Image denoising is a prerequisite for downstream tasks in many fields. Low-dose and photon-counting computed tomography (CT) denoising can optimize diagnostic performance at minimized radiation dose. Supervised deep denoising methods are popular but require paired clean or noisy samples that are often unavailable in practice. Limited by the independent noise assumption, current unsupervised denoising methods cannot process correlated noises as in CT images. Here we propose the first-of-its-kind similarity-based unsupervised deep denoising approach, referred to as Noise2Sim, that works in a nonlocal and nonlinear fashion to suppress not only independent but also correlated noises. Theoretically, Noise2Sim is asymptotically equivalent to supervised learning methods under mild conditions. Experimentally, Nosie2Sim recovers intrinsic features from noisy low-dose CT and photon-counting CT images as effectively as or even better than supervised learning methods on practical datasets visually, quantitatively and statistically. Noise2Sim is a general unsupervised denoising approach and has great potential in diverse applications.
National University of Singapore（新加坡国立大学）；Nanyang Technological University（南洋理工大学）；Yale-NUS College（耶鲁-新加坡国立大学学院）
Night images suffer not only from low light, but also from uneven distributions of light. Most existing night visibility enhancement methods focus mainly on enhancing low-light regions. This inevitably leads to over enhancement and saturation in bright regions, such as those regions affected by light effects (glare, floodlight, etc). To address this problem, we need to suppress the light effects in bright regions while, at the same time, boosting the intensity of dark regions. With this idea in mind, we introduce an unsupervised method that integrates a layer decomposition network and a light-…
Guangzhou University（广州大学）；Tencent（腾讯）；Nanjing University of Science and Technology（南京理工大学）；Xiamen University（厦门大学）
In recent years, Face Image Quality Assessment (FIQA) has become an indispensable part of the face recognition system to guarantee the stability and reliability of recognition performance in an unconstrained scenario. For this purpose, the FIQA method should consider both the intrinsic property and the recognizability of the face image. Most previous works aim to estimate the sample-wise embedding uncertainty or pair-wise similarity as the quality score, which only considers the information from partial intra-class. However, these methods ignore the valuable information from the inter-class, w…
Huazhong University of Science and Technology（华中科技大学）；Peking University（北京大学）；University of Electronic Science and Technology of China（电子科技大学）
Image deraining is a typical low-level image restoration task, which aims at decomposing the rainy image into two distinguishable layers: the clean image layer and the rain layer. Most of the existing learning-based deraining methods are supervisedly trained on synthetic rainy-clean pairs. The domain gap between the synthetic and real rains makes them less generalized to different real rainy scenes. Moreover, the existing methods mainly utilize the property of the two layers independently, while few of them have considered the mutually exclusive relationship between the two layers. In this wor…
Hierarchical Similarity Learning for Aliasing Suppression Image Super-Resolution
School of Software, Dalian University of Technology（大连理工大学软件学院）；International School of Information Science and Engineering, Dalian University of Technology（大连理工大学国际信息与工程学院）；School of Electronic and Computer Engineering, Peking University Shenzhen Graduate School（北京大学深圳研究生院电子与计算机工程学院）
Ryerson University（瑞尔森大学）；University of Toronto（多伦多大学）；Institute for Biomedical Engineering, Science and Technology (iBEST)（生物医学工程、科学与技术研究所（iBEST））
We propose FlowReg, a deep learning-based framework for unsupervised image registration for neuroimaging applications. The system is composed of two architectures that are trained sequentially: FlowReg-A which affinely corrects for gross differences between moving and fixed volumes in 3D followed by FlowReg-O which performs pixel-wise deformations on a slice-by-slice basis for fine tuning in 2D. The affine network regresses the 3D affine matrix based on a correlation loss function that enforces global similarity. The deformable network operates on 2D image slices based on the optical flow netw…
Multi-modality image registration is one of the most underlined processes in medical image analysis. Recently, convolutional neural networks (CNNs) have shown significant potential in deformable registration. However, the lack of voxel-wise ground truth challenges the training of CNNs for an accurate registration. In this work, we propose a cross-modality similarity metric, based on the KL-divergence of image variables, and implement an efficient estimation method using a CNN. This estimation network, referred to as KLDivNet, can be trained unsupervisedly. We then embed the KLDivNet into a reg…
In this paper, we introduce an adaptive unsupervised learning framework, which utilizes natural images to train filter sets. The applicability of these filter sets is demonstrated by evaluating their performance in two contrasting applications - image quality assessment and texture retrieval. While assessing image quality, the filters need to capture perceptual differences based on dissimilarities between a reference image and its distorted version. In texture retrieval, the filters need to assess similarity between texture images to retrieve closest matching textures. Based on experiments, we…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
Contrastive learning is one of the fastest growing research areas in machine learning due to its ability to learn useful representations without labeled data. However, contrastive learning is susceptible to feature suppression, i.e., it may discard important information relevant to the task of interest, and learn irrelevant features. Past work has addressed this limitation via handcrafted data augmentations that eliminate irrelevant information. This approach however does not work across all datasets and tasks. Further, data augmentations fail in addressing feature suppression in multi-attribu…
Yandex（扬达克斯）；Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）
We tackle the problem of unsupervised visual descriptors compression, which is a key ingredient of large-scale image retrieval systems. While the deep learning machinery has benefited literally all computer vision pipelines, the existing state-of-the-art compression methods employ shallow architectures, and we aim to close this gap by our paper. In more detail, we introduce a DNN architecture for the unsupervised compressed-domain retrieval, based on multi-codebook quantization. The proposed architecture is designed to incorporate both fast data encoding and efficient distances computation via…
MVTec Software GmbH（MVTec软件有限公司）；University of Amsterdam（阿姆斯特丹大学）
Convolutional autoencoders have emerged as popular methods for unsupervised defect segmentation on image data. Most commonly, this task is performed by thresholding a pixel-wise reconstruction error based on an $\ell^p$ distance. This procedure, however, leads to large residuals whenever the reconstruction encompasses slight localization inaccuracies around edges. It also fails to reveal defective regions that have been visually altered when intensity values stay roughly consistent. We show that these problems prevent these approaches from being applied to complex real-world scenarios and that…
Wuhan University of Technology（武汉理工大学）；Institute of Geographic Sciences and Natural Resources Research, CAS（中国科学院地理科学与资源研究所）；Institute of High Performance Computing, A*Star (Agency for Science, Technology and Research)（新加坡科技研究局高性能计算研究所）
To achieve reliable mining results for massive vessel trajectories, one of the most important challenges is how to efficiently compute the similarities between different vessel trajectories. The computation of vessel trajectory similarity has recently attracted increasing attention in the maritime data mining research community. However, traditional shape- and warping-based methods often suffer from several drawbacks such as high computational cost and sensitivity to unwanted artifacts and non-uniform sampling rates, etc. To eliminate these drawbacks, we propose an unsupervised learning method…
Singapore-MIT Alliance（新加坡-麻省理工学院联盟）；Nanyang Technological University（南洋理工大学）；National University of Singapore（新加坡国立大学）
In this paper, a similarity-driven cluster merging method is proposed for unsuper-vised fuzzy clustering. The cluster merging method is used to resolve the problem of cluster validation. Starting with an overspecified number of clusters in the data, pairs of similar clusters are merged based on the proposed similarity-driven cluster merging criterion. The similarity between clusters is calculated by a fuzzy cluster similarity matrix, while an adaptive threshold is used for merging. In addition, a modified generalized ob- jective function is used for prototype-based fuzzy clustering. The functi…
Shandong Normal University（山东师范大学）；Shandong University（山东大学）；University of Electronic Science and Technology of China（电子科技大学）；National University of Singapore（新加坡国立大学）
In this paper, we investigate the research problem of unsupervised multi-view feature selection. Conventional solutions first simply combine multiple pre-constructed view-specific similarity structures into a collaborative similarity structure, and then perform the subsequent feature selection. These two processes are separate and independent. The collaborative similarity structure remains fixed during feature selection. Further, the simple undirected view combination may adversely reduce the reliability of the ultimate similarity structure for feature selection, as the view-specific similarit…
Unsupervised domain adaptation (UDA) transfers knowledge from a label-rich source domain to a different but related fully-unlabeled target domain. To address the problem of domain shift, more and more UDA methods adopt pseudo labels of the target samples to improve the generalization ability on the target domain. However, inaccurate pseudo labels of the target samples may yield suboptimal performance with error accumulation during the optimization process. Moreover, once the pseudo labels are generated, how to remedy the generated pseudo labels is far from explored. In this paper, we propose a…
Experimental evidence indicates that simple models outperform complex deep networks on many unsupervised similarity tasks. We provide a simple yet rigorous explanation for this behaviour by introducing the concept of an optimal representation space, in which semantically close symbols are mapped to representations that are close under a similarity measure induced by the model's objective function. In addition, we present a straightforward procedure that, without any retraining or architectural modifications, allows deep recurrent models to perform equally well (and sometimes better) when compa…
School of Computing and Augmented Intelligence（计算与增强智能学院）；Arizona State University（亚利桑那州立大学）；Baidu Research（百度研究院）
Similarity-based clustering methods separate data into clusters according to the pairwise similarity between the data, and the pairwise similarity is crucial for their performance. In this paper, we propose {\em Clustering by Discriminative Similarity (CDS)}, a novel method which learns discriminative similarity for data clustering. CDS learns an unsupervised similarity-based classifier from each data partition, and searches for the optimal partition of the data by minimizing the generalization error of the learnt classifiers associated with the data partitions. By generalization analysis via …
Graph-based extractive document summarization relies on the quality of the sentence similarity graph. Bag-of-words or tf-idf based sentence similarity uses exact word matching, but fails to measure the semantic similarity between individual words or to consider the semantic structure of sentences. In order to improve the similarity measure between sentences, we employ off-the-shelf deep embedding features and tf-idf features, and introduce a new text similarity metric. An improved sentence similarity graph is built and used in a submodular objective function for extractive summarization, which…
Autodesk AI Lab（Autodesk人工智能实验室）；UCL（伦敦大学学院）；Autodesk Research（Autodesk研究院）
Boundary Representations (B-Reps) are the industry standard in 3D Computer Aided Design/Manufacturing (CAD/CAM) and industrial design due to their fidelity in representing stylistic details. However, they have been ignored in the 3D style research. Existing 3D style metrics typically operate on meshes or pointclouds, and fail to account for end-user subjectivity by adopting fixed definitions of style, either through crowd-sourcing for style labels or hand-crafted features. We propose UVStyle-Net, a style similarity measure for B-Reps that leverages the style signals in the second order statist…
University of Tehran（德黑兰大学）；Amirkabir University of Technology（阿米尔卡比尔理工大学）
Feature selection methods have an important role on the readability of data and the reduction of complexity of learning algorithms. In recent years, a variety of efforts are investigated on feature selection problems based on unsupervised viewpoint due to the laborious labeling task on large datasets. In this paper, we propose a novel approach on unsupervised feature selection initiated from the subspace clustering to preserve the similarities by representation learning of low dimensional subspaces among the samples. A self-expressive model is employed to implicitly learn the cluster similarit…
Structure-aware Hybrid-order Similarity Learning for Multi-view Unsupervised Feature Selection
Joint Laboratory of Data Science and Business Intelligence, School of Statistics and Data Science, Southwestern University of Finance and Economics（数据科学与商务智能联合实验室，统计与数据科学学院，西南财经大学）；Big Data Laboratory on Financial Security and Behavior, SWUFE (Laboratory of Philosophy and Social Sciences, Ministry of Education)（金融安全与行为大数据实验室，SWUFE（哲学社会科学实验室，教育部））；Department of Electrical Engineering and Computer Science, University of Kansas（电气与计算机工程系，堪萨斯大学）；School of Computing and Artificial Intelligence, Southwest Jiaotong University（计算与人工智能学院，西南交通大学）
Classifying samples as in-distribution or out-of-distribution (OOD) is a challenging problem of anomaly detection and a strong test of the generalisation power for models of the in-distribution. In this paper, we present a simple and generic framework, {\it SemSAD}, that makes use of a semantic similarity score to carry out anomaly detection. The idea is to first find for any test example the semantically closest examples in the training set, where the semantic relation between examples is quantified by the cosine similarity between feature vectors that leave semantics unchanged under transfor…
Air Force Engineering University（空军工程大学）；Northwestern Polytechnical University（西北工业大学）
In the field of data mining, how to deal with high-dimensional data is an inevitable problem. Unsupervised feature selection has attracted more and more attention because it does not rely on labels. The performance of spectral-based unsupervised methods depends on the quality of constructed similarity matrix, which is used to depict the intrinsic structure of data. However, real-world data contain a large number of noise samples and features, making the similarity matrix constructed by original data cannot be completely reliable. Worse still, the size of similarity matrix expands rapidly as th…
We introduce a new paradigm to investigate unsupervised learning, reducing unsupervised learning to supervised learning. Specifically, we mitigate the subjectivity in unsupervised decision-making by leveraging knowledge acquired from prior, possibly heterogeneous, supervised learning tasks. We demonstrate the versatility of our framework via comprehensive expositions and detailed experiments on several unsupervised problems such as (a) clustering, (b) outlier detection, and (c) similarity prediction under a common umbrella of meta-unsupervised-learning. We also provide rigorous PAC-agnostic bo…
There is increasing realization in neuroscience that information is represented in the brain, e.g., neocortex, hippocampus, in the form sparse distributed codes (SDCs), a kind of cell assembly. Two essential questions are: a) how are such codes formed on the basis of single trials, and how is similarity preserved during learning, i.e., how do more similar inputs get mapped to more similar SDCs. I describe a novel Modular Sparse Distributed Code (MSDC) that provides simple, neurally plausible answers to both questions. An MSDC coding field (CF) consists of Q WTA competitive modules (CMs), each …
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
In this work, we tackle the problem of transform-invariant unsupervised learning in the space of Covariance matrices and applications thereof. We begin by introducing the Spectral Polytope Covariance Matrix (SPCM) Similarity function; a similarity function for Covariance matrices, invariant to any type of transformation. We then derive the SPCM-CRP mixture model, a transform-invariant non-parametric clustering approach for Covariance matrices that leverages the proposed similarity function, spectral embedding and the distance-dependent Chinese Restaurant Process (dd-CRP) (Blei and Frazier, 201…
In this paper we consider the binary similarity problem that consists in determining if two binary functions are similar only considering their compiled form. This problem is know to be crucial in several application scenarios, such as copyright disputes, malware analysis, vulnerability detection, etc. The current state-of-the-art solutions in this field work by creating an embedding model that maps binary functions into vectors in $\mathbb{R}^{n}$. Such embedding model captures syntactic and semantic similarity between binaries, i.e., similar binary functions are mapped to points that are clo…
Southwestern University of Finance and Economics（西南财经大学）；JD Intelligent Cities Research（京东智能城市研究院）；JD Intelligent Cities Business Unit（京东智能城市事业部）；University of Central Florida（中佛罗里达大学）；Southwest Jiaotong University（西南交通大学）
Multi-view unsupervised feature selection (MUFS) has been demonstrated as an effective technique to reduce the dimensionality of multi-view unlabeled data. The existing methods assume that all of views are complete. However, multi-view data are usually incomplete, i.e., a part of instances are presented on some views but not all views. Besides, learning the complete similarity graph, as an important promising technology in existing MUFS methods, cannot achieve due to the missing views. In this paper, we propose a complementary and consensus learning-based incomplete multi-view unsupervised fea…
Unsupervised approaches to extractive summarization usually rely on a notion of sentence importance defined by the semantic similarity between a sentence and the document. We propose new metrics of relevance and redundancy using pointwise mutual information (PMI) between sentences, which can be easily computed by a pre-trained language model. Intuitively, a relevant sentence allows readers to infer the document content (high PMI with the document), and a redundant sentence can be inferred from the summary (high PMI with the summary). We then develop a greedy sentence selection algorithm to max…
UC San Diego（加利福尼亚大学圣地亚哥分校）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；Princeton University（普林斯顿大学）；IBM Research（IBM研究院）
The fruit fly Drosophila's olfactory circuit has inspired a new locality sensitive hashing (LSH) algorithm, FlyHash. In contrast with classical LSH algorithms that produce low dimensional hash codes, FlyHash produces sparse high-dimensional hash codes and has also been shown to have superior empirical performance compared to classical LSH algorithms in similarity search. However, FlyHash uses random projections and cannot learn from data. Building on inspiration from FlyHash and the ubiquity of sparse expansive representations in neurobiology, our work proposes a novel hashing algorithm BioHas…
This paper describes an unsupervised machine learning methodology capable of target tracking and background suppression via a novel dual-model approach. ``Jekyll`` produces a video bit-mask describing an estimate of the locations of moving objects, and ``Hyde`` outputs a pseudo-background frame to subtract from the original input image sequence. These models were trained with a custom-modified version of Cross Entropy Loss. Simulated data were used to compare the performance of Jekyll and Hyde against a more traditional supervised Machine Learning approach. The results from these comparisons s…
Joint Multi-view Unsupervised Feature Selection and Graph Learning
South China Agricultural University（华南农业大学）；Key Laboratory of Smart Agricultural Technology in Tropical South China, Ministry of Agriculture and Rural Affairs（农业农村部华南热带智慧农业技术重点实验室）；Sun Yat-sen University（中山大学）；Guangdong Key Laboratory of Information Security Technology（广东省信息安全技术重点实验室）；South China Normal University（华南师范大学）
Faculty of Computer Science and Management（计算机科学与管理学院）
The role of spatial data in tackling city-related tasks has been growing in recent years. To use them in machine learning models, it is often necessary to transform them into a vector representation, which has led to the development in the field of spatial data representation learning. There is also a growing variety of spatial data types for which representation learning methods are proposed. Public transport timetables have so far not been used in the task of learning representations of regions in a city. In this work, a method is developed to embed public transport availability information …
The main source of various religious teachings is their sacred texts which vary from religion to religion based on different factors like the geographical location or time of the birth of a particular religion. Despite these differences, there could be similarities between the sacred texts based on what lessons it teaches to its followers. This paper attempts to find the similarity using text mining techniques. The corpus consisting of Asian (Tao Te Ching, Buddhism, Yogasutra, Upanishad) and non-Asian (four Bible texts) is used to explore findings of similarity measures like Euclidean, Manhatt…
UC Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）；Chinese University of Hong Kong（香港中文大学）；Amazon Rekognition（亚马逊 Rekognition）
Neural net classifiers trained on data with annotated class labels can also capture apparent visual similarity among categories without being directed to do so. We study whether this observation can be extended beyond the conventional domain of supervised learning: Can we learn a good feature representation that captures apparent similarity among instances, instead of classes, by merely asking the feature to be discriminative of individual instances? We formulate this intuition as a non-parametric classification problem at the instance-level, and use noise-contrastive estimation to tackle the …
University of Bari（巴里大学）；Dublin City University（都柏林城市大学）
This paper describes the system proposed for the SemEval-2020 Task 1: Unsupervised Lexical Semantic Change Detection. We focused our approach on the detection problem. Given the semantics of words captured by temporal word embeddings in different time periods, we investigate the use of unsupervised methods to detect when the target word has gained or loosed senses. To this end, we defined a new algorithm based on Gaussian Mixture Models to cluster the target similarities computed over the two periods. We compared the proposed approach with a number of similarity-based thresholds. We found that…
In NLP, a large volume of tasks involve pairwise comparison between two sequences (e.g. sentence similarity and paraphrase identification). Predominantly, two formulations are used for sentence-pair tasks: bi-encoders and cross-encoders. Bi-encoders produce fixed-dimensional sentence representations and are computationally efficient, however, they usually underperform cross-encoders. Cross-encoders can leverage their attention heads to exploit inter-sentence interactions for better performance but they require task fine-tuning and are computationally more expensive. In this paper, we present a…
Joint Laboratory of Data Science and Business Intelligence, School of Statistics, Southwestern University of Finance and Economics（西南财经大学统计学院数据科学与商务智能联合实验室）；University of Kansas（堪萨斯大学）；School of Computing and Artificial Intelligence, Southwest Jiaotong University（西南交通大学计算机与人工智能学院）
Zhejiang University（浙江大学）；AI Lab, School of Engineering（工程学院AI实验室）；Westlake University（西湖大学）
Unsupervised attributed graph representation learning is challenging since both structural and feature information are required to be represented in the latent space. Existing methods concentrate on learning latent representation via reconstruction tasks, but cannot directly optimize representation and are prone to oversmoothing, thus limiting the applications on downstream tasks. To alleviate these issues, we propose a novel graph embedding framework named Deep Manifold Attributed Graph Embedding (DMAGE). A node-to-node geodesic similarity is proposed to compute the inter-node similarity betw…
Unsupervised Seismic Footprint Removal With Physical Prior Augmented Deep Autoencoder
University of Electronic Science and Technology of China（电子科技大学）；Sichuan University（四川大学）；Sinopec Geophysical Research Institute（中国石油化工股份有限公司石油物探技术研究院）
KAIST（韩国科学技术院）；Kyungpook National University（庆北国立大学）
Self-supervised contrastive learning offers a means of learning informative features from a pool of unlabeled data. In this paper, we delve into another useful approach -- providing a way of selecting a core-set that is entirely unlabeled. In this regard, contrastive learning, one of a large number of self-supervised methods, was recently proposed and has consistently delivered the highest performance. This prompted us to choose two leading methods for contrastive learning: the simple framework for contrastive learning of visual representations (SimCLR) and the momentum contrastive (MoCo) lear…
Joint Laboratory of Data Science and Business Intelligence, School of Statistics and Data Science, Southwestern University of Finance and Economics（数据科学与商业智能联合实验室，统计与数据科学学院，西南财经大学）；Department of Electrical Engineering and Computer Science, University of Kansas（电气与计算机工程系，堪萨斯大学）；School of Computing and Artificial Intelligence, Southwest Jiaotong University（计算与人工智能学院，西南交通大学）
Unsupervised learning of time series data, also known as temporal clustering, is a challenging problem in machine learning. Here we propose a novel algorithm, Deep Temporal Clustering (DTC), to naturally integrate dimensionality reduction and temporal clustering into a single end-to-end learning framework, fully unsupervised. The algorithm utilizes an autoencoder for temporal dimensionality reduction and a novel temporal clustering layer for cluster assignment. Then it jointly optimizes the clustering objective and the dimensionality reduction objec tive. Based on requirement and application, …
Singapore University of Technology and Design（新加坡科技设计大学）；DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；ZJU-UIUC Institue（浙江大学-伊利诺伊大学厄巴纳-香槟分校联合学院）
BERT is inefficient for sentence-pair tasks such as clustering or semantic search as it needs to evaluate combinatorially many sentence pairs which is very time-consuming. Sentence BERT (SBERT) attempted to solve this challenge by learning semantically meaningful representations of single sentences, such that similarity comparison can be easily accessed. However, SBERT is trained on corpus with high-quality labeled sentence pairs, which limits its application to tasks where labeled data is extremely scarce. In this paper, we propose a lightweight extension on top of BERT and a novel self-super…
National University of Singapore（新加坡国立大学）；Tsinghua University（清华大学）；Shenzhen Kivita Innovative Drug Discovery Institute（深圳科维塔创新药物发现研究院）；Zhejiang University（浙江大学）；China Pharmaceutical University（中国药科大学）；Shanghai Kudai Network Technology Co., Ltd.（上海酷袋网络科技有限公司）
Unsupervised clustering has broad applications in data stratification, pattern investigation and new discovery beyond existing knowledge. In particular, clustering of bioactive molecules facilitates chemical space mapping, structure-activity studies, and drug discovery. These tasks, conventionally conducted by similarity-based methods, are complicated by data complexity and diversity. We ex-plored the superior learning capability of deep autoencoders for unsupervised clustering of 1.39 mil-lion bioactive molecules into band-clusters in a 3-dimensional latent chemical space. These band-clusters…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Georgia Institute of Technology（佐治亚理工学院）；Google Research（谷歌研究院）；U.S. Army Research Laboratory（美国陆军研究实验室）
Unsupervised text embedding has shown great power in a wide range of NLP tasks. While text embeddings are typically learned in the Euclidean space, directional similarity is often more effective in tasks such as word similarity and document clustering, which creates a gap between the training stage and usage stage of text embedding. To close this gap, we propose a spherical generative model based on which unsupervised word and paragraph embeddings are jointly learned. To learn text embeddings in the spherical space, we develop an efficient optimization algorithm with convergence guarantee base…
