Gaussian Process Latent Variable Model Research Articles

Extended Poisson Gaussian-Process Latent Variable Model for Unsupervised Neural Decoding.

Dimension reduction on neural activity paves a way for unsupervised neural decoding by dissociating the measurement of internal neural pattern reactivation from the measurement of external variable tuning. With assumptions only on the smoothness of latent dynamics and of internal tuning curves, the Poisson gaussian-process latent variable model (P-GPLVM; Wu etal., 2017) is a powerful tool to discover the low-dimensional latent structure for high-dimensional spike trains. However, when given novel neural data, the original model lacks a method to infer their latent trajectories in the learned latent space, limiting its ability for estimating the neural reactivation. Here, we extend the P-GPLVM to enable the latent variable inference of new data constrained by previously learned smoothness and mapping information. We also describe a principled approach for the constrained latent variable inference for temporally compressed patterns of activity, such as those found in population burst events during hippocampal sharp-wave ripples, as well as metrics for assessing the validity of neural pattern reactivation and inferring the encoded experience. Applying these approaches to hippocampal ensemble recordings during active maze exploration, we replicate the result that P-GPLVM learns a latent space encoding the animal's position. We further demonstrate that this latent space can differentiate one maze context from another. By inferring the latent variables of new neural data during running, certain neural patterns are observed to reactivate, in accordance with the similarity of experiences encoded by its nearby neural trajectories in the training data manifold. Finally, reactivation of neural patterns can be estimated for neural activity during population burst events as well, allowing the identification for replay events of versatile behaviors and more general experiences. Thus, our extension of the P-GPLVM framework for unsupervised analysis of neural activity can be used to answer critical questions related to scientific discovery.

Gaussian Process Latent Variable Model-Based Multi-Output Modeling of Incomplete Data

The rapid development of sensor technologies allows the acquisition of high dimensional sensing data. Multi-output modeling techniques have been developed to leverage the data for decision making. However, the data often contain segments of missing values, which cause great information loss and thus affect the modeling performance. This study explores the missing pattern and the correlation structure of missing segments and maximally exploits useful information in the data to improve multi-output modeling accuracy. Specifically, a new multi-output modeling method is developed based on Gaussian Process Latent Variable Model (GPLVM). A decision score is developed to seek an optimal modeling strategy and then a tailored Expectation-Maximization (EM) algorithm based on GPLVM is designed to estimate the missing segments while optimizing model parameters. The proposed method demonstrates superior performance in both a simulation study and a case study, which makes it a powerful tool to enable process automation. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In real-life applications, missing values are constantly present in multi-output sensing data, which greatly affects data-driven decision-making. Modeling of such data becomes more difficult when consecutive observations within or across different outputs are missing. Existing methods often discard the missing values and extract information only from the available observations. However, the pattern of missing values may contain important messages that can potentially boost the modeling performance. This research develops a new framework based on GPLVM to model multi-output data with segmented missing patterns. A tailored EM algorithm is developed to iteratively impute the missing values and optimize model parameters. In addition, a decision score that quantifies both the missing pattern and correlation is designed to determine an optimal modeling strategy. The proposed method can benefit many applications across different industries that require modeling of multi-output incomplete data, especially when the data have many segments of missing observations.

Multi-Modal Gaussian Process Latent Variable Model With Semi-Supervised Label Dequantization

Multi-Modal Gaussian Process Latent Variable Model With Semi-Supervised Label Dequantization

Visual Emotion Recognition Through Multimodal Cyclic-Label Dequantized Gaussian Process Latent Variable Model

A multimodal cyclic-label dequantized Gaussian process latent variable model (mCDGP) for visual emotion recognition is presented in this paper. Although the emotion is followed by various emotion models that describe cyclic interactions between them, they should be represented as precise labels respecting the emotions’ continuity. Traditional feature integration approaches, however, are incapable of reflecting circular structures to the common latent space. To address this issue, mCDGP uses the common latent space and the cyclic-label dequantization by maximizing the probability function utilizing the cyclic-label feature as one of the observed features. The likelihood maximization problem provides limits to preserve the emotions’ circular structures. Then mCDGP increases the number of dimensions of the common latent space by translating the rough label to the detailed one by label dequantization, with a focus on emotion continuity. Furthermore, label dequantization improves the ability to express label features by retaining circular structures, making accurate visual emotion recognition possible. The main contribution of this paper is the implementation of feature integration through the use of cyclic-label dequantization.

Nonlinear probabilistic virtual sample generation using Gaussian process latent variable model and fitting for rubber material

The development of material informatics has led to an increasingly deep intersection between material science and machine learning (ML). However, limited data volume significantly hampers the application of machine learning models in materials science. To address this challenge, we propose a virtual sample generation algorithm using Gaussian process (VSG-GP) latent variable model and nonlinear fitting for rubber material. The VSG-GP algorithm is versatile, nonlinear, probabilistic, and interpretable. As we known, we are the first to use the Gaussian process latent variable model (GPLVM) and nonlinear fitting to solve the issue of sample generation in material science. Specifically, GPLVM fits the observed few samples to obtain its low-dimensional representation and latent distribution. Furthermore, if the samples contain labels, we use the Gaussian process regression (GPR) to nonlinearly fit the labels. We consider the polymerized styrene butadiene rubber (SBR) dataset and employ the algorithm to generate virtual samples for the SBR. Our investigation shows that the virtual samples follow the same distribution as the original data. The prediction performances of learning models on the virtual samples indicate the value of the labels generated by GPR. Compared with existing methods, experimental results verify the effectiveness and competitiveness of the proposed algorithm. The accuracies of XGBoost and random forest models on the dataset of original and virtual samples are improved by 39% and 43%, respectively.

A Gaussian Process Decoder with Spectral Mixtures and a Locally Estimated Manifold for Data Visualization

Dimensionality reduction plays an important role in interpreting and visualizing high-dimensional data. Previous methods for data visualization overestimate the local structure and lack the consideration of global preservation. In this study, we develop a Gaussian process latent variable model (GP-LVM) for data visualization. GP-LVMs are one of the frameworks of principal component analysis and preserve the global structure effectively. The drawbacks of GP-LVMs are the absence of local structure preservation and the use of low-expressive kernel functions. Therefore, we introduce regularization for local preservation and an expressive kernel function into GP-LVMs to overcome these limitations. As a result, we reflect the global and local structures in low-dimensional representations, improving the reliability and visibility of embeddings. We conduct qualitative and quantitative experiments comparing baselines and state-of-the-art methods on image and text datasets.

Kernel-based Nonlinear Manifold Learning for EEG-based Functional Connectivity Analysis and Channel Selection with Application to Alzheimer’s Disease

Dynamical, causal, and cross-frequency coupling analysis using the electroencephalogram (EEG) has gained significant attention for diagnosing and characterizing neurological disorders. Selecting important EEG channels is crucial for reducing computational complexity in implementing these methods and improving classification accuracy. In neuroscience, measures of (dis) similarity between EEG channels are often used as functional connectivity (FC) features, and important channels are selected via feature selection. Developing a generic measure of (dis) similarity is important for FC analysis and channel selection. In this study, learning of (dis) similarity information within the EEG is achieved using kernel-based nonlinear manifold learning. The focus is on FC changes and, thereby, EEG channel selection. Isomap and Gaussian Process Latent Variable Model (Isomap-GPLVM) are employed for this purpose. The resulting kernel (dis) similarity matrix is used as a novel measure of linear and nonlinear FC between EEG channels. The analysis of EEG from healthy controls (HC) and patients with mild to moderate Alzheimer’s disease (AD) are presented as a case study. Classification results are compared with other commonly used FC measures. Our analysis shows significant differences in FC between bipolar channels of the occipital region and other regions (i.e. parietal, centro-parietal, and fronto-central) between AD and HC groups. Furthermore, our results indicate that FC changes between channels along the fronto-parietal region and the rest of the EEG are important in diagnosing AD. Our results and its relation to functional networks are consistent with those obtained from previous studies using fMRI, resting-state fMRI and EEG.

Optimized Gaussian-Process-Based Probabilistic Latent Variable Modeling Framework for Distributed Nonlinear Process Monitoring

Plant-wide multiunit processes generally contain numerous variables, complex relations, and strong nonlinearity, making the monitoring of such processes challenging. This work proposes a new Gaussian-process-based probabilistic latent variable (GPPLV) modeling framework for distributed monitoring of multiunit nonlinear processes. A Gaussian-process latent variable model is first established to extract the dominant features of a local unit. Using the extracted features, a correlation between the local unit and its neighboring units are then modeled through a Gaussian-process regression (GPR) model. The genetic algorithm is used to determine the ideal independent variables from the neighboring units and optimize the hyperparameters of the GPR model simultaneously. Residuals are generated and monitoring statistics are constructed using an established GPPLV model. Experimental studies on three processes: 1) a numerical example; 2) the Tennessee Eastman benchmark process; and 3) a laboratory distillation process show that compared to some common distributed process monitoring models, the proposed method performs better in showing the nature of different faults and shows higher fault detection rate for large-scale multiunit processes.

Fingerprinting methods for origin and variety assessment of rice: development, validation and data fusion experiments

Several analytical techniques, i.e. Near Infrared (NIR) and Mid-Infrared (MIR) spectroscopy, Hyper Spectral Imaging (HSI), Gas Chromatography coupled to Mass Spectrometry (GC-MS) and Proton-transfer Reaction Time-of-Flight Mass spectrometry (PTR-TOF-MS), combined with chemometrics, are examined to evaluate their potential for solving food authenticity questions on the case of rice. In total, 237 rice samples were analyzed in this study to examine origin and variety assessment and sourced from producing countries (Italy, Spain, Vietnam, Pakistan and Thailand). The Gaussian Process Latent Variable Model (GP-LVM) was applied as technique to obtain a meaningful compressed representation of the data in a two-dimensional space followed by a classification with a nearest neighbour algorithm. What concerns the origin assessment, GC-MS results score good across all countries, resulting in the most accurate method, with prediction rates ranging from 86% to 94%. Data fusion experiments of combination of GC-MS with NIR, or GC-MS with HSI resulted in prediction rates on origin assessment of more than 90%. Variety assessment was performed with these analytical techniques. Using single techniques, HSI achieved prediction values above 90% for all classes (96%–99%). For data fusion experiments of variety assessment combining GC-MS and NIR all prediction values were equal or higher than 92%; for GC-MS combined with HSI, prediction values were all 98%. The presented methodology can be successfully used for the authentication of rice.

Fast and reliable prediction of scramjet flowfields via Gaussian process latent variable model and deep learning

Fast and accurate prediction of high-speed flowfields is of particular interest to researchers in fluid science and engineering to enable efficient design exploration and knowledge discovery. The reliability of prediction is another important metric for the performance of prediction models. While predictive modeling approaches with and without reduced-order modeling (ROM) via machine learning techniques have been proposed, they are inherently subject to loss of information for ROM-based approaches and substantial computational costs in modeling for non-ROM-based approaches. This paper proposes an accurate ROM-based predictive framework with minimum information loss enabled by incorporating Gaussian process latent variable modeling (GPLVM) and deep learning. The stochastic nature of GPLVM allows for uncertainty quantification that indicates the degree of prediction error or reliability of prediction without requiring validation data. The applicability for supersonic/hypersonic viscous flowfields has been examined for two cases including axisymmetric intakes and two-dimensional fuel injection in scramjet engines by comparison with other predictive models. Comparable or superior prediction accuracy over the other models has been achieved by the proposed approaches, demonstrating its high potential to serve as a new competent, data-driven technique for fast, accurate, and reliable prediction of scramjet flowfields.

Surface Approximation by Means of Gaussian Process Latent Variable Models and Line Element Geometry

The close relation between spatial kinematics and line geometry has been proven to be fruitful in surface detection and reconstruction. However, methods based on this approach are limited to simple geometric shapes that can be formulated as a linear subspace of line or line element space. The core of this approach is a principal component formulation to find a best-fit approximant to a possibly noisy or impartial surface given as an unordered set of points or point cloud. We expand on this by introducing the Gaussian process latent variable model, a probabilistic non-linear non-parametric dimensionality reduction approach following the Bayesian paradigm. This allows us to find structure in a lower dimensional latent space for the surfaces of interest. We show how this can be applied in surface approximation and unsupervised segmentation to the surfaces mentioned above and demonstrate its benefits on surfaces that deviate from these. Experiments are conducted on synthetic and real-world objects.

WITHDRAWN: Effects of the penetration height of ethylene transverse jets on flame stabilization behavior in a Mach 2 supersonic crossflow

Effects of fuel jet penetration height on supersonic combustion behaviors were investigated experimentally in a supersonic combustion ramjet model combustor at a Mach speed of 2 and at a stagnation temperature of 1900K. The jet-to-crossflow momentum flux ratio was varied to control the fuel-jet penetration height, using several injectors with different orifice diameters: 2, 3, and 4 mm. First, transverse nitrogen jets were observed to identify a relationship between the fuel jet penetration height and the momentum flux ratio by focusing Schlieren photography. Then, supersonic combustion behaviors of ethylene were investigated through combustion pressure measurements. Simultaneously, time-resolved images of CH* chemiluminescence and shadowgraphs were recorded with high-speed video cameras. Furthermore, a morphology of supersonic combustion modes was investigated for various equivalence ratios and fuel penetration heights in a two-dimensional latent space trained by the shared Gaussian process latent variable models (SGPLVM), considering CH* chemiluminescence images and the shock parameters. The results indicated that the penetration height of nitrogen jets was a function of the jet momentum flux ratio; this function was expressed by a fitting curve. Five typical combustion modes were identified based on time-resolved CH* chemiluminescence images, shadowgraphs, and pressure profiles. Even for a given equivalence ratio, different combustion modes were observed depending on the fuel penetration height. For an injection diameter of 3 and 4 mm, cavity shear-layer and jet-wake stabilized combustions were observed as the scram modes. On the other hand, although the cavity shear-layer and lifted-shear-layer stabilized combustions were observed, no jet-wake stabilized combustion was observed for an orifice diameter of 2mm. Fuel penetration heights above the cavity aft wall were expected to affect the combustion behavior. Finally, a morphology of the supersonic combustion modes was clearly shown in the two-dimensional latent space of the SGPVLM.

Effects of the penetration height of ethylene transverse jets on flame stabilization behavior in a Mach 2 supersonic crossflow

Effects of fuel jet penetration height on supersonic combustion behaviors were investigated experimentally in a supersonic combustion ramjet model combustor at a Mach speed of 2 and at a stagnation temperature of 1900 K. The jet-to-crossflow momentum flux ratio was varied to control the fuel-jet penetration height, using several injectors with different orifice diameters: 2, 3, and 4 mm. First, transverse nitrogen jets were observed to identify a relationship between the fuel jet penetration height and the momentum flux ratio by focusing Schlieren photography. Then, supersonic combustion behaviors of ethylene were investigated through combustion pressure measurements. Simultaneously, time-resolved images of CH* chemiluminescence and shadowgraphs were recorded with high-speed video cameras. Furthermore, a morphology of supersonic combustion modes was investigated for various equivalence ratios and fuel penetration heights in a two-dimensional latent space trained by the shared Gaussian process latent variable models (SGPLVM), considering CH* chemiluminescence images and the shock parameters. The results indicated that the penetration height of nitrogen jets was a function of the jet momentum flux ratio; this function was expressed by a fitting curve. Five typical combustion modes were identified based on time-resolved CH* chemiluminescence images, shadowgraphs, and pressure profiles. Even for a given equivalence ratio, different combustion modes were observed depending on the fuel penetration height. For an injection diameter of 3 and 4 mm, cavity shear-layer and jet-wake stabilized combustions were observed as the scram modes. On the other hand, although the cavity shear-layer and lifted-shear-layer stabilized combustions were observed, no jet-wake stabilized combustion was observed for an orifice diameter of 2 mm. Fuel penetration heights above the cavity aft wall were expected to affect the combustion behavior. Finally, a morphology of the supersonic combustion modes was clearly shown in the two-dimensional latent space of the SGPVLM.

A Generative Model for Quasar Spectra

We build a multi-output generative model for quasar spectra and the properties of their black hole engines, based on a Gaussian process latent-variable model. This model treats every quasar as a vector of latent properties such that the spectrum and all physical properties of the quasar are associated with non-linear functions of those latent parameters; the Gaussian process kernel functions define priors on the function space. Our generative model is trained with a justifiable likelihood function that allows us to treat heteroscedastic noise and missing data correctly, which is crucial for all astrophysical applications. It can simultaneously predict unobserved spectral regions and the physical properties of quasars in held-out test data. We apply the model to rest-frame ultraviolet and optical quasar spectra for which precise black hole masses (based on reverberation-mapping measurements) are available. Unlike reverberation-mapping studies that require multi-epoch data, our model predicts black hole masses from single-epoch spectra—even with limited spectral coverage. We demonstrate the capabilities of the model by predicting black hole masses and unobserved spectral regions. We find that we predict black hole masses at close to the best possible accuracy.

Authenticity analysis of oregano: development, validation and fitness for use of several food fingerprinting techniques

Several analytical techniques, i.e. spectroscopic techniques as Near Infrared (NIR) and Mid-Infrared (MIR), Hyper Spectral Imaging (HSI), Gas Chromatography coupled to Mass Spectrometry (GC–MS) and Proton-transfer Reaction Time-of-Flight Mass spectrometry (PTR-TOF-MS), combined with chemometrics, are examined to evaluate their potential to solve different food authenticity questions on the case of oregano. In total, 102 oregano samples from one harvest season were analyzed for origin and variety assessment, 159 samples for adulteration-assessment and 72 samples for batch-to-batch control. The Gaussian Process Latent Variable Model (GP-LVM) was applied as technique to obtain a reduced two-dimensional space. A Random Forest Regression algorithm was used as regression model for the adulteration assessment. Prediction rates of more than 89% could be achieved for origin assessment. For variety assessment, prediction rates of more than 78% could be obtained. Batch-to-batch control could be successfully performed with NIR and PTR-TOF-MS. Detection of adulteration could be successfully performed from 10% on with HSI, NIR and PTR-TOF-MS.

UNSUPERVISED CLUSTERING AND ANALYSIS OF CONTRACTION-DEPENDENT FETAL HEART RATE SEGMENTS.

The computer-aided interpretation of fetal heart rate (FHR) and uterine contraction (UC) has not been developed well enough for wide use in delivery rooms. The main challenges still lie in the lack of unclear and nonstandard labels for cardiotocography (CTG) recordings, and the timely prediction of fetal state during monitoring. Rather than traditional supervised approaches to FHR classification, this paper demonstrates a way to understand the UC-dependent FHR responses in an unsupervised manner. In this work, we provide a complete method for FHR-UC segment clustering and analysis via the Gaussian process latent variable model, and density-based spatial clustering. We map the UC-dependent FHR segments into a space with a visual dimension and propose a trajectory-based FHR interpretation method. Three metrics of FHR trajectory are defined and an open-access CTG database is used for testing the proposed method.

Adaptive surrogate models with partially observed information

Surrogate models have been developed to replace expensive physical models and reduce the computational cost in various engineering applications, such as reliability analysis and uncertainty quantification. Gaussian process (GP) model exhibits superior performance among surrogate models with a distinguishing feature of estimating the uncertainty. However, fully observed datasets are generally required to establish a GP model, which is often scarce and expensive to obtain in complex engineering systems. Partially overserved information is often available and relatively plentiful in the collected datasets, which often contain data from different sources that have multi-fidelity or dimensionality and missing values. Therefore, correctly accounting for the partially observed information is important in order to take advantage of all available information and increase the prediction performance of the surrogate model to be developed. This paper presents a new method for modeling system performance with partially observed information, which integrates the Bayesian Gaussian process latent variable model (BGPLVM) with adaptive sampling to iteratively select new partially observable training sample points to improve the modeling efficiency. A novel adaptive sampling approach considering the missing frame and information cost of the partially observed information is proposed to iteratively select new training sample points and refine the model. To the best of the authors' knowledge, this is the first work designing adaptive sampling and adaptive surrogate modeling approaches for a dataset containing missing values. The numerical experiments demonstrated that the adaptive surrogate modeling method can effectively use all available information including both fully observed and partially observed data points. The developed methodology provides an accurate and cost-effective solution to take advantage of extra partially observed information in developing surrogate models.

Investigating Shape Variation Using Generalized Procrustes Analysis and Machine Learning

The biological investigation of a population’s shape diversity using digital images is typically reliant on geometrical morphometrics, which is an approach based on user-defined landmarks. In contrast to this traditional approach, the progress in deep learning has led to numerous applications ranging from specimen identification to object detection. Typically, these models tend to become black boxes, which limits the usage of recent deep learning models for biological applications. However, the progress in explainable artificial intelligence tries to overcome this limitation. This study compares the explanatory power of unsupervised machine learning models to traditional landmark-based approaches for population structure investigation. We apply convolutional autoencoders as well as Gaussian process latent variable models to two Nile tilapia datasets to investigate the latent structure using consensus clustering. The explanatory factors of the machine learning models were extracted and compared to generalized Procrustes analysis. Hypotheses based on the Bayes factor are formulated to test the unambiguity of population diversity unveiled by the machine learning models. The findings show that it is possible to obtain biologically meaningful results relying on unsupervised machine learning. Furthermore we show that the machine learning models unveil latent structures close to the true population clusters. We found that 80% of the true population clusters relying on the convolutional autoencoder are significantly different to the remaining clusters. Similarly, 60% of the true population clusters relying on the Gaussian process latent variable model are significantly different. We conclude that the machine learning models outperform generalized Procrustes analysis, where 16% of the population cluster was found to be significantly different. However, the applied machine learning models still have limited biological explainability. We recommend further in-depth investigations to unveil the explanatory factors in the used model.

Investigating Explanatory Factors of Machine Learning Models for Plant Classification.

Recent progress in machine learning and deep learning has enabled the implementation of plant and crop detection using systematic inspection of the leaf shapes and other morphological characters for identification systems for precision farming. However, the models used for this approach tend to become black-box models, in the sense that it is difficult to trace characters that are the base for the classification. The interpretability is therefore limited and the explanatory factors may not be based on reasonable visible characters. We investigate the explanatory factors of recent machine learning and deep learning models for plant classification tasks. Based on a Daucus carota and a Beta vulgaris image data set, we implement plant classification models and compare those models by their predictive performance as well as explainability. For comparison we implemented a feed forward convolutional neuronal network as a default model. To evaluate the performance, we trained an unsupervised Bayesian Gaussian process latent variable model as well as a convolutional autoencoder for feature extraction and rely on a support vector machine for classification. The explanatory factors of all models were extracted and analyzed. The experiments show, that feed forward convolutional neuronal networks ( and mean accuracy) outperforms the Bayesian Gaussian process latent variable pipeline ( and mean accuracy) as well as the convolutional autoenceoder pipeline ( and mean accuracy) based approaches in terms of classification accuracy, even though not significant for Beta vulgaris images. Additionally, we found that the neuronal network used biological uninterpretable image regions for the plant classification task. In contrast to that, the unsupervised learning models rely on explainable visual characters. We conclude that supervised convolutional neuronal networks must be used carefully to ensure biological interpretability. We recommend unsupervised machine learning, careful feature investigation, and statistical feature analysis for biological applications.

Gaussian process latent variable model factorization for context-aware recommender systems

• We propose a novel algorithm named GPLVMF to achieve a Bayesian factorization for Context-aware recommendation. • We model the non-zero mean in Gaussian processes to capture the bias and provide a generalized variational inference. • Our method can flexibly deal with both real-valued and categorical contexts. • Both scaled conjugate gradient and stochastic gradient descent optimization methods are applied to the model. • Experiment results show that GPLVMF not only improves accuracy but can automatically infer the influence of context. Context-aware recommender systems (CARS) have gained increasing attention due to their ability to utilize contextual information. Compared to traditional recommender systems, CARS are, in general, able to generate more accurate recommendations. The latent factors approach accounts for a large proportion of CARS. Recently, a non-linear Gaussian Process (GP) based factorization method was proven to outperform the state-of-the-art methods in CARS. Despite its effectiveness, standard GP model-based methods can suffer from over-fitting and may not be able to determine the impact of each context automatically. In order to address such shortcomings, we propose a Gaussian Process Latent Variable Model Factorization (GPLVMF) method, where we apply an appropriate prior to the original GP model. Our work is primarily inspired by the Gaussian Process Latent Variable Model (GPLVM), which is a non-linear dimensionality reduction method. Since the traditional maximum likelihood approach for standard GP cannot be applied to our model, we adopt a variational inference method to solve the GPLVMF model. As a result, we improve the results on the real datasets significantly as well as capturing the importance of each context. In addition to the general advantages, our method flexibly provides two contributions regarding recommender system settings: (1) addressing the influence of bias by setting a non-zero mean function, and (2) utilizing real-valued contexts by fixing the latent space with real values.