t-SNE and hierarchical clustering are popular methods of exploratory data analysis, particularly in biology. Building on recent advances in speeding up t-SNE and obtaining finer-grained structure, we combine the two to create tree-SNE, a hierarchical clustering and visualization algorithm based on stacked one-dimensional t-SNE embeddings. We also introduce alpha-clustering, which recommends the optimal cluster assignment, without foreknowledge of the number of clusters, based off of the cluster stability across multiple scales. We demonstrate the effectiveness of tree-SNE and alpha-clustering on images of handwritten digits, mass cytometry (CyTOF) data from blood cells, and single-cell RNA-sequencing (scRNA-seq) data from retinal cells. Furthermore, to demonstrate the validity of the visualization, we use alpha-clustering to obtain unsupervised clustering results competitive with the state of the art on several image data sets. Software is available at https://github.com/isaacrob/treesne.
We present GraphTSNE, a novel visualization technique for graph-structured data based on t-SNE. The growing interest in graph-structured data increases the importance of gaining human insight into such datasets by means of visualization. Among the most popular visualization techniques, classical t-SNE is not suitable on such datasets because it has no mechanism to make use of information from the graph structure. On the other hand, visualization techniques which operate on graphs, such as Laplacian Eigenmaps and tsNET, have no mechanism to make use of information from node features. Our propos…
Hyperbolic space can naturally embed hierarchies that often exist in real-world data and semantics. While high-dimensional hyperbolic embeddings lead to better representations, most hyperbolic models utilize low-dimensional embeddings, due to non-trivial optimization and visualization of high-dimensional hyperbolic data. We propose CO-SNE, which extends the Euclidean space visualization tool, t-SNE, to hyperbolic space. Like t-SNE, it converts distances between data points to joint probabilities and tries to minimize the Kullback-Leibler divergence between the joint probabilities of high-dimen…
Modern datasets and models are notoriously difficult to explore and analyze due to their inherent high dimensionality and massive numbers of samples. Existing visualization methods which employ dimensionality reduction to two or three dimensions are often inefficient and/or ineffective for these datasets. This paper introduces t-SNE-CUDA, a GPU-accelerated implementation of t-distributed Symmetric Neighbor Embedding (t-SNE) for visualizing datasets and models. t-SNE-CUDA significantly outperforms current implementations with 50-700x speedups on the CIFAR-10 and MNIST datasets. These speedups e…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
A first line of attack in exploratory data analysis is data visualization, i.e., generating a 2-dimensional representation of data that makes clusters of similar points visually identifiable. Standard Johnson-Lindenstrauss dimensionality reduction does not produce data visualizations. The t-SNE heuristic of van der Maaten and Hinton, which is based on non-convex optimization, has become the de facto standard for visualization in a wide range of applications. This work gives a formal framework for the problem of data visualization - finding a 2-dimensional embedding of clusterable data that cor…
t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering and visualization method proposed by van der Maaten & Hinton in 2008, has rapidly become a standard tool in a number of natural sciences. Despite its overwhelming success, there is a distinct lack of mathematical foundations and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters; more precisely, we prove that t-SNE in the `early exaggeration' phase, an optimization technique proposed by van der Maaten & Hinton (2008) and van…
Duke University（杜克大学）；University of Rochester（罗切斯特大学）
Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have demonstrated impressive visualization performance on many real world datasets. One tension that has always faced these methods is the trade-off between preservation of global structure and preservation of local structure: these methods can either handle one or the other, but not both. In this work, our main goal is to understand what aspects of DR methods are important for preserving both local and global structure: it is difficult to design a better method without a true understanding of the choices we make in our algori…
Georgia Institute of Technology（佐治亚理工学院）；Northwestern University（西北大学）；Ansys Inc.（Ansys公司）；Temple University（天普大学）
The optimization of electric machines at multiple operating points is crucial for applications that require frequent changes on speeds and loads, such as the electric vehicles, to strive for the machine optimal performance across the entire driving cycle. However, the number of objectives that would need to be optimized would significantly increase with the number of operating points considered in the optimization, thus posting a potential problem in regards to the visualization techniques currently in use, such as in the scatter plots of Pareto fronts, the parallel coordinates, and in the pri…
We propose an algorithm, HPREF (Hierarchical Partitioning by Repeated Features), that produces a hierarchical partition of a set of clusterings of a fixed dataset, such as sets of clusterings produced by running a clustering algorithm with a range of parameters. This gives geometric structure to such sets of clustering, and can be used to visualize the set of results one obtains by running a clustering algorithm with a range of parameters.
Kyoto University（京都大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；NAVER Corporation（NAVER公司）
Multimodal relational data analysis has become of increasing importance in recent years, for exploring across different domains of data, such as images and their text tags obtained from social networking services (e.g., Flickr). A variety of data analysis methods have been developed for visualization; to give an example, t-Stochastic Neighbor Embedding (t-SNE) computes low-dimensional feature vectors so that their similarities keep those of the observed data vectors. However, t-SNE is designed only for a single domain of data but not for multimodal data; this paper aims at visualizing multimod…
University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）；University of Colorado Boulder（科罗拉多大学博尔德分校）
A fundamental task in machine learning involves visualizing high-dimensional data sets that arise in high-impact application domains. When considering the context of large imbalanced data, this problem becomes much more challenging. In this paper, the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm is used to reduce the dimensions of an earthquake engineering related data set for visualization purposes. Since imbalanced data sets greatly affect the accuracy of classifiers, we employ Synthetic Minority Oversampling Technique (SMOTE) to tackle the imbalanced nature of such data set…
On the continuum limit of t-SNE for data visualization
School of Mathematics, University of Minnesota（明尼苏达大学数学学院）；Department of Mathematics, University of Utah（犹他大学数学系）；Department of Mathematics, North Carolina State University（北卡罗来纳州立大学数学系）
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Modern methods for data visualization via dimensionality reduction, such as t-SNE, usually have performance issues that prohibit their application to large amounts of high-dimensional data. In this work, we propose NCVis -- a high-performance dimensionality reduction method built on a sound statistical basis of noise contrastive estimation. We show that NCVis outperforms state-of-the-art techniques in terms of speed while preserving the representation quality of other methods. In particular, the proposed approach successfully proceeds a large dataset of more than 1 million news headlines in se…
University of Melbourne（墨尔本大学）；Walter and Eliza Hall Institute for Medical Research（沃尔特与伊丽莎·霍尔医学研究所）
Non-parametric dimensionality reduction techniques, such as t-SNE and UMAP, are proficient in providing visualizations for datasets of fixed sizes. However, they cannot incrementally map and insert new data points into an already provided data visualization. We present Self-Organizing Nebulous Growths (SONG), a parametric nonlinear dimensionality reduction technique that supports incremental data visualization, i.e., incremental addition of new data while preserving the structure of the existing visualization. In addition, SONG is capable of handling new data increments, no matter whether they…
Mercedes-Benz Research and Development India Pvt. Ltd.（梅赛德斯-奔驰印度私人有限公司研发中心）；Centre of Studies in Resources Engineering, Indian Institute of Technology Bombay（印度理工学院孟买分校资源工程研究中心）
Hyperspectral Imagery (and Remote Sensing in general) captured from UAVs or satellites are highly voluminous in nature due to the large spatial extent and wavelengths captured by them. Since analyzing these images requires a huge amount of computational time and power, various dimensionality reduction techniques have been used for feature reduction. Some popular techniques among these falter when applied to Hyperspectral Imagery due to the famed curse of dimensionality. In this paper, we propose a novel approach, LEt-SNE, which combines graph based algorithms like t-SNE and Laplacian Eigenmaps…
Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）；Duke University（杜克大学）
We introduce a nonlinear method for directly embedding large, sparse, stochastic graphs into low-dimensional spaces, without requiring vertex features to reside in, or be transformed into, a metric space. Graph data and models are prevalent in real-world applications. Direct graph embedding is fundamental to many graph analysis tasks, in addition to graph visualization. We name the novel approach SG-t-SNE, as it is inspired by and builds upon the core principle of t-SNE, a widely used method for nonlinear dimensionality reduction and data visualization. We also introduce t-SNE-$Π$, a high-perf…
We propose a unified view on two widely used data visualization techniques: Self-Organizing Maps (SOMs) and Stochastic Neighbor Embedding (SNE). We show that they can both be derived from a common mathematical framework. Leveraging this formulation, we propose to compare SOM and SNE quantitatively on two datasets, and discuss possible avenues for future work to take advantage of both approaches.
Centre d’Estudis Avançats de Blanes (CEAB-CSIC)（布兰斯高等研究中心 (CEAB-CSIC)）；Centre de Recerca Ecològica i Aplicacions Forestals (CREAF)（生态研究与应用林学中心 (CREAF)）；Institució Catalana de Recerca i Estudis Avançats, ICREA（加泰罗尼亚研究与高级研究机构 (ICREA)）
Dimensional data reduction methods are fundamental to explore and visualize large data sets. Basic requirements for unsupervised data exploration are simplicity, flexibility and scalability. However, current methods show complex parameterizations and strong computational limitations when exploring large data structures across scales. Here, we focus on the t-SNE algorithm and show that a simplified parameter setup with a single control parameter, namely the perplexity, can effectively balance local and global data structure visualization. We also designed a chunk\&mix protocol to efficiently pa…
University of Pennsylvania（宾夕法尼亚大学）；Stanford University（斯坦福大学）
This paper investigates the theoretical foundations of the t-distributed stochastic neighbor embedding (t-SNE) algorithm, a popular nonlinear dimension reduction and data visualization method. A novel theoretical framework for the analysis of t-SNE based on the gradient descent approach is presented. For the early exaggeration stage of t-SNE, we show its asymptotic equivalence to power iterations based on the underlying graph Laplacian, characterize its limiting behavior, and uncover its deep connection to Laplacian spectral clustering, and fundamental principles including early stopping as im…
Generating interpretable visualizations from complex data is a common problem in many applications. Two key ingredients for tackling this issue are clustering and representation learning. However, current methods do not yet successfully combine the strengths of these two approaches. Existing representation learning models which rely on latent topological structure such as self-organising maps, exhibit markedly lower clustering performance compared to recent deep clustering methods. To close this performance gap, we (a) present a novel way to fit self-organizing maps with probabilistic cluster …
Cluster and then Embed: A Modular Approach for Visualization
Department of Mathematics, University of California, San Diego（数学系，加州大学圣地亚哥分校）；Halıcıoğlu Data Science Institute, University of California, San Diego（Halıcıoğlu数据科学研究所，加州大学圣地亚哥分校）
We use graphical methods to probe neural nets that classify images. Plots of t-SNE outputs at successive layers in a network reveal increasingly organized arrangement of the data points. They can also reveal how a network can diminish or even forget about within-class structure as the data proceeds through layers. We use class-specific analogues of principal components to visualize how succeeding layers separate the classes. These allow us to sort images from a given class from most typical to least typical (in the data) and they also serve as very useful projection coordinates for data visual…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces to extract hidden and useful information or facilitate visual understanding and interpretation of the data. However, few of them take into consideration the potential cluster information contained implicitly in the high-dimensional data. In this paper, we propose LaptSNE, a new graph-layout nonlinear dimensionality reduction method based on t-SNE, one of the best techniques for visualizing high-dimensional data as 2D scatter plots. Specifically, LaptSNE leverages the eigenvalue information o…
Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
Convex clustering is a promising new approach to the classical problem of clustering, combining strong performance in empirical studies with rigorous theoretical foundations. Despite these advantages, convex clustering has not been widely adopted, due to its computationally intensive nature and its lack of compelling visualizations. To address these impediments, we introduce Algorithmic Regularization, an innovative technique for obtaining high-quality estimates of regularization paths using an iterative one-step approximation scheme. We justify our approach with a novel theoretical result, gu…
UC Berkeley（加州大学伯克利分校）；Massachusetts Institute of Technology（麻省理工学院）
t-SNE is a popular tool for embedding multi-dimensional datasets into two or three dimensions. However, it has a large computational cost, especially when the input data has many dimensions. Many use t-SNE to embed the output of a neural network, which is generally of much lower dimension than the original data. This limits the use of t-SNE in unsupervised scenarios. We propose using \textit{random} projections to embed high dimensional datasets into relatively few dimensions, and then using t-SNE to obtain a two dimensional embedding. We show that random projections preserve the desirable clu…
Norwegian University of Science and Technology（挪威科技大学）；Aalto University（阿尔托大学）；Finnish Geospatial Research Institute（芬兰地理空间研究所）；University of Oslo（奥斯陆大学）；Wellcome Sanger Institute（威康桑格研究所）；University of Helsinki（赫尔辛基大学）
Neighbor Embedding (NE) aims to preserve pairwise similarities between data items and has been shown to yield an effective principle for data visualization. However, even the best existing NE methods such as Stochastic Neighbor Embedding (SNE) may leave large-scale patterns hidden, for example clusters, despite strong signals being present in the data. To address this, we propose a new cluster visualization method based on the Neighbor Embedding principle. We first present a family of Neighbor Embedding methods that generalizes SNE by using non-normalized Kullback-Leibler divergence with a sca…
The paper presents an O(N log N)-implementation of t-SNE -- an embedding technique that is commonly used for the visualization of high-dimensional data in scatter plots and that normally runs in O(N^2). The new implementation uses vantage-point trees to compute sparse pairwise similarities between the input data objects, and it uses a variant of the Barnes-Hut algorithm - an algorithm used by astronomers to perform N-body simulations - to approximate the forces between the corresponding points in the embedding. Our experiments show that the new algorithm, called Barnes-Hut-SNE, leads to substa…
EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings
Department of Computer Science, Columbia University（哥伦比亚大学计算机科学系）；Department of Statistics & Data Science, Wu Tsai Institute, Yale University（耶鲁大学统计与数据科学系、吴氏研究所）；Department of Machine Learning, University of Fribourg（弗里堡大学机器学习系）；Department of Mathematics, Columbia University（哥伦比亚大学数学系）；Irving Institute for Cancer Dynamics, Columbia University（哥伦比亚大学癌症动力学研究所）
Norwegian University of Science and Technology（挪威科技大学）；Aalto University（阿尔托大学）；Finnish Geospatial Research Institute（芬兰地理空间研究所）；University of Oslo（奥斯陆大学）；University of Helsinki（赫尔辛基大学）；Wellcome Sanger Institute（桑格研究所）
Cluster visualization is an essential task for nonlinear dimensionality reduction as a data analysis tool. It is often believed that Student t-Distributed Stochastic Neighbor Embedding (t-SNE) can show clusters for well clusterable data, with a smaller Kullback-Leibler divergence corresponding to a better quality. There was even theoretical proof for the guarantee of this property. However, we point out that this is not necessarily the case -- t-SNE may leave clustering patterns hidden despite strong signals present in the data. Extensive empirical evidence is provided to support our claim. Fi…
Vironova AB（Vironova公司）；KTH Royal Institute of Technology（瑞典皇家理工学院）
In this paper we propose an adaptive approach for clustering and visualization of data by an orthogonalization process. Starting with the data points being represented by a Markov process using the diffusion map framework, the method adaptively increase the orthogonality of the clusters by applying a feedback mechanism inspired by the Gromov-Wasserstein distance. This mechanism iteratively increases the spectral gap and refines the orthogonality of the data to achieve a clustering with high specificity. By using the diffusion map framework and representing the relation between data points usin…
In recent years convolutional neural networks (CNN) have shown striking progress in various tasks. However, despite the high performance, the training and prediction process remains to be a black box, leaving it a mystery to extract what neurons learn in CNN. In this paper, we address the problem of interpreting a CNN from the aspects of the input image's focus and preference, and the neurons' domination, activation and contribution to a concrete final prediction. Specifically, we use two techniques - visualization and clustering - to tackle the problems above. Visualization means the method o…
NEC Labs America（美国NEC实验室）；National Research Council Canada（加拿大国家研究委员会）；Duke University（杜克大学）
Parametric embedding methods such as parametric t-SNE (pt-SNE) have been widely adopted for data visualization and out-of-sample data embedding without further computationally expensive optimization or approximation. However, the performance of pt-SNE is highly sensitive to the hyper-parameter batch size due to conflicting optimization goals, and often produces dramatically different embeddings with different choices of user-defined perplexities. To effectively solve these issues, we present parametric t-distributed stochastic exemplar-centered embedding methods. Our strategy learns embedding …
Purdue School of Engineering and Technology, IUPUI（印第安纳大学与普渡大学印第安纳波利斯联合分校普渡工程与技术学院）
Document clustering is a text mining technique used to provide better document search and browsing in digital libraries or online corpora. A lot of research has been done on biomedical document clustering that is based on using existing ontology. But, associations and co-occurrences of the medical concepts are not well represented by using ontology. In this research, a vector representation of concepts of diseases and similarity measurement between concepts are proposed. They identify the closest concepts of diseases in the context of a corpus. Each document is represented by using the vector …
Beijing University of Posts and Telecommunications（北京邮电大学）；Tsinghua University（清华大学）
T-SNE is a well-known approach to embedding high-dimensional data and has been widely used in data visualization. The basic assumption of t-SNE is that the data are non-constrained in the Euclidean space and the local proximity can be modelled by Gaussian distributions. This assumption does not hold for a wide range of data types in practical applications, for instance spherical data for which the local proximity is better modelled by the von Mises-Fisher (vMF) distribution instead of the Gaussian. This paper presents a vMF-SNE embedding algorithm to embed spherical data. An iterative process …
EPFL, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Visualizing high-dimensional data has been a focus in data analysis communities for decades, which has led to the design of many algorithms, some of which are now considered references (such as t-SNE for example). In our era of overwhelming data volumes, the scalability of such methods have become more and more important. In this work, we present a method which allows to apply any visualization or embedding algorithm on very large datasets by considering only a fraction of the data as input and then extending the information to all data points using a graph encoding its global similarity. We s…
Westlake University（西湖大学）；Westlake Institute for Advanced Study（西湖高等研究院）
High dimensional data analysis for exploration and discovery includes three fundamental tasks: dimensionality reduction, clustering, and visualization. When the three associated tasks are done separately, as is often the case thus far, inconsistencies can occur among the tasks in terms of data geometry and others. This can lead to confusing or misleading data interpretation. In this paper, we propose a novel neural network-based method, called Consistent Representation Learning (CRL), to accomplish the three associated tasks end-to-end and improve the consistencies. The CRL network consists of…
We present a new method GTSNE to visualize high-dimensional data points in the two dimensional map. The technique is a variation of t-SNE that produces better visualizations by capturing both the local neighborhood structure and the macro structure in the data. This is particularly important for high-dimensional data that lie on continuous low-dimensional manifolds. We illustrate the performance of GTSNE on a wide variety of datasets and compare it the state of art methods, including t-SNE and UMAP. The visualizations produced by GTSNE are better than those produced by the other techniques on …
Institute for Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM)（信息与通信技术、电子与应用数学研究所（ICTEAM））；Center for Operations Research and Econometrics (CORE)（运筹学与计量经济学中心（CORE））；Université catholique de Louvain（天主教鲁汶大学）
A main challenge in mining network-based data is finding effective ways to represent or encode graph structures so that it can be efficiently exploited by machine learning algorithms. Several methods have focused in network representation at node/edge or substructure level. However, many real life challenges such as time-varying, multilayer, chemical compounds and brain networks involve analysis of a family of graphs instead of single one opening additional challenges in graph comparison and representation. Traditional approaches for learning representations relies on hand-crafting specialized…
Aalto University（阿尔托大学）；Australian National University（澳大利亚国立大学）；University of Oslo（奥斯陆大学）；University of Helsinki（赫尔辛基大学）；Norwegian University of Science and Technology（挪威科技大学）
Stochastic Neighbor Embedding (SNE) methods minimize the divergence between the similarity matrix of a high-dimensional data set and its counterpart from a low-dimensional embedding, leading to widely applied tools for data visualization. Despite their popularity, the current SNE methods experience a crowding problem when the data include highly imbalanced similarities. This implies that the data points with higher total similarity tend to get crowded around the display center. To solve this problem, we introduce a fast normalization method and normalize the similarity matrix to be doubly stoc…
PayAnalytics；Technical University of Denmark（丹麦技术大学）
Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to…
We propose methods for the analysis of hierarchical clustering that fully use the multi-resolution structure provided by a dendrogram. Specifically, we propose a loss for choosing between clustering methods, a feature importance score and a graphical tool for visualizing the segmentation of features in a dendrogram. Current approaches to these tasks lead to loss of information since they require the user to generate a single partition of the instances by cutting the dendrogram at a specified level. Our proposed methods, instead, use the full structure of the dendrogram. The key insight behind …
Traditionally it had been a problem that researchers did not have access to enough spatial data to answer pressing research questions or build compelling visualizations. Today, however, the problem is often that we have too much data. Spatially redundant or approximately redundant points may refer to a single feature (plus noise) rather than many distinct spatial features. We use a machine learning approach with density-based clustering to compress such spatial data into a set of representative features.
We propose a visualization technique that utilizes neural network embeddings and a generative network to reconstruct original data. This method allows for independent manipulation of individual image embeddings through its non-parametric structure, providing more flexibility than traditional autoencoder approaches. We have evaluated the effectiveness of this technique in data visualization and compared it to t-SNE and VAE methods. Furthermore, we have demonstrated the scalability of our method through visualizations on the ImageNet dataset. Our technique has potential applications in human-in-…
Clustering and visualizing high-dimensional (HD) data are important tasks in a variety of fields. For example, in bioinformatics, they are crucial for analyses of single-cell data such as mass cytometry (CyTOF) data. Some of the most effective algorithms for clustering HD data are based on representing the data by nodes in a graph, with edges connecting neighbouring nodes according to some measure of similarity or distance. However, users of graph-based algorithms are typically faced with the critical but challenging task of choosing the value of an input parameter that sets the size of neighb…
Centre d’Estudis Avançats de Blanes (CEAB-CSIC)（布拉内斯高等研究中心（CEAB-CSIC））；Centre de Recerca Ecològica i Aplicacions Forestals (CREAF)（生态研究与林业应用中心（CREAF））
We introduce an improved unsupervised clustering protocol specially suited for large-scale structured data. The protocol follows three steps: a dimensionality reduction of the data, a density estimation over the low dimensional representation of the data, and a final segmentation of the density landscape. For the dimensionality reduction step we introduce a parallelized implementation of the well-known t-Stochastic Neighbouring Embedding (t-SNE) algorithm that significantly alleviates some inherent limitations, while improving its suitability for large datasets. We also introduce a new adaptiv…
