Negative Weights Research Articles

SHEEP, a Signed Hamiltonian Eigenvector Embedding for Proximity

Signed network embedding methods allow for a low-dimensional representation of nodes and primarily focus on partitioning the graph into clusters, hence losing information on continuous node attributes. Here, we introduce a spectral embedding algorithm for understanding proximal relationships between nodes in signed graphs, where edges can take either positive or negative weights. Inspired by a physical model, we construct our embedding as the minimum energy configuration of a Hamiltonian dependent on the distance between nodes and locate the optimal embedding dimension. We show through a series of experiments on synthetic and empirical networks, that our method (SHEEP) can recover continuous node attributes showcasing its main advantages: re-configurability into a computationally efficient eigenvector problem, retrieval of ground state energy which can be used as a statistical test for the presence of strong balance, and measure of node extremism, computed as the distance to the origin in the optimal embedding.

Self-satisfaction described by Zuckerman's alternative five factors of personality.

This study investigates the relationships between an individual's self-satisfaction within different life areas, which correspond to Bracken's self-concept primary domains (competence, family, social, physical, emotional, and academic domains), and Zuckerman's Alternative Five-Factor Model of Personality (AFFM). It is supposed that the AFFM, as a psychobiological personality model which allows causal explanations, could provide a comprehensive insight into the nature of satisfaction with self. The study included 489 adults (64% women), between 18 and 60 years old, who completed the short Self-Satisfaction Scale (SC-6) and the Zuckerman-Kuhlman-Aluja Personality Questionnaire (ZKA-PQ). Relations between personality factors and facets from the ZKA-PQ and self-satisfaction measures were analyzed using correlational and multiple regression analysis. The relationship of self-satisfaction with gender and age was also analyzed. Based on responses to scales, 11% to 43% of the variance in self-satisfaction responses was predicted by personality, age, and sex. Extraversion had positive predictive weights for each self-satisfaction dimension. Neuroticism had negative predictive weights which were significant except for satisfaction with family. Sensation seeking negatively predicted satisfaction with competencies, family, academic aspects, emotions, and overall satisfaction. Aggression had small negative predictor weights for satisfaction with social and academic aspects. Activity had significant positive predictor weights for competencies, academic aspects, emotions, and overall satisfaction. The findings suggest that personality predicts satisfaction with aspects of the self, and that the AFFM provides an adequate theoretical framework, which includes a lower level of personality traits in the explanation of the nature of a person's satisfaction, in general or related to specific life contexts.

Distributed State Estimation for Linear Systems in Networks With Antagonistic Interactions.

This article deals with the distributed state estimation problem for linear systems in networks with cooperative interactions and antagonistic interactions, where the cooperative interactions and the antagonistic interactions are characterized by the positive weights and the negative weights, respectively. Due to the coexistence of the cooperative interactions and the antagonistic interactions, the existing methods based on the non-negatively weighted graph become not applicable. First, a partition method of the nodes and a decomposition form of the system matrices are introduced. Then a new form of distributed observers is proposed in the network with cooperative interactions and antagonistic interactions. Necessary and sufficient conditions, which guarantee the existence of proposed distributed observers, are presented. In particular, when the communication network is cooperative, the proposed method is suitable for two cases, that is, the nodes in each independent strongly connected component are jointly detectable, or each node is jointly detectable with its neighbors. Finally, three examples are simulated to illustrate the effectiveness of the proposed methods.

Enhancing Information Maximization With Distance-Aware Contrastive Learning for Source-Free Cross-Domain Few-Shot Learning.

Existing Cross-Domain Few-Shot Learning (CDFSL) methods require access to source domain data to train a model in the pre-training phase. However, due to increasing concerns about data privacy and the desire to reduce data transmission and training costs, it is necessary to develop a CDFSL solution without accessing source data. For this reason, this paper explores a Source-Free CDFSL (SF-CDFSL) problem, in which CDFSL is addressed through the use of existing pretrained models instead of training a model with source data, avoiding accessing source data. However, due to the lack of source data, we face two key challenges: effectively tackling CDFSL with limited labeled target samples, and the impossibility of addressing domain disparities by aligning source and target domain distributions. This paper proposes an Enhanced Information Maximization with Distance-Aware Contrastive Learning (IM-DCL) method to address these challenges. Firstly, we introduce the transductive mechanism for learning the query set. Secondly, information maximization (IM) is explored to map target samples into both individual certainty and global diversity predictions, helping the source model better fit the target data distribution. However, IM fails to learn the decision boundary of the target task. This motivates us to introduce a novel approach called Distance-Aware Contrastive Learning (DCL), in which we consider the entire feature set as both positive and negative sets, akin to Schrödinger's concept of a dual state. Instead of a rigid separation between positive and negative sets, we employ a weighted distance calculation among features to establish a soft classification of the positive and negative sets for the entire feature set. We explore three types of negative weights to enhance the performance of CDFSL. Furthermore, we address issues related to IM by incorporating contrastive constraints between object features and their corresponding positive and negative sets. Evaluations of the 4 datasets in the BSCD-FSL benchmark indicate that the proposed IM-DCL, without accessing the source domain, demonstrates superiority over existing methods, especially in the distant domain task. Additionally, the ablation study and performance analysis confirmed the ability of IM-DCL to handle SF-CDFSL. The code will be made public at https://github.com/xuhuali-mxj/IM-DCL.

Separation of pair and single top quark production in tWb-associated final state using a neural network

The paper presents a method for separating the contribution of pair and single top quark production in tWb-associated final state using a neural network. The proposed method makes possible to calculate such processes in a gauge-invariant way with fully taking into account interference contributions and dividing the phase space into single-resonant and double-resonant regions, which is necessary to increase the accuracy of the search for possible deviations from the predictions of the Standard Model in these processes. To train the neural network, the optimized set of observables is used to separate single-resonance and double-resonance contributions to the overall process. The use of this method allows us to avoid the disadvantages inherent in the schemes used in collider physics for calculating the processes of tWb-associated top quark production with the removal of Feynman diagrams, which leads to violation of gauge invariance, or the addition of a subtraction scheme, which leads to the appearance of negative weights for the part of simulated events. The proposed method can be used to increase the efficiency of searching for deviations from the predictions of the Standard Model in the interaction of the top quark with the W boson and b quark.

New Third-Order Finite Volume Unequal-Sized WENO Lagrangian Schemes for Solving Euler Equations

In this paper, new third-order finite volume unequal-sized weighted essentially non-oscillatory (US-WENO) Lagrangian schemes are designed to solve Euler equations in two and three dimensions. The spatial reconstruction procedures are implemented by using a convex combination of a quadratic polynomial with several linear polynomials specified on unequal-sized stencils, so the new US-WENO Lagrangian schemes can achieve the designed third-order accuracy and maintain an essentially non-oscillatory property near strong discontinuities in multi-dimensions. Unlike the traditional WENO reconstruction procedures specified on unstructured meshes, the linear weights of these new two-dimensional and three-dimensional US-WENO spatial reconstructions can be selected as any positive numbers as long as their summation equals one and they are not related to the local mesh topology or the location of quadrature points. Moreover, the linear weights do not have to be recalculated even if the grid moves with the fluid, avoiding the appearance of negative linear weights, thus improving computation efficiency and robustness in multi-dimensional Lagrangian numerical simulations. Finally, extensive benchmark numerical cases are employed to display the excellent capability of the presented US-WENO Lagrangian schemes.

Complexity of Vocal Vibrato in Opera and Jazz Recordings: Insights From Entropy and Recurrence Analyses

Vibrato is an oscillation in frequency, intensity, and timbre of the singing voice. Previous studies have found a relationship between its periodicity and perceived quality. The diversity of vibrato enriches the music and singer's performances but create challenges for quantifying and capturing the characteristics that contribute to achieving these expressive goals. Vibrato tones have been addressed using rate, extent, jitter, and shimmer; however, these do not necessarily capture relevant complex time-varying features. This paper applies techniques from disciplines that specialize in periodicity and complexity to provide insight into vibrato characteristics not yet understood. This study aimed to assess whether nonlinear metrics are relevant features in characterizing and illustrating differences in vibrato behavior in opera and jazz singing, as well as considering the relationship of nonlinear metrics to other vibrato parameters. Vibrato tones from published music material of world-class singers from opera and jazz were analyzed with entropy, recurrence, and the established parameters of rate, extent, jitter, and shimmer. Dimensionality reduction was employed to consider the relationship and significance of each of the metrics in collectively characterizing vibrato. The principal component, explaining 40% of variability, had positive weights of determinism and line length derived from recurrence while having negative weights of rate, shimmer, and sample entropy. Using these components, the vibrato tones from opera compared to jazz singing were found to be more regular and had lower rate and extent, and it was possible to spotlight singers and notes with high periodicity. Our study shows that nonlinear metrics applied to vibrato tones provide a valuable tool for observing and quantifying regularity in vibrato tones. The results of this study highlight the potential for more detailed descriptions of vibrato characteristics that may support categorization of individuals, genres, or musical expression in the future and could be applicable to pedagogical techniques.

Breast cancer risk for the joint exposure to metals and metalloids in women: Results from the EPIC-Spain cohort

Environmental factors play a role in breast cancer development. While metals and metalloids (MMs) include some carcinogens, their association with breast cancer depends on the element studied. Most studies focus on individual MMs, but the combined effects of metal mixtures remain unclear. The aim of this study was to analyze the relationship between the joint exposure to MMs and the risk of developing female breast cancer.We conducted a case-control study within the multicenter prospective EPIC-Spain cohort. Study population comprised 292 incident cases and 286 controls. Plasma concentrations of 16 MMs were quantified at recruitment. Potential confounders were collected using a questionnaire and anthropometric measurements. Mixed-effects logistic regression models were built to explore the effect of individual MMs. Quantile-based g computation models were applied to identify the main mixture components and to estimate the joint effect of the metal mixture.The geometric means were highest for Cu (845.6 ng/ml) and Zn (604.8 ng/ml). Cases had significantly higher Cu concentrations (p = 0.010) and significantly lower Zn concentrations (p < 0.001). Cu (+0.42) and Mn (+0.13) showed the highest positive weights, whereas Zn (−0.61) and W (−0.16) showed the highest negative weights. The joint effect of the metal mixture was estimated at an OR = 4.51 (95%CI = 2.32–8.79), suggesting a dose-response relationship. No evidence of non-linearity or non-additivity was found.An unfavorable exposure profile, primarily characterized by high Cu and low Zn levels, could lead to a significant increase in the risk of developing female breast cancer. Further studies are warranted to confirm these findings.

Interpretable fusion methodology of health indices with an application to industrial turbine cavitation condition monitoring.

Using health indices (HIs) to characterize machine conditions is greatly helpful to prevent machine failures and their subsequent catastrophe. Fusion and interpretation of the main contributions of HIs to machine condition monitoring are still challenging. In this paper, an interpretable fusion methodology of HIs is proposed for machine condition monitoring. The proposed methodology begins with elements of statistical learning for classification, following by an essence of how HIs are fused with their associated linear weights to realize machine condition monitoring. One main contribution of this paper gives a theoretical justification for positive and negative weights of the proposed fusion methodology for understanding their importance for machine condition monitoring and making the proposed methodology physically interpretable. In order to be suitable for two practical situations, in which whether faulty data are available or not, two solutions including an offline solution with healthy and faulty datasets and an online solution with only available healthy datasets are suggested to estimate interpretable weights of the proposed methodology. Finally, industrial turbine cavitation status data collected from our group are used to verify the proposed methodology and show its superiority to two existing popular machine fault diagnosis methods. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 2)'.

The metabolic spatial covariance pattern of definite idiopathic normal pressure hydrocephalus: an FDG PET study with principal components analysis

Identification of patients with idiopathic normal pressure hydrocephalus (iNPH) in a collective with suspected neurodegenerative disease is essential. This study aimed to determine the metabolic spatial covariance pattern of iNPH on FDG PET using an established technique based on scaled subprofile model principal components analysis (SSM-PCA).We identified 11 patients with definite iNPH. By applying SSM-PCA to the FDG PET data, they were compared to 48 age-matched healthy controls to determine the whole-brain voxel-wise metabolic spatial covariance pattern of definite iNPH (iNPH-related pattern, iNPHRP). The iNPHRP score was compared between groups of patients with definite iNPH, possible iNPH (N = 34), Alzheimer’s (AD, N = 38), and Parkinson’s disease (PD, N = 35) applying pairwise Mann–Whitney U tests and correction for multiple comparisons.SSM-PCA of FDG PET revealed an iNPHRP that is characterized by relative negative voxel weights at the vicinity of the lateral ventricles and relative positive weights in the paracentral midline region. The iNPHRP scores of patients with definite iNPH were substantially higher than in patients with AD and PD (both p < 0.05) and non-significantly higher than those of patients with possible iNPH. Subject scores of the iNPHRP discriminated definite iNPH from AD and PD with 96% and 100% accuracy and possible iNPH from AD and PD with 83% and 86% accuracy.We defined a novel metabolic spatial covariance pattern of iNPH that might facilitate the differential diagnosis of iNPH versus other neurodegenerative disorders. The knowledge of iNPH-associated alterations in the cerebral glucose metabolism is of high relevance as iNPH constitutes an important differential diagnosis to dementia and movement disorders.

A new way of reducing negative weights in MC@NLO

We introduce a new technique, that we dub Born spreading, aimed at reducing the number of negative-weight S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {S}}$$\\end{document} events in the MC@NLO matching of NLO calculations with parton-shower simulations. We show that such a technique, based on a re-distribution of Born matrix elements in the radiative phase space, achieves a sizeable reduction of negative-weight events at little computational cost. The method does not induce any biases in physical distributions.

Generalizing some key results from "alternative weighting schemes when performing matching-adjusted indirect comparisons".

A recent paper proposed an alternative weighting scheme when performing matching-adjusted indirect comparisons. This alternative approach follows the conventional one in matching the covariate means across two studies but differs in that it maximizes the effective sample size when doing so. The appendix of this paper showed, assuming there is one covariate and negative weights are permitted, that the resulting weights are linear in the covariates. This explains how the alternative method achieves a larger effective sample size and results in a metric that quantifies the difficulty of matching on particular covariates. We explain how these key results generalize to the case where there are multiple covariates, giving rise to a new metric that can be used to quantify the impact of matching on multiple covariates.

Evaluation of 12 precipitation products and comparison of 8 multi-model averaging methods for estimating precipitation in the Qilian Mountains, Northwest China

Mountains are the water towers of the world, so it is critical to obtain accurate precipitation data for mountainous areas. Due to the complex topography of high mountainous areas, precipitation ground stations are sparse and unevenly distributed in such areas, so precipitation products such as remote sensing and reanalysis products are used to obtain gridded precipitation data for these areas. However, no single precipitation product performs best in all areas of mountainous regions. Therefore, this study first evaluated the performance of 12 precipitation products in estimating precipitation in the Qilian Mountains at the station scale and sub-basin scale, and then compared the performance of precipitation estimates for the Qilian Mountains generated by 8 multi-model averaging methods. The evaluation results for 29 meteorological stations in the Qilian Mountains showed that the China Meteorological Forcing Dataset product was the best-performing precipitation product, while the Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks-Cloud Classification System-Climate Data Record product was the worst-performing precipitation product. The evaluation results for 18 sub-basins showed that at these sub-basins, the WorldClim was the best-performing precipitation product, while the High Asia Refined analysis was the worst-performing precipitation product. Thus, station-scale evaluations may not necessarily be applicable to the basin scale. Multi-model averaging methods effectively improved the accuracy of precipitation estimates both at station scale and at sub-basin scale. The Granger-Ramanathan variant C was the best multi-model averaging method for estimating precipitation at station scale. As the Granger-Ramanathan methods allow negative weights, they are not recommended to interpolate the Granger-Ramanathan weight values of stations to grids. The Bayesian model averaging (BMA) was found to be the most suitable multi-model averaging method for estimating precipitation in the Qilian Mountains by interpolation of weight values of stations to grids. The precipitation estimates generated by BMA show that the mean annual precipitation in the Qilian Mountains from 2001 to 2018 was approximately 336.1 mm, and the annual precipitation during this period increased linearly by 2.4 mm per year.

Uncovering the relationship between gut microbial dysbiosis, metabolomics, and dietary intake in type 2 diabetes mellitus and in healthy volunteers: a multi-omics analysis

Type 2 Diabetes Mellitus has reached epidemic levels globally, and several studies have confirmed a link between gut microbial dysbiosis and aberrant glucose homeostasis among people with diabetes. While the assumption is that abnormal metabolomic signatures would often accompany microbial dysbiosis, the connection remains largely unknown. In this study, we investigated how diet changed the gut bacteriome, mycobiome and metabolome in people with and without type 2 Diabetes.1 Differential abundance testing determined that the metabolites Propionate, U8, and 2-Hydroxybutyrate were significantly lower, and 3-Hydroxyphenyl acetate was higher in the high fiber diet compared to low fiber diet in the healthy control group. Next, using multi-omics factor analysis (MOFA2), we attempted to uncover sources of variability that drive each of the different groups (bacterial, fungal, and metabolite) on all samples combined (control and DM II). Performing variance decomposition, ten latent factors were identified, and then each latent factor was tested for significant correlations with age, BMI, diet, and gender. Latent Factor1 was the most significantly correlated. Remarkably, the model revealed that the mycobiome explained most of the variance in the DM II group (12.5%) whereas bacteria explained most of the variance in the control group (64.2% vs. 10.4% in the DM II group). The latent Factor1 was significantly correlated with dietary intake (q < 0.01). Further analyses of the impact of bacterial and fungal genera on Factor1 determined that the nine bacterial genera (Phocaeicola, Ligilactobacillus, Mesosutterella, Acidaminococcus, Dorea A, CAG-317, Caecibacter, Prevotella and Gemmiger) and one fungal genus (Malassezia furfur) were found to have high factor weights (absolute weight > 0.6). Alternatively, a linear regression model was fitted per disease group for each genus to visualize the relationship between the factor values and feature abundances, showing Xylose with positive weights and Propionate, U8, and 2-Hydroxybutyrate with negative weights. This data provides new information on the microbially derived changes that influence metabolic phenotypes in response to different diets and disease conditions in humans.

Prenatal exposure to metal mixtures and lung function in children from the New Hampshire birth cohort study

Prenatal exposure to metals/metalloids, even at common US population levels, may pose risks to fetal health, and affect children's lung function. Yet, the combined effects of simultaneous prenatal exposures on children's lung function remain largely unexplored. This study analyzed 11 metals (As speciation, Cd, Co, Cu, Mo, Ni, Pb, Sb, Se, Sn, Zn) in maternal urine during weeks 24–28 of gestation and evaluated lung function, including forced vital capacity (FVC) and forced expiratory volume in the first second of expiration (FEV1), in 316 US mother-child pairs at around age 7. We used Bayesian Kernel Machine Regression (BKMR), weighted quantile sum regression (WQSR), and multiple linear regression to examine the association between metal mixture exposure and children's lung function, adjusting for maternal smoking, child age, sex, and height. In BKMR models assessing combined exposure effects, limited evidence of metal non-linearity or interactions was found. Nevertheless, Co, As species, and Pb showed a negative association, while Mo exhibited a positive association with children's FVC and FEV1, with other metals held constant at their medians. The weighted index, from WQSR analysis assessing the cumulative impact of all metals, highlighted prenatal Mo with the highest positive weight, and Co, As, and Sb with the most substantial negative weights on children's FVC and FEV1. Urinary Co and Pb were negatively associated with FVC (β = −0.09, 95% confidence interval (CI) (−0.18; −0.01) and β = −0.07, 95% CI (−0.13; 0.00), respectively). Co was also negatively associated with FEV1 (β = −0.09, 95% CI (−0.18; 0.00). There was a negative association between As and FVC, and a positive association between Mo and both FVC and FEV1, though with wide confidence intervals. Our findings suggest that prenatal trace element exposures may impact children's lung function, emphasizing the importance of reducing toxic exposures and maintaining adequate nutrient levels.

H∞ Bipartite Synchronization Control of Markov Jump Cooperation-Competition Networks With Reaction-Diffusions.

This article is concerned with the bipartite synchronization problem of coupled switching neural networks with cooperative-competitive interactions and reaction-diffusion terms. Different from the existing literature, the networked systems under investigation possess the relationship of cooperation and competition among nodes. Notably, the switching topology is described by a signed graph subject to the Markov jump process with the coexistence of positive and negative interaction weights. Specifically, a positive weight indicates an alliance relationship between two nodes and a negative one shows an adversary relationship. This article aims to design a bipartite synchronization controller for the aforementioned networks with the switching topology such that a prescribed H∞ bipartite synchronization is satisfied. Then, some sufficient criteria to ensure the stochastic stability of bipartite synchronization error systems are established in view of an appropriate Lyapunov function. Finally, two simulation examples are presented to verify the validity of the proposed bipartite synchronization control method.

Efficient negative-weight elimination in large high-multiplicity Monte Carlo event samples

We demonstrate that cell resampling can eliminate the bulk of negative event weights in large event samples of high multiplicity processes without discernible loss of accuracy in the predicted observables. The application of cell resampling to much larger data sets and higher multiplicity processes such as vector boson production with up to five jets has been made possible by improvements in the method paired with drastic enhancement of the computational efficiency of the implementation.

Are Negative Weights in Combining Forecasts So Bad?

Purpose of the study. In this paper, we consider the problem of negativity of weight coefficients when combining forecasts. Combining forecasts as a method has long ago proved itself in practice as a good way to improve forecast accuracy. However, in the literature little attention is paid to the issue of negative weights during aggregation, although the cases of obtaining such weights in practice are quite common. The reasons why this may happen are not considered or analyzed. Often, when obtaining weights less than zero, such weights are reset to zero, thus excluding the information contained in the particular forecasting method from the combination, which may reduce the accuracy of the combined forecast. In this regard, it is important to understand why when combining forecasts, negative weight can be obtained and determine options for how to avoid such situations in combining without losing accuracy.Materials and methods. It is proposed to consider various approaches to eliminate excluded weights when combining forecasts, including truncation of weight coefficients or imposing restrictions on them, including the option of sequential combining of forecasts. Results. The result is a list of reasons why negative weights can be obtained when combining forecasts, what risks they have and how to avoid them.Conclusion. Based on the results obtained, it can be concluded that the negative weights themselves when combining forecasts can be triggers for identifying problems when combining. However, it is dangerous to retain them, as they can lead to uncertain prediction results and degrade the accuracy of the resulting combined forecast. The proposed methods of work allow you to bypass the negativity of the weights without a strong deterioration in forecasting.

Spatial averaging method based on adaptive weight for imaging photoplethysmography.

Imaging photoplethysmography (iPPG) is a non-contact measuring technology for several physiological parameters reflecting personal health status without a special sensor. However, the pulse signal obtained using the iPPG usually is contaminated by various noises, and the intensity of the interesting pulse signal is relatively weak compared to the noises, emphasizing the necessity of obtaining high-quality pulse signals to measure physiological parameters correctly. Various regions of the face harbor distinct pulse information. We propose a spatial averaging method based on adaptive weights, which can obtain high-quality pulse signals by applying different weights to facial sub-regions of interest (sub-ROIs; sROIs). First, the facial ROI is divided into seven sROIs and the coarse heart rate (HR) is calculated from them. Next, the signal-to-noise ratio (SNR) of the raw signal obtained from each sROI is calculated using the coarse HR, and then a high-quality pulse signal is obtained by assigning positive or negative weights to each sROI based on the SNRs. We compare our method with others through the quality analysis of the obtained pulse signals using the self-collected database and the public database PURE. The comparison results show that the proposed method can provide a better pulse signal compared to other methods under various resolutions and states. This proposed method can obtain the pulse signal with better quality, which is helpful to accurately measure physiological parameters, such as HR and HR variability.

Research Progress of Neural Synapses Based on Memristors

The memristor, characterized by its nano-size, nonvolatility, and continuously adjustable resistance, is a promising candidate for constructing brain-inspired computing. It operates based on ion migration, enabling it to store and retrieve electrical charges. This paper reviews current research on synapses using digital and analog memristors. Synapses based on digital memristors have been utilized to construct positive, zero, and negative weights for artificial neural networks, while synapses based on analog memristors have demonstrated their ability to simulate the essential functions of neural synapses, such as short-term memory (STM), long-term memory (LTM), spike-timing-dependent plasticity (STDP), spike-rate-dependent plasticity (SRDP), and paired-pulse facilitation (PPF). Furthermore, synapses based on analog memristors have shown potential for performing advanced functions such as experiential learning, associative learning, and nonassociative learning. Finally, we highlight some challenges of building large-scale artificial neural networks using memristors.