Minimum Euclidean Distance Research Articles

Hybrid Naïve Bayes Gaussian mixture models and SAR polarimetry based automatic flooded vegetation studies using PALSAR-2 data

Flood mapping using Synthetic Aperture Radar (SAR) data impose limitations in fully distinguishing flood under vegetation due to false double bounce returns from inundated tree trunks along with seasonal heterogeneities devised from changing land cover settings. In addition, rapid mapping of flooded vegetation is challenging during near real time applications. In this paper a fully automatic novel supervised classification approach called polarimetric Naïve Bayes is proposed that combines polarimetric information with series of Gaussian mixture models in Naïve Bayes framework to detect various flooded vegetation classes. It also allows the user to choose class configuration and eliminates creation of Region of Interest (ROI) for supervised training. The proposed approach uses scattering information from pre monsoon PolSAR dataset in training step to create ROIs for buildings and other features. In the next step series of Gaussian Mixtures are used for density estimation for different features in Bayesian multiclass problem. The newly developed classifier applied on 2016 Assam flood event resulted in precise mapping of at least three different vegetation classes under flood such as submerged vegetation, wetlands and floating vegetation. Under optimal class configuration, the approach showed better performance compared to other supervised techniques applied on the same data set such as MLE, Mahalanobis, Minimum Euclidean distance, and SVM classifications in delineating flood, submerged vegetation, wetlands and floating vegetation with Producer’s Accuracy of 98.6%, 81.1%, 94% and 51.5% respectively and combined Overall accuracy of 95.5% for flooded vegetation class. This method also detected multiple vegetation classes with better accuracy compared to similar methods.

Advancements in Blind Source Separation for EEG Artifact Removal: A comparative analysis of Variational Mode Decomposition and Discrete Wavelet Transform approaches

Electroencephalography (EEG) is a vital tool for elucidating cerebral processes; however, it is inherently vulnerable to physiological interference, including cardiac rhythm, ocular movement, and muscular activity. To guarantee the reliability of essential neuronal data, it is imperative to implement efficacious denoising methodologies. This study compares the efficacy of two advanced blind source separation (BSS) techniques applied to EEG signals: variational mode decomposition-based BSS (VMD-BSS) and discrete wavelet transform-based BSS (DWT-BSS). The efficacy of these methods is rigorously assessed using performance metrics such as the Euclidean Distance (ED) and the Spearman Correlation Coefficient (SCC), which evaluate the precision of signal reconstruction and the correlation between the original and denoised signals, respectively. The findings indicate that both methods yield robust results, with minimal Euclidean distances of 704.04 for VMD-BSS and 703.64 for DWT-BSS, and a strong correlation coefficient of 0.82. The results demonstrate the effectiveness of the proposed techniques in removing artifacts while preserving essential neural information in EEG recordings. Furthermore, the proposed techniques are benchmarked against previous studies, considering factors such as signal properties, computational complexity, frequency localization, and flexibility. These findings highlight the importance of customized parameter selection tailored to the specific characteristics of EEG datasets and research objectives.

Research on bond-slip behavior of corroded square rebar-concrete in historical reinforced concrete buildings

Corrosion of rebar often leads to the degradation of the bond-slip performance of reinforced concrete members, which is a serious issue in historical buildings. The bond-slip behavior of square rebar, which is widely used in historical reinforced concrete buildings, is significantly different from that of round rebar. In this paper, a bond-slip model of corroded square rebar reinforced concrete was developed. First, 18 square rebar reinforced concrete cube pull-out specimens are made, and the corrosion of rebars is accelerated by electrochemical corrosion experiments. Second, a universal global optimization method was used to construct the bond slip expression and the four-fold line bond-slip model with different corrosion rates was obtained by using the minimum Euclidean distance point. Third, the calculation formulas for the bond strength reduction coefficient and slip ratio of corroded square rebar-concrete in historical reinforced concrete buildings are derived. Finally, the bond-slip constitutive model of corroded square rebar-concrete in historical reinforced concrete buildings is established. The results show that the model is more accurate, simple, effective, and easy to use in practical engineering. The research results provide an important theoretical basis for the structural evaluation and conservation of historical reinforced concrete buildings.

A comparison of fMRI presurgical mapping techniques with intraoperative brain mapping-based validation

Abstract Resting-state functional MRI (rsfMRI) could enable preoperative risk assessment and intraoperative guidance for patients who cannot undergo task-based fMRI (tbfMRI). To ascertain rsfMRI’s applicability, we investigated differences in accuracy between tbfMRI with a voxel size of 1.8 x 1.8 x 3.2 mm³ and rsfMRI acquired with single-echo (sTE) with a voxel size of 2 x 2 x 2.2 mm³ and multiecho (mTE) with a voxel size of 3 x 3 x 3 mm³ using intraoperative mapping with direct electrical stimulations (DES) as the ground truth. Functional sensory-motor mapping results of hands and feet were spatially compared relative to positive (pDES, functional effect) and negative (nDES, no functional effect) coordinates in 16 preoperative patients. A general linear model analysis was used for tbfMRI, and seed-based analysis (SBA) for rsfMRI. Minimum Euclidean distances between fMRI and DES were calculated and compared between fMRI methods. Receiver-operating characteristic (ROC) curves were used to compare accuracy and determine distance cutoffs for fMRI agreement with DES, and binary agreement rates were compared at different cutoffs. Two-part mixed-effects linear models were used to compare fMRI methods while accounting for unequal intersubject DES repetition. Only minor differences were found between fMRI methods in unthresholded distances (mean differences ~2 mm). ROCs and binary agreement measures showed comparable accuracy for tbfMRI and sTE-rsfMRI at 2 mm, but mildly worse for sTE-rsfMRI at 3 mm and mTE-rsfMRI. However, differences in relative accuracy between sTE-rsfMRI and mTE-rsfMRI were minor when the same distance cutoff was applied to all methods. This was also reflected in comparing binary agreement rates and confirmed by the two-part linear models, which showed no significant differences between fMRI methods and a significant effect of DES response. A similar accuracy for SBA rsfMRI functional sensory-motor mapping compared with tbfMRI for the hands and feet indicates that rsfMRI may be suitable for presurgical mapping. The differences in relative accuracy between sTE-rsfMRI and mTE-rsfMRI warrant further investigation in a larger sample.

Accurate and Efficient Schemes for Mapping Out Two Isolated Seams of Conical Intersections with Neural Networks Solely Based on Adiabatic Energies: A Case Study of H+ + NO(X2Π) → H + NO+(X1Σ+).

A new scheme in the neural network (NN) diabatization approach that solely utilizes the adiabatic energies for constructing the global diabatic potential energy matrices (PEMs) of the molecular systems with two isolated seams of conical intersections (CIs) is proposed. Taking a prototype charge transfer reaction H+ + NO(X2Π) → H + NO+(X1Σ+), where two seams of CIs are located at the different linear geometries N-O-H and O-N-H, for example, the diabatization with the new scheme including a diabatic state constraint is shown to map out the topographies of both two linear CIs with 100% of the success rate in 10 different trainings, while the diabatization without such constraint hardly represents CIs, in which the avoided crossings appear instead. Simultaneously, we propose a scheme to separate the whole reactive space into three different regions and define the minimal Euclidean distances for each region to efficiently sample the energy points for the NN trainings. Through adjusting the minimal Euclidean distances, the number of the adiabatic energy data needed in the construction of diabatic PEM can be heavily decreased, lowered from ∼2000 to ∼280 energy points, which is much less than the number (>22 000) of ab initio energy points used in earlier spline diabatic PEM. Further quantum dynamic calculations show that the reaction probabilities, vibrational state distributions, and vibrational state resolved differential cross sections are well reproduced on the new NN diabatic PEM, validating these schemes for constructing the diabatic PEM.

Определение селекционной ценности сортов и гибридов подсолнечника с помощью кластерного и факторного анализов

Abstract. The creation of highly productive sunflower varieties and hybrids adapted to the conditions of cultivation regions depends on the use of source material with high genetic diversity. The purpose of the study is to study the variability of agronomic characters in sunflower varieties and hybrids created in various breeding centers, their correlation, and to group genotypes according to a set of economically valuable indicators, using multivariate statistics methods. Methods. The research was carried out on the Right bank of the Saratov region on the experimental field of the Federal Center of Agriculture Research of the South-East Region in 2021–2023. The objects of research were varieties and new experimental hybrids bred by epy Federal Center of Agriculture Research of the South-East Region (31 varieties and hybrids) and V. S. Pustovoyt All-Russian Research Institute of Oil Crops (12 hybrids). To evaluate the breeding material, generally accepted methodological recommendations were used. Results. 43 varieties and hybrids of sunflower were assessed according to 8 agronomic characters. It was established a low degree of variability in yield (V=9.4%); as well as of plant height (V = 7.8 %); anthode diameter (V = 5.2 %); in unit (V = 5.3 %); and of oil content (V = 5.7 %). Average variability was revealed in the weight of 1000 seeds (V = 16.0 %); anthode area (V = 11.1 %); oil crop (V = 15.8 %). Correlation analysis made it possible to identify 8 significant connections at the 5.0 % level. High correlation coefficients were established between oilseed yield and unit (r = 0.7), as well as between anthode diameter and anthode area (r = 0.83). As a result of factor analysis, 8 characteristics were reduced to 5 factors (loading above 5.0 %) with a total variance of 90.95 %. Clustering by minimum Euclidean distances allowed the studied genotypes to be grouped into 10 clusters, including from 1 to 16 genotypes at the 33rd step. The identified clusters can be considered as independent groups for inclusion in the selection process. Scientific novelty. An attempt was made to group varieties and new experimental sunflower hybrids created in various breeding centers according to the similarities and differences of the main agronomic characters in the arid conditions of the Saratov region.

Seismic retrofitting optimization model using fiber-reinforced polymer jacketing and NSGA-III

Fiber-reinforced polymer (FRP) is widely used for retrofitting structural elements due to its easy application. However, establishing a retrofit strategy is challenging due to conflicting objectives, such as cost and performance level, requiring optimization for effective decision-making. This study proposes a many-objective optimization model for seismic retrofitting using FRP and the Non-dominated Sorting Genetic Algorithm (NSGA)-III. The model's efficacy was demonstrated through two numerical examples: a reinforced concrete building retrofitted with FRP jackets and a masonry-infilled reinforced concrete building retrofitted with FRP bracings. Each example included three objective functions and multiple constraints. Nonlinear static pushover analysis provided optimal strategies to enhance base shear and energy dissipation while minimizing costs and retrofitting locations. Among the Pareto-optimal solutions, the optimal solution with the minimum Euclidean distance was selected. NSGA-III offered a wider distribution and more Pareto-optimal solutions compared to NSGA-II, demonstrating its potential in addressing many-objective problems related to retrofit decision-making.

Optimization framework for multi-fidelity surrogate model based on adaptive addition strategy—A case study of self-excited oscillation cavity

This study proposes a multi-fidelity efficient global optimization framework for the structural optimization of self-excited oscillation cavity. To construct a high-precision multi-fidelity surrogate model to correlate the structural parameters of a self-excited oscillation cavity with the gas precipitation and energy consumption characteristics by effectively fuzing the information of different fidelity levels, choosing different correlation functions and hyper-parameter estimation methods, and learning the correlation between the data. The optimization framework determines various sampling methods and quantities by calculating the minimum Euclidean distance between sample points and sensitivity index. To enhance computational efficiency, a multi-fidelity sample library is established by utilizing both precise and coarse computational fluid dynamics grids. The expected improvement criterion-based algorithm for global optimization is employed as an additive strategy to incorporate additional data points into the model. This approach considers both local and global search of the model, thereby enhancing sample accuracy while reducing computation time. Moreover, the utilization of the highly generalized Non-dominated Sorting Genetic Algorithm-II (NSGA-II) for identifying the Pareto optimal solution set enhances convergence speed. The proposed optimization framework in this study achieves a remarkable level of model accuracy and provides optimal solutions even with a limited sample size. It can be widely used in engineering optimization problems.

Optimization of three-dimensional superposed constellation with probabilistic shaping for MIMO visible light communication systems.

Superposed constellation technology has received increasing attention due to its robustness to the correlation of multiple-input multiple-output (MIMO) channels in visible-light communication (VLC) systems. In this study, a novel superposed three-dimensional 32-quadrature amplitude modulation (3D-32QAM) constellation scheme combined with probabilistic shaping is proposed for MIMO VLC systems. Tetrahedron-shaped 3D-4QAM and cube-shaped 3D-8QAM signals are transmitted from two separate LEDs, which are superposed at the receiver to obtain a cross-shaped 32QAM signal. In addition to improving the minimum Euclidean distance (MED) by using 3D constellations, the proposed scheme focuses on exploring the advantages of superposed 3D constellations in probabilistic shaping. Both global and local probabilistic shaping can be achieved in terms of superposition of 3D constellations, which increase the MED and reduce the risk of nonlinear distortion of light-emitting diodes, respectively. Experimental results demonstrated that the proposed scheme achieved a better normalized generalized mutual information performance and significantly improved the dynamic range of the driving peak-to-peak voltage when compared with the traditional superposed two-dimensional 32QAM constellation schemes.

Interspectral Error Tracking and Compensation of DSDT in Satellite Internet of Things

With the rapid growth of satellite Internet of Things (SIoT) services, existing frequency band resources are insufficient to meet future business demands. To effectively address this issue, it is necessary to enhance the utilization of existing frequency resources. However, idle frequency resources are typically scattered across multiple bands and vary in bandwidth size. Direct Spectrum Division Transmission (DSDT), dividing a complete signal into sub-spectrum signals for transmission in idle frequency bands, can take the use of fragmented spectrum resources for satellite communication. Nevertheless, the performance of DSDT depends heavily on accurate synchronization toward multiple sub-spectrums. In this paper, an algorithm for error synchronization tracking and compensation is proposed by utilizing the focusing nature of constellation. All sub-spectrums are weighed by the minimum Euclidean distance of the constellation to compensate for amplitude–frequency–phase errors simultaneously. Simulations and experimental verification demonstrate synchronization performance and feasibility of proposed method in a multi-radio frequency channels environment.

Signal points allocation for generalized enhanced spatial modulation

AbstractIn this paper, making full use of the spatial domain of transmit antennas (TAs) and keeping the characteristic of the squared minimum Euclidean distance (MED) between the transmitted spatial vectors (TSVs), generalized enhanced spatial modulation with signal points allocation (GESM‐SPA) is proposed to expand the size of signal spaces for enhancing the spectral efficiency and the reliability of communications. In the GESM‐SPA, according to the number of active TAs, signal constellation points (CPs) from the QAM or secondary QAM constellations are allocated and then modulated on the corresponding active TAs with the selected antenna index (AI) vector. Through this design, which further exploits the spatial domain with the variability of active TAs, the squared MED between the TSVs is increased in comparison with the existing traditional systems. More specifically, in view of the disadvantage of the classic ESM system, the constellation groups (CGs) with priority given to the QAM CPs are constructed to further maximize the squared MED. Then, the AI vector subsets corresponding to the obtained CGs are designed to be candidate for the specified AI vector set with the AI information. The squared MED and the average bit error probability (BEP) are analyzed. In simulation results using Monte Carlo, the GESM‐SPA outperforms the existing classic systems in terms of the bit error rate performance.

Multi-fidelity deep learning for aerodynamic shape optimization using convolutional neural network

Aerodynamic shape design is essential for improving aircraft performance and efficiency. First, this study introduces a data-driven optimization framework utilizing a multi-fidelity convolutional neural network (MFCNN) for aerodynamic shape optimization. To achieve better optimization results with reduced computational cost, the framework dynamically incorporates new data in each optimization cycle. Specifically, it constantly involves the optimal solution from previous cycle as a new high-fidelity sample and employs a low-fidelity infilling strategy that maximizes the minimum Euclidean distance for selecting new low-fidelity samples. Moreover, a standard synthetic benchmark is used to elaborate the procedure of optimization and show the capability and effectiveness of the framework. Finally, the framework is applied to two aerodynamic shape optimization problems: maximizing the lift-to-drag ratio for the Royal Aircraft Establishment 2822 (RAE2822) airfoils and minimizing the cruise drag coefficient for the three-dimensional (3D) drooped and scarfed non-axisymmetric nacelles. The framework increases the lift-to-drag ratio by 51.21% over the baseline and achieves an 18.79% reduction in the cruise drag coefficient for nacelle optimization, outperforming traditional multi-fidelity deep neural network optimization framework. Sufficiently utilizing the implicit relations between different fidelity levels of data through defined local perceptual fields and convolution, our MFCNN-based optimization framework signifies a step forward in the efficiency and accuracy of aerodynamic shape optimization.

Joint equalization of frequency offset and phase noise using two-stage cascaded extended Kalman filter for discrete spectrum 16/64APSK NFDM systems.

For the discrete spectrum nonlinear frequency division multiplexing (DS-NFDM) 16/64 amplitude phase shift keying (APSK) system, the inevitable laser impairments including frequency offset (FO) and carrier phase noise (CPN) would cause different rotations of the received signal constellations. In addition, the combined effect of FO and amplifier spontaneous emission (ASE) noise induces the eigenvalue shift, accordingly the residual channel impairment (RCI) is inevitably yielded. To address the above problems, we deduce the joint impairment model of FO, CPN and RCI, and then propose a joint equalization scheme using two-stage cascaded extended Kalman filter (TSC-EKF) for these impairments. It performs frequency offset compensation in the first stage, subsequently carries out joint equalization of CPN and RCI in the second stage. Meanwhile, the minimum Euclidean distance and phase difference between the received symbols and the ideal 16/64APSK constellations are ingeniously fused to calculate the innovations of TSC-EKF. The effectiveness has been verified by 2 GBaud DS-NFDM 16/64 APSK simulations and DS-NFDM 16APSK transmission experiments. The results demonstrate that when performing the joint equalization of FO, CPN and RCI, the maximum FOE range of TSC-EKF scheme achieves 1.2 and 9.6 times as that of nonlinear frequency domain (NFD) scheme and fast Fourier transform -Like (FFT-Like) scheme, respectively. Furthermore, its maximum LW tolerance reaches 3.3 times as that of the M-th power scheme. Importantly, the complexity of TSC-EKF is 63.4% as that of NFD scheme and on an order of O(N).

Urban distribution network planning and zoning design based on improved K-means clustering in low carbon background

Abstract Aiming at the problem that the traditional algorithm is difficult to adapt to the chaotic planning of urban distribution network and the uneven distribution of load points, an improved K-means clustering algorithm based zoning planning method of urban distribution network is proposed. Firstly, considering the effect of the capacity margin on the partition, the weighted factor is introduced to improve the Euclidean distance. Secondly, considering the selection of power supply units according to the distribution characteristics of the substation, the distance between the cluster center and the substation is calculated. Finally, a distribution network zoning planning model with the maximum number of inter-station power supply units and the minimum total Euclidean distance of power distribution as the objective function is constructed. Finally, taking the transformation of a distribution network in Fujian Province as a practical engineering example, the results show that the average difference of load partition obtained by the improved K-means clustering algorithm is reduced by 34.35% compared with the traditional method.

Nonlinear Model Predictive Control-based Collision Avoidance for Mobile Robot

This work proposes an efficient and safe single-layer Nonlinear Model Predictive Control (NMPC) system based on LiDAR to solve the problem of autonomous navigation in cluttered environments with previously unidentified static and dynamic obstacles of any shape. Initially, LiDAR sensor data is collected. Then, the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, is used to cluster the (Lidar) points that belong to each obstacle together. Moreover, a Minimum Euclidean Distance (MED) between the robot and each obstacle with the aid of a safety margin is utilized to implement safety-critical obstacle avoidance rather than existing methods in the literature that depend on enclosing the obstacles with a circle or minimum bounding ellipse. After that, to impose avoidance constraints with feasibility guarantees and without compromising stability, an NMPC for set-point stabilization is taken into consideration with a design strategy based on terminal inequality and equality constraints. Consequently, numerous obstacles can be avoided at the same time efficiently and rapidly through unstructured environments with narrow corridors. Finally, a case study with an omnidirectional wheeled mobile robot (OWMR) is presented to assess the proposed NMPC formulation for set-point stabilization. Furthermore, the efficacy of the proposed system is tested by experiments in simulated scenarios using a robot simulator named CoppeliaSim in combination with MATLAB which utilizes the CasADi Toolbox, and Statistics and Machine Learning Toolbox. Two simulation scenarios are considered to show the performance of the proposed framework. The first scenario considers only static obstacles while the second scenario is more challenging and contains static and dynamic obstacles. In both scenarios, the OWMR successfully reached the target pose (1.5m, 1.5m, 0°) with a small deviation. Four performance indices are utilized to evaluate the set-point stabilization performance of the proposed control framework including the steady-state error in the posture vector which is less than 0.02 meters for position and 0.012 for orientation, and the integral of norm squared actual control inputs which is 19.96 and 21.74 for the first and second scenarios respectively. The proposed control framework shows a positive performance in a narrow-cluttered environment with unknown obstacles.

Interleaving based SCMA Codebook Design Using Arnold’s Cat Chaotic Map

Non-orthogonal multiple access (NOMA) technology is a significant topic of the present 5G investigation. A possible NOMA approach is Sparse Code Multiple Access (SCMA), increasing next-generation communication's ability to handle multiple users. In SCMA, codebook designing and optimization are crucial components that define the performance of multiple access systems. In this work, the complete design procedure of the SCMA codebook based on interleaving is explained and simulated. Furthermore, a chaotic interleaving based onArnold’sCat map is proposed to reduce the computational complexity. The performance of the SCMA codebook based on interleaving is evaluated through comparison with selected codebooks for SCMA multiplexing. The measures used for performance evaluation include Bit Error rate (BER), Peak to Average Power Ratio(PAPR), and minimum EuclidianDistance(MED). The results show that the proposed codebook with chaotic interleaving achieves comparable performance to the traditional codebook based on interleaving and less MED and higher BER compared to computer-generated and 16-star QAM codebook design methods but with reduced complexity.

Modification of the Leslie Model on Population Growth in the Bangka Belitung Islands Province

The Leslie model is one of the applications of Algebra in solving a population growth model. The birth rate and survival rate of a population are constituents of the Leslie Matrix. The advantage of this model is that it only requires data on the total female population. This study aims to modify the classic Leslie Model by adding correction values to matrix elements, especially birth rates and survival rates. The correction value is obtained from the minimum Euclidean distance for each birth rate and survival rate for each population age group. The Euclidean distance is used because it requires simple calculations. Based on the modified results, the Perron value obtained from the Leslie matrix is 0.9. If the constant value is zero, then the modification of the Leslie model will be the same as the classic Leslie model.

Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications.

Orthogonal frequency division multiplexing with index modulation (OFDM-IM) has great potential for the implementation of high spectral-efficiency underwater acoustic (UWA) communications. However, general receivers consisting of the optimal maximum likelihood detection suffer from high computational load, which prohibits real-time data transmissions in underwater scenarios. In this paper, we propose a detection based on a vector approximate message passing (VAMP) algorithm for UWA OFDM-IM communications. Firstly, a VAMP framework with a non-loopy factor graph for index detection is formulated. Secondly, by utilizing the sparsity inherently existing in OFDM-IM symbols, a novel shrinkage function is derived based on the minimum mean square error criterion, which guarantees better posterior estimation. To reduce the errors from estimated non-existing indices, one trick is utilized to search the elements from the look-up table with the minimal Euclidean distance for the replacement of erroneously estimated indices. Experiments verify the advantages of the proposed detector in terms of low complexity, robustness and effectiveness compared with the state-of-art benchmarks.

MaOEA/D with adaptive external population guided weight vector adjustment

In order to make multiobjective evolutionary algorithm based on decomposition (MOEA/D) have better performance in dealing with many-objective optimization problems (MaOPs) with different Pareto fronts (PFs), this paper proposes a MOEA/D with adaptive external population guided weight vector adjustment (MaOEA/D-AEW). The algorithm modifies the MOEA/D method of constructing mating pools by calculating the utility function value of individuals in the population to produce high-quality offspring. For the maintenance of non-dominated solutions in the external population (EP), solutions exceeding the capacity of the EP are removed by the method of minimum Euclidean distance combined with the aggregation function value to ensure the convergence and diversity of non-dominated solutions in the EP. For the addition of weight vectors, a new sparsity evaluating method (SL+) is proposed to determine the sparsity of weight vectors by multiplying parallel distances between individuals. For the deletion of the weight vector, the minimum Euclidean distance combined with the aggregation function value is used to delete the weight vector corresponding to invalid or crowded individuals in the population. Comparing MaOEA/D-AEW with six state-of-the-art algorithms on regular PF and irregular PF test problems respectively, simulation results show that the proposed algorithm can adapt well to MaOPs with different PFs and significantly outperforms other comparative algorithms in terms of convergence and diversity of solution sets.

Prototype-Based Interpretation of the Functionality of Neurons in Winner-Take-All Neural Networks.

Prototype-based learning (PbL) using a winner-take-all (WTA) network based on minimum Euclidean distance (ED-WTA) is an intuitive approach to multiclass classification. By constructing meaningful class centers, PbL provides higher interpretability and generalization than hyperplane-based learning (HbL) methods based on maximum inner product (IP-WTA) and can efficiently detect and reject samples that do not belong to any classes. In this article, we first prove the equivalence of IP-WTA and ED-WTA from a representational power perspective. Then, we show that naively using this equivalence leads to unintuitive ED-WTA networks in which the centers have high distances to data that they represent. We propose ±ED-WTA that models each neuron with two prototypes: one positive prototype, representing samples modeled by that neuron, and a negative prototype, representing the samples erroneously won by that neuron during training. We propose a novel training algorithm for the ±ED-WTA network, which cleverly switches between updating the positive and negative prototypes and is essential to the emergence of interpretable prototypes. Unexpectedly, we observed that the negative prototype of each neuron is indistinguishably similar to the positive one. The rationale behind this observation is that the training data that are mistaken for a prototype are indeed similar to it. The main finding of this article is this interpretation of the functionality of neurons as computing the difference between the distances to a positive and a negative prototype, which is in agreement with the BCM theory. Our experiments show that the proposed ±ED-WTA method constructs highly interpretable prototypes that can be successfully used for explaining the functionality of deep neural networks (DNNs), and detecting outlier and adversarial examples.