduces two methods that find compact deep feature models for approximating images in set based face recognition problems. The proposed method treats each image set as a nonlinear face manifold that is composed of linear components. To find linear components of the face manifold, we first split image sets into subsets containing face images which share similar appearances. Then, our first proposed method approximates each subset by using the center of the deep feature representations of images in those subsets. Centers modeling the subsets are learned by using distance metric learning. The second proposed method uses discriminative common vectors to represent image features in the subsets, and entire subset is approximated with an affine hull in this approach. Discriminative common vectors are subset centers that are projected onto a new feature space where the combined within-class variances coming from all subsets are removed. Our proposed methods can also be considered as distance metric learning methods using triplet loss function where the learned subcluster centers are the selected anchors. This procedure yields to applying distance metric learning to quantized data and brings many advantages over using classical distance metric learning methods. We tested proposed methods on various face recognition problems using image sets and some visual object classification problems. Experimental results show that the proposed methods achieve the state-of-the-art accuracies on the most of the tested image datasets.

EVALUATION OF MARE FAMILIES OF ORLOV’S TROTTER OF UKRAINIAN POPULATION HORSES

It was found that the domestic population of horses of the Orlov’s Trotter breed is 465 individuals, including mares of reproductive age - only 193, which indicates a significant limitation of this breed. The most numerous uterine families by number of successors: Vorozheyka and Fatma (8.3% in the uterine composition), Rumba (7.8% in the uterine composition) and Cadetka (7.3% in the uterine composition). Only 12 mares (6.2%) belong to a small number of uterine families, numbering 1-2 mares (Aristocratka, Kaluga, Pusti, Damochka, Zib, Lelya, Levitsa, Panika, Pika). The analyzed uterine families number from 3 to 10 generations from the ancestor to the representatives of the modern uterine composition. Evaluation of mare families on a set of breeding traits found that the highest liveliness at a classical distance of 1600 m on average characterized mares belonging to the uterine families of Lena (2.11,4 ± 0,65 min), Uh-Kakaya (2.12.2±0,28 min), Dosujaya (2.13,4±0,18 min), Barvikha (2.15,3±0,41 min), Vorojeyka (2.15,7±0,15 min), Cadetka (2.15,8± 0,22 min). The quietest mares belong to the parent families of Povest, Evil, Molodytsa, Kaleria. Mares from the mare families of Lena, Vorojeyka and Molodytsa differ in terms of measurement indicators. They are characterized by high stature, elongated body shape, typical of the harness type of horses, bulky chest. In terms of wrist circumference, all assessed mares do not exceed 20.8 cm, which indicates an insufficient level of skeletal development and, accordingly, is not negatively affected by agility. By dividing into gradations by liveliness classes, it was established that mares of the Orlov’s Trotter breed of the Ukrainian population of liveliness class 2.05 min and more vigorous were not detected. Of all tested mares (76.2 %), only 12.2 % have a high liveliness class of 2.10 min or more. However, mares of low liveliness class (2.30,1 min and quieter) are few - 10.9 %. The predominant number of mares have an average liveliness class of 2.10,1-2.20 min (55.1 %) and 2.20,1-2,30 min (21.8 %). Almost a quarter of mares (23.8 %) remain untested, which negatively affects the selection process. Most mares of high vitality class belong to the parent families: Barvikha, Dan’, Dosuzhaya, Rumba, Uh-Kakaya.

Average stride length and stride rate of Thoroughbreds and Quarter Horses during racing.

The main factors influencing speed in athletes are stride length (SL) and stride rate (SR). However, conflict remains whether SL or SR is the key determinant of higher speeds. Quarter Horses (QH) generally reach higher speeds in their races than do Thoroughbreds (TB). However, the influence of SL and SR on this greater speed is unclear. Therefore, the main objective of this study was to compare SL and SR in QH and TB raced in short (sprint) and long (classic) distances. We hypothesized that QH have a higher SR in comparison to TB, and SR decreases as distance increases. Two race distances were analyzed for each breed: QH races of 100.6 and 402.3 m, and TB races of 1,207.0 and 2,011.7 m. Data from 20 horses were obtained, consisting of five horses from each race distance (10 QH and 10 TB). Five individuals watched recordings of each race three times counting the number of strides taken by each winning horse. The SR was calculated using the average number of strides over a given race duration, and SL was determined by calculating the total number of strides over the distance covered. Speed was calculated by dividing the distance by the time of the winning horse. The PROC Mixed Procedure was used to identify statistical differences between breeds, and between distances within the same breed. Results showed that although the SL of the TB was longer in comparison with the QH (P < 0.001), the average SR in QH was higher than in TB (2.88 vs. 2.34 + 0.03 strides/s; P < 0.001). Furthermore, QH classic distance demonstrated a faster speed than TB at either distance (P < 0.001). In conclusion, QH achieve a higher SR in comparison to TB (between 14% and 20% more than TB), confirming the importance of SR in achieving high racing speeds.

UAV-Assisted Three-Dimensional Spectrum Mapping Driven by Spectrum Data and Channel Model

As the number of civil aerial vehicles increase explosively, spectrum scarcity and security become an increasingly challenge in both the airspace and terrestrial space. To address this difficulty, this paper presents an unmanned aerial vehicle-assisted (UAV-assisted) spectrum mapping system and a spectrum data reconstruction algorithm driven by spectrum data and channel model are proposed. The reconstruction algorithm, which includes a model-driven spectrum data inference method and a spectrum data completion method with uniformity decision mechanism, can reconstruct limited and incomplete spectrum data to a three-dimensional (3D) spectrum map. As a result, spectrum scarcity and security can be achieved. Spectrum mapping is a symmetry-based digital twin technology. By employing an uniformity decision mechanism, the proposed completion method can effectively interpolate spatial data even when the collected data are unevenly distributed. The effectiveness of the proposed mapping scheme is evaluated by comparing its results with the ray-tracing simulated data of the campus scenario. Simulation results show that the proposed reconstruction algorithm outperforms the classical inverse distance weighted (IDW) interpolation method and the tensor completion method by about 12.5% and 92.3%, respectively, in terms of reconstruction accuracy when the collected spectrum data are regularly missing, unevenly distributed and limited.

Energy efficient routing and task scheduling for autonomous transfer vehicles in intra logistics

Autonomous Transfer Vehicles (ATVs) are becoming increasingly prevalent in intra logistics. Industry 4.0 is bringing us closer to the efficient routing and scheduling of autonomous multi robot systems which perform transportation tasks. In this study, an energy efficient routing and scheduling system is proposed to minimize the total energy that the vehicles spend. Not only travelled distance but also the load of the vehicle is considered between two points. The routes of vehicles are obtained by using the proposed Hybrid Simulated Annealing Algorithm. An algorithm for the initial solution is also proposed for determining of the minimum number of vehicles for pickup and delivery requests. The performance of the algorithm is compared with the best solutions of the test problems in the literature. Besides, the proposed energy efficient routing and task scheduling model is compared with the classical distance model for routing and scheduling with backhauls. An analysis of trade-offs between energy and distance is proposed for intra logistics.

Estimating the fundamental niche: Accounting for the uneven availability of existing climates in the calibration area

Studies that question important conceptual and methodological aspects of the field of ecological niche modeling (and species distribution modeling) have cast doubts on whether it is possible to estimate the fundamental niche of a species using presence-only data. The main limitation in niche estimation is that presence data come from the realized niche, which is only a subset of the fundamental niche. Most of the existing methods lack the ability to overcome this limitation and therefore fit niches that resemble the realized niche. To obtain a more accurate estimate of the fundamental niche, we propose using the geographic region that is accessible to a species (based on its dispersal ability) to determine a sampling distribution, in environmental space, from which we can quantify the likelihood of observing a particular environmental combination in a sample of presence points. We incorporate this sampling distribution into a multivariate normal model (i.e., a Mahalanobis distance model) by creating a weight function that modifies the probabilities of observing different environmental combinations in a sample of presences. This modification accounts for the uneven availability of environmental conditions. We show that the parameters of this weighted-normal model can be approximated with a maximum likelihood estimation approach; and then used to draw ellipsoids (confidence regions) that represent the fundamental niche of the species. We illustrate the application of our model with two worked examples. First, we use presence data for an invasive species and an accessible area that includes only its native range to evaluate whether the fitted model predicts confirmed establishments of the species outside its native range. Second, we use presence data for closely related species with known accessible areas to demonstrate how the different dispersal abilities of the species constrain a classic Mahalanobis distance model. Overall, we show that accounting for the distribution of environmental conditions that are accessible to a species indeed affects the estimation of the ellipsoids used to model its fundamental niche.

On Bipartite Distinct Distances in the Plane

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

Band-based similarity indices for gene expression classification and clustering

The concept of depth induces an ordering from centre outwards in multivariate data. Most depth definitions are unfeasible for dimensions larger than three or four, but the Modified Band Depth (MBD) is a notable exception that has proven to be a valuable tool in the analysis of high-dimensional gene expression data. This depth definition relates the centrality of each individual to its (partial) inclusion in all possible bands formed by elements of the data set. We assess (dis)similarity between pairs of observations by accounting for such bands and constructing binary matrices associated to each pair. From these, contingency tables are calculated and used to derive standard similarity indices. Our approach is computationally efficient and can be applied to bands formed by any number of observations from the data set. We have evaluated the performance of several band-based similarity indices with respect to that of other classical distances in standard classification and clustering tasks in a variety of simulated and real data sets. However, the use of the method is not restricted to these, the extension to other similarity coefficients being straightforward. Our experiments show the benefits of our technique, with some of the selected indices outperforming, among others, the Euclidean distance.