Conventional uncertainty quantification methods usually lacks the capability of dealing with high-dimensional problems due to the curse of dimensionality. This paper presents a semi-supervised learning framework for dimension reduction and reliability analysis. An autoencoder is first adopted for mapping the high-dimensional space into a low-dimensional latent space, which contains a distinguishable failure surface. Then a deep feedforward neural network (DFN) is utilized to learn the mapping relationship and reconstruct the latent space, while the Gaussian process (GP) modeling technique is used to build the surrogate model of the transformed limit state function. During the training process of the DFN, the discrepancy between the actual and reconstructed latent space is minimized through semi-supervised learning for ensuring the accuracy. Both labeled and unlabeled samples are utilized for defining the loss function of the DFN. Evolutionary algorithm is adopted to train the DFN, then the Monte Carlo simulation method is used for uncertainty quantification and reliability analysis based on the proposed framework. The effectiveness is demonstrated through a mathematical example.
University of Amsterdam（阿姆斯特丹大学）；Radboud University Medical Center（拉德堡德大学医学中心）
The accurate estimation of predictive uncertainty carries importance in medical scenarios such as lung node segmentation. Unfortunately, most existing works on predictive uncertainty do not return calibrated uncertainty estimates, which could be used in practice. In this work we exploit multi-grader annotation variability as a source of 'groundtruth' aleatoric uncertainty, which can be treated as a target in a supervised learning problem. We combine this groundtruth uncertainty with a Probabilistic U-Net and test on the LIDC-IDRI lung nodule CT dataset and MICCAI2012 prostate MRI dataset. We f…
The problem of fully supervised classification is that it requires a tremendous amount of annotated data, however, in many datasets a large portion of data is unlabeled. To alleviate this problem semi-supervised learning (SSL) leverages the knowledge of the classifier on the labeled domain and extrapolates it to the unlabeled domain which has a supposedly similar distribution as annotated data. Recent success on SSL methods crucially hinges on thresholded pseudo labeling and thereby consistency regularization for the unlabeled domain. However, the existing methods do not incorporate the uncert…
National University of Defense Technology（国防科技大学）；Chinese Academy of Military Science（中国人民解放军军事科学院）
The surrogate model-based uncertainty quantification method has drawn much attention in many engineering fields. Polynomial chaos expansion (PCE) and deep learning (DL) are powerful methods for building a surrogate model. However, PCE needs to increase the expansion order to improve the accuracy of the surrogate model, which causes more labeled data to solve the expansion coefficients, and DL also requires a lot of labeled data to train the deep neural network (DNN). First of all, this paper proposes the adaptive arbitrary polynomial chaos (aPC) and proves two properties about the adaptive exp…
We present a new data-driven model to reconstruct nonlinear flow from spatially sparse observations. The model is a version of a conditional variational auto-encoder (CVAE), which allows for probabilistic reconstruction and thus uncertainty quantification of the prediction. We show that in our model, conditioning on the measurements from the complete flow data leads to a CVAE where only the decoder depends on the measurements. For this reason we call the model as Semi-Conditional Variational Autoencoder (SCVAE). The method, reconstructions and associated uncertainty estimates are illustrated o…
Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification
Rational Intelligence Lab, CISPA Helmholtz Center for Information Security Department of Statistics, LMU Munich（理性智能实验室，CISPA海德堡信息安全中心统计学系，慕尼黑大学）；Department of Statistics, LMU Munich（统计学系，慕尼黑大学）；Department of Computer Science, The University of Manchester（计算机科学系，曼彻斯特大学）
University of California Los Angeles（加利福尼亚大学洛杉矶分校）；Caltech（加州理工学院）；University of Edinburgh（爱丁堡大学）
Classification of high dimensional data finds wide-ranging applications. In many of these applications equipping the resulting classification with a measure of uncertainty may be as important as the classification itself. In this paper we introduce, develop algorithms for, and investigate the properties of, a variety of Bayesian models for the task of binary classification; via the posterior distribution on the classification labels, these methods automatically give measures of uncertainty. The methods are all based around the graph formulation of semi-supervised learning. We provide a unified…
Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
Semi-supervised learning algorithms reduce the high cost of acquiring labeled training data by using both labeled and unlabeled data during learning. Deep Convolutional Networks (DCNs) have achieved great success in supervised tasks and as such have been widely employed in the semi-supervised learning. In this paper we leverage the recently developed Deep Rendering Mixture Model (DRMM), a probabilistic generative model that models latent nuisance variation, and whose inference algorithm yields DCNs. We develop an EM algorithm for the DRMM to learn from both labeled and unlabeled data. Guided b…
A Comprehensive Framework for Uncertainty Quantification of Voxel-wise Supervised Models in IVIM MRI
Department of Acute Brain（急性脑和心血管损伤部门）；Department of Neuroscience（神经科学部门）；Istituto di Ricerche Farmacologiche Mario Negri IRCCS（马里奥·内格里药理研究所）；Milan（米兰）；Italy（意大利）
We present a simple and effective approach for posterior uncertainty quantification in deep operator networks (DeepONets); an emerging paradigm for supervised learning in function spaces. We adopt a frequentist approach based on randomized prior ensembles, and put forth an efficient vectorized implementation for fast parallel inference on accelerated hardware. Through a collection of representative examples in computational mechanics and climate modeling, we show that the merits of the proposed approach are fourfold. (1) It can provide more robust and accurate predictions when compared against…
Supervised Machine Learning Methods with Uncertainty Quantification for Exoplanet Atmospheric Retrievals from Transmission Spectroscopy
Physics Department, University of Florida, Gainesville, FL 32611, USA（佛罗里达大学物理系）；Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA（阿拉巴马大学物理与天文学系）
We formulate a new problem at the intersectionof semi-supervised learning and contextual bandits,motivated by several applications including clini-cal trials and ad recommendations. We demonstratehow Graph Convolutional Network (GCN), a semi-supervised learning approach, can be adjusted tothe new problem formulation. We also propose avariant of the linear contextual bandit with semi-supervised missing rewards imputation. We thentake the best of both approaches to develop multi-GCN embedded contextual bandit. Our algorithmsare verified on several real world datasets.
Labeled data used for training activity recognition classifiers are usually limited in terms of size and diversity. Thus, the learned model may not generalize well when used in real-world use cases. Semi-supervised learning augments labeled examples with unlabeled examples, often resulting in improved performance. However, the semi-supervised methods studied in the activity recognition literatures assume that feature engineering is already done. In this paper, we lift this assumption and present two semi-supervised methods based on convolutional neural networks (CNNs) to learn discriminative h…
Graduate School of Frontier Sciences, The University of Tokyo（东京大学前沿科学研究生院）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）
Maximizing the area under the receiver operating characteristic curve (AUC) is a standard approach to imbalanced classification. So far, various supervised AUC optimization methods have been developed and they are also extended to semi-supervised scenarios to cope with small sample problems. However, existing semi-supervised AUC optimization methods rely on strong distributional assumptions, which are rarely satisfied in real-world problems. In this paper, we propose a novel semi-supervised AUC optimization method that does not require such restrictive assumptions. We first develop an AUC opti…
Center of Informatics - CIn, Universidade Federal de Pernambuco（信息中心 - 伯南布哥联邦大学）
There has been an increasing interest in semi-supervised learning in the recent years because of the great number of datasets with a large number of unlabeled data but only a few labeled samples. Semi-supervised learning algorithms can work with both types of data, combining them to obtain better performance for both clustering and classification. Also, these datasets commonly have a high number of dimensions. This article presents a new semi-supervised method based on self-organizing maps (SOMs) for clustering and classification, called Semi-Supervised Self-Organizing Map (SS-SOM). The method…
We explain Poisson learning on graph-based semi-supervised learning to see if it could avoid the problem of global information loss problem as Laplace-based learning methods on large graphs. From our analysis, Poisson learning is simply Laplace regularization with thresholding, cannot overcome the problem.
Delft University of Technology（代尔夫特理工大学）；Leiden University Medical Center（莱顿大学医学中心）
In this paper, we introduce a package for semi-supervised learning research in the R programming language called RSSL. We cover the purpose of the package, the methods it includes and comment on their use and implementation. We then show, using several code examples, how the package can be used to replicate well-known results from the semi-supervised learning literature.
Rice University（莱斯大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We introduce Negative Sampling in Semi-Supervised Learning (NS3L), a simple, fast, easy to tune algorithm for semi-supervised learning (SSL). NS3L is motivated by the success of negative sampling/contrastive estimation. We demonstrate that adding the NS3L loss to state-of-the-art SSL algorithms, such as the Virtual Adversarial Training (VAT), significantly improves upon vanilla VAT and its variant, VAT with Entropy Minimization. By adding the NS3L loss to MixMatch, the current state-of-the-art approach on semi-supervised tasks, we observe significant improvements over vanilla MixMatch. We cond…
National Research University Higher School of Economics（国家研究型高等经济大学）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Yandex School of Data Analysis（Yandex数据分析学院）；Samsung AI Center Moscow（三星莫斯科人工智能中心）；University of Amsterdam（阿姆斯特丹大学）
This paper proposes a semi-conditional normalizing flow model for semi-supervised learning. The model uses both labelled and unlabeled data to learn an explicit model of joint distribution over objects and labels. Semi-conditional architecture of the model allows us to efficiently compute a value and gradients of the marginal likelihood for unlabeled objects. The conditional part of the model is based on a proposed conditional coupling layer. We demonstrate performance of the model for semi-supervised classification problem on different datasets. The model outperforms the baseline approach bas…
Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）；Leiden University Medical Center（莱顿大学医学中心）
Semi-supervised learning is an important and active topic of research in pattern recognition. For classification using linear discriminant analysis specifically, several semi-supervised variants have been proposed. Using any one of these methods is not guaranteed to outperform the supervised classifier which does not take the additional unlabeled data into account. In this work we compare traditional Expectation Maximization type approaches for semi-supervised linear discriminant analysis with approaches based on intrinsic constraints and propose a new principled approach for semi-supervised l…
An Axiomatic Assessment of Entropy- and Variance-based Uncertainty Quantification in Regression
LMU Munich（慕尼黑莱布尼茨大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；DLR-German Aerospace Center University of Tromsø（德国航空航天中心图勒大学）；German Research Center for Artificial Intelligence (DFKI, DSA)（德国人工智能研究中心）
University of Chicago, Booth School of Business（芝加哥大学布斯商学院）
We propose a computationally efficient method to construct nonparametric, heteroscedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. Our approach provides an alternative to the now-standard conformal prediction for uncertainty quantification, with novel theoretical insights and computational advantages. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, has strong non-asymptotic coverage properties, and is easy to implement using standard convex programs. Our approach can be viewed as a…
Graz University of Technology（格拉茨技术大学）；Austrian Research Institute for Artificial Intelligence（奥地利人工智能研究所）；University of Cambridge（剑桥大学）
In several domains obtaining class annotations is expensive while at the same time unlabelled data are abundant. While most semi-supervised approaches enforce restrictive assumptions on the data distribution, recent work has managed to learn semi-supervised models in a non-restrictive regime. However, so far such approaches have only been proposed for linear models. In this work, we introduce semi-supervised parameter learning for Sum-Product Networks (SPNs). SPNs are deep probabilistic models admitting inference in linear time in number of network edges. Our approach has several advantages, a…
Deep neural networks demonstrated their ability to provide remarkable performances on a wide range of supervised learning tasks (e.g., image classification) when trained on extensive collections of labeled data (e.g., ImageNet). However, creating such large datasets requires a considerable amount of resources, time, and effort. Such resources may not be available in many practical cases, limiting the adoption and the application of many deep learning methods. In a search for more data-efficient deep learning methods to overcome the need for large annotated datasets, there is a rising research …
Delft University of Technology（代尔夫特理工大学）；Leiden University Medical Center（莱顿大学医学中心）；University of Copenhagen（哥本哈根大学）
For semi-supervised techniques to be applied safely in practice we at least want methods to outperform their supervised counterparts. We study this question for classification using the well-known quadratic surrogate loss function. Using a projection of the supervised estimate onto a set of constraints imposed by the unlabeled data, we find we can safely improve over the supervised solution in terms of this quadratic loss. Unlike other approaches to semi-supervised learning, the procedure does not rely on assumptions that are not intrinsic to the classifier at hand. It is theoretically demonst…
The ever-increasing size of modern data sets combined with the difficulty of obtaining label information has made semi-supervised learning one of the problems of significant practical importance in modern data analysis. We revisit the approach to semi-supervised learning with generative models and develop new models that allow for effective generalisation from small labelled data sets to large unlabelled ones. Generative approaches have thus far been either inflexible, inefficient or non-scalable. We show that deep generative models and approximate Bayesian inference exploiting recent advances…
We explore a novel approach to semi-supervised learning. This approach is contrary to the common approach in that the unlabeled examples serve to "muffle," rather than enhance, the guidance provided by the labeled examples. We provide several variants of the basic algorithm and show experimentally that they can achieve significantly higher AUC than boosted trees, random forests and logistic regression when unlabeled examples are available.
Learning from data sequentially arriving, possibly in a non i.i.d. way, with changing task distribution over time is called continual learning. Much of the work thus far in continual learning focuses on supervised learning and some recent works on unsupervised learning. In many domains, each task contains a mix of labelled (typically very few) and unlabelled (typically plenty) training examples, which necessitates a semi-supervised learning approach. To address this in a continual learning setting, we propose a framework for semi-supervised continual learning called Meta-Consolidation for Cont…
Reaching the performance of fully supervised learning with unlabeled data and only labeling one sample per class might be ideal for deep learning applications. We demonstrate for the first time the potential for building one-shot semi-supervised (BOSS) learning on Cifar-10 and SVHN up to attain test accuracies that are comparable to fully supervised learning. Our method combines class prototype refining, class balancing, and self-training. A good prototype choice is essential and we propose a technique for obtaining iconic examples. In addition, we demonstrate that class balancing methods subs…
Robust and Reliable AI for Predictive Quality in Semiconductor Materials Manufacturing with MLOps and Uncertainty Quantification
Merck Group（默克集团）；Versum Materials US LLC（Versum Materials美国分公司）；Merck Electronics KGaA（默克电子（KGaA））；Technical University of Darmstadt（达姆施塔特技术大学）；Otto von Guericke University Magdeburg（奥托·冯·格里克大学马格德堡分校）
Recent advances in semi-supervised learning have shown tremendous potential in overcoming a major barrier to the success of modern machine learning algorithms: access to vast amounts of human-labeled training data. Previous algorithms based on consistency regularization can harness the abundance of unlabeled data to produce impressive results on a number of semi-supervised benchmarks, approaching the performance of strong supervised baselines using only a fraction of the available labeled data. In this work, we challenge the long-standing success of consistency regularization by introducing se…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We introduce the SaaS Algorithm for semi-supervised learning, which uses learning speed during stochastic gradient descent in a deep neural network to measure the quality of an iterative estimate of the posterior probability of unknown labels. Training speed in supervised learning correlates strongly with the percentage of correct labels, so we use it as an inference criterion for the unknown labels, without attempting to infer the model parameters at first. Despite its simplicity, SaaS achieves state-of-the-art results in semi-supervised learning benchmarks.
One of the greatest obstacles in the adoption of deep neural networks for new applications is that training the network typically requires a large number of manually labeled training samples. We empirically investigate the scenario where one has access to large amounts of unlabeled data but require labeling only a single prototypical sample per class in order to train a deep network (i.e., one-shot semi-supervised learning). Specifically, we investigate the recent results reported in FixMatch for one-shot semi-supervised learning to understand the factors that affect and impede high accuracies…
KTH Royal Institute of Technology（KTH皇家理工学院）；School of Computer Science（计算机科学学院）
We propose the application of a semi-supervised learning method to improve the performance of acoustic modelling for automatic speech recognition based on deep neural net- works. As opposed to unsupervised initialisation followed by supervised fine tuning, our method takes advantage of both unlabelled and labelled data simultaneously through mini- batch stochastic gradient descent. We tested the method with varying proportions of labelled vs unlabelled observations in frame-based phoneme classification on the TIMIT database. Our experiments show that the method outperforms standard supervised …
Virginia Tech（弗吉尼亚理工大学）；Oklahoma State University（俄克拉荷马州立大学）
Radio emitter recognition in dense multi-user environments is an important tool for optimizing spectrum utilization, identifying and minimizing interference, and enforcing spectrum policy. Radio data is readily available and easy to obtain from an antenna, but labeled and curated data is often scarce making supervised learning strategies difficult and time consuming in practice. We demonstrate that semi-supervised learning techniques can be used to scale learning beyond supervised datasets, allowing for discerning and recalling new radio signals by using sparse signal representations based on …
Department of Computer and Information Science University of Delaware（特拉华大学计算机与信息科学系）
Adversarial training is a technique of improving model performance by involving adversarial examples in the training process. In this paper, we investigate adversarial training with multiple adversarial examples to benefit the relation extraction task. We also apply adversarial training technique in semi-supervised scenarios to utilize unlabeled data. The evaluation results on protein-protein interaction and protein subcellular localization task illustrate adversarial training provides improvement on the supervised model, and is also effective on involving unlabeled data in the semi-supervised…
In this paper, we study statistical properties of semi-supervised learning, which is considered as an important problem in the community of machine learning. In the standard supervised learning, only the labeled data is observed. The classification and regression problems are formalized as the supervised learning. In semi-supervised learning, unlabeled data is also obtained in addition to labeled data. Hence, exploiting unlabeled data is important to improve the prediction accuracy in semi-supervised learning. This problems is regarded as a semiparametric estimation problem with missing data. …
We consider the estimation problem in high-dimensional semi-supervised learning. Our goal is to investigate when and how the unlabeled data can be exploited to improve the estimation of the regression parameters of linear model in light of the fact that such linear models may be misspecified in data analysis. We first establish the minimax lower bound for parameter estimation in the semi-supervised setting, and show that this lower bound cannot be achieved by supervised estimators using the labeled data only. We propose an optimal semi-supervised estimator that can attain this lower bound and …
Technical University Munich（慕尼黑工业大学）；Helmholtz AI（亥姆霍兹人工智能研究院）
In computational pathology, we often face a scarcity of annotations and a large amount of unlabeled data. One method for dealing with this is semi-supervised learning which is commonly split into a self-supervised pretext task and a subsequent model fine-tuning. Here, we compress this two-stage training into one by introducing S5CL, a unified framework for fully-supervised, self-supervised, and semi-supervised learning. With three contrastive losses defined for labeled, unlabeled, and pseudo-labeled images, S5CL can learn feature representations that reflect the hierarchy of distance relations…
A common assumption in semi-supervised learning is that the labeled, unlabeled, and test data are drawn from the same distribution. However, this assumption is not satisfied in many applications. In many scenarios, the data is collected sequentially (e.g., healthcare) and the distribution of the data may change over time often exhibiting so-called covariate shifts. In this paper, we propose an approach for semi-supervised learning algorithms that is capable of addressing this issue. Our framework also recovers some popular methods, including entropy minimization and pseudo-labeling. We provide…
National University of Singapore（新加坡国立大学）；Institute for Infocomm Research, A*STAR（新加坡科技研究局资讯通信研究院）；Singapore University of Technology and Design（新加坡科技设计大学）
Deep learning approaches achieve state-of-the-art performance for classifying radiology images, but rely on large labelled datasets that require resource-intensive annotation by specialists. Both semi-supervised learning and active learning can be utilised to mitigate this annotation burden. However, there is limited work on combining the advantages of semi-supervised and active learning approaches for multi-label medical image classification. Here, we introduce a novel Consistency-based Semi-supervised Evidential Active Learning framework (CSEAL). Specifically, we leverage predictive uncertai…
In this paper we analyze the graph-based approach to semi-supervised learning under a manifold assumption. We adopt a Bayesian perspective and demonstrate that, for a suitable choice of prior constructed with sufficiently many unlabeled data, the posterior contracts around the truth at a rate that is minimax optimal up to a logarithmic factor. Our theory covers both regression and classification.
Quantification using Permutation-Invariant Networks based on Histograms
Artificial Intelligence Center（人工智能中心）；University of Oviedo（奥维耶多大学）；Istituto di Scienza e Tecnologie dell’Informazione（信息科学与技术研究所）；Consiglio Nazionale delle Ricerche（国家研究委员会）
We present a new flavor of Variational Autoencoder (VAE) that interpolates seamlessly between unsupervised, semi-supervised and fully supervised learning domains. We show that unlabeled datapoints not only boost unsupervised tasks, but also the classification performance. Vice versa, every label not only improves classification, but also unsupervised tasks. The proposed architecture is simple: A classification layer is connected to the topmost encoder layer, and then combined with the resampled latent layer for the decoder. The usual evidence lower bound (ELBO) loss is supplemented with a supe…
Palo Alto Research Center（帕洛阿尔托研究中心）；Verizon Labs（威瑞森实验室）
We introduce a new model for building conditional generative models in a semi-supervised setting to conditionally generate data given attributes by adapting the GAN framework. The proposed semi-supervised GAN (SS-GAN) model uses a pair of stacked discriminators to learn the marginal distribution of the data, and the conditional distribution of the attributes given the data respectively. In the semi-supervised setting, the marginal distribution (which is often harder to learn) is learned from the labeled + unlabeled data, and the conditional distribution is learned purely from the labeled data.…
Alma Mater Studiorun - University of Bologna（博洛尼亚大学）
Recent works demonstrated the usefulness of temporal coherence to regularize supervised training or to learn invariant features with deep architectures. In particular, enforcing smooth output changes while presenting temporally-closed frames from video sequences, proved to be an effective strategy. In this paper we prove the efficacy of temporal coherence for semi-supervised incremental tuning. We show that a deep architecture, just mildly trained in a supervised manner, can progressively improve its classification accuracy, if exposed to video sequences of unlabeled data. The extent to which,…
École Polytechnique（巴黎综合理工学院）；University of California, Berkeley（加州大学伯克利分校）
Class labels are often imperfectly observed, due to mistakes and to genuine ambiguity among classes. We propose a new semi-supervised deep generative model that explicitly models noisy labels, called the Mislabeled VAE (M-VAE). The M-VAE can perform better than existing deep generative models which do not account for label noise. Additionally, the derivation of M-VAE gives new theoretical insights into the popular M1+M2 semi-supervised model.
Anomaly Detection has several important applications. In this paper, our focus is on detecting anomalies in seller-reviewer data using tensor decomposition. While tensor-decomposition is mostly unsupervised, we formulate Bayesian semi-supervised tensor decomposition to take advantage of sparse labeled data. In addition, we use Polya-Gamma data augmentation for the semi-supervised Bayesian tensor decomposition. Finally, we show that the Pólya-Gamma formulation simplifies calculation of the Fisher information matrix for partial natural gradient learning. Our experimental results show that our se…
We used the Ladder Network [Rasmus et al. (2015)] to perform Hyperspectral Image Classification in a semi-supervised setting. The Ladder Network distinguishes itself from other semi-supervised methods by jointly optimizing a supervised and unsupervised cost. In many settings this has proven to be more successful than other semi-supervised techniques, such as pretraining using unlabeled data. We furthermore show that the convolutional Ladder Network outperforms most of the current techniques used in hyperspectral image classification and achieves new state-of-the-art performance on the Pavia Un…
Large amounts of labeled data are typically required to train deep learning models. For many real-world problems, however, acquiring additional data can be expensive or even impossible. We present semi-supervised deep kernel learning (SSDKL), a semi-supervised regression model based on minimizing predictive variance in the posterior regularization framework. SSDKL combines the hierarchical representation learning of neural networks with the probabilistic modeling capabilities of Gaussian processes. By leveraging unlabeled data, we show improvements on a diverse set of real-world regression tas…
Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）
Improvement guarantees for semi-supervised classifiers can currently only be given under restrictive conditions on the data. We propose a general way to perform semi-supervised parameter estimation for likelihood-based classifiers for which, on the full training set, the estimates are never worse than the supervised solution in terms of the log-likelihood. We argue, moreover, that we may expect these solutions to really improve upon the supervised classifier in particular cases. In a worked-out example for LDA, we take it one step further and essentially prove that its semi-supervised version …
