The Grassmannian Atlas: A General Framework for Exploring Linear Projections of High‐Dimensional Data

Liu, S.; Bremer, Peer Timo; Jayaraman, J. J.; Wang, B.; Summa, Brian; Pascucci, Valerio

doi:10.1111/cgf.12876

Cited by 13 publications

(7 citation statements)

References 25 publications

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“…Instead, we introduce a new approach to approximate a linear projection with a small number of axis‐aligned projections via generalization of sparse representations to the Riemannian space of linear projections [TG07], coupled with a greedy dimension selection technique. Furthermore, independent of how well the user can interpret a single linear projection, as corroborated by several recent related works [NM13, LWT*15, WM17, LT16, LBJ*16], an effective exploration of most high dimensional data requires a diverse set of views. To this end, we extend the decomposition approach to the case of multiple linear projections with additional constraint to reduce duplicated axis‐aligned presentation.…”

Section: Introductionmentioning

confidence: 87%

“…For a given measure, the global extrema provides the user with only one view of the data. In order to address this limitation, the Grassmannian Atlas framework [LBJ*16], adopts tools from topological data analysis, in lieu of the global optimization, and identifies multiple local extrema of the quality measure, thus identifying a complementary set of linear projections. Recently, subspace clustering/selection based methods [LWT*15, TMF*12, NM13, YRWG13, WM17], have enabled structure‐driven exploration, particularly while identifying important linear projections.…”

Section: Related Workmentioning

confidence: 99%

“…In addition, we introduce a new optimization approach to construct multiple linear projections that are both accurate and diverse. Unlike existing approaches, we do not focus exclusively on distinctive 2D patterns [LT16], which may lead to projections with poor embedding quality, nor do we require a dense sampling of all possible projections [LBJ*16], or expensive subspace clustering computation [NM13, LWT*15, WM17]. Instead, we introduce an iterative optimization to explore the Grassmannian (i.e., the space of all linear projections) with any convex embedding objective, e.g., principal component analysis (PCA), locality preserving projection (LPP).…”

Section: Introductionmentioning

confidence: 99%

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Exploring High‐Dimensional Structure via Axis‐Aligned Decomposition of Linear Projections

Thiagarajan

Liu

Ramamurthy

et al. 2018

Computer Graphics Forum

Self Cite

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Dim40 Dim44 Dim8 Dim23 (a) (b)Axis-Aligned Linear Dim51 Figure 1: The proposed method decomposes carefully selected linear projections into a set of axis-aligned projections, which are easier to understand yet still retain the structure insight captured by the original linear projections. In the seawater temperature forecasting example shown above, the first axis-aligned projection captures the dominant periodic structure, while the second one highlights an additional loop that is different from one observed in the first projection. The relationship between linear projections and their axis-aligned decompositions is encoded as a bipartite graph (shown in (a)). Dynamic projection transitions are used to illustrate the structural correspondence between the linear and axis-aligned projections (shown in (b)). AbstractTwo-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to bivariate relationships. Linear project can be considered as a compromise between complexity and interpretability, as they allow explicit axes labels, yet provide significantly more degrees of freedom compared to axis-aligned projections. Nevertheless, interpreting the axes directions, which are often linear combinations of many non-trivial components, remains difficult. To address this problem we introduce a structure aware decomposition of (multiple) linear projections into sparse sets of axis-aligned projections, which jointly capture all information of the original linear ones. In particular, we use tools from Dempster-Shafer theory to formally define how relevant a given axis-aligned project is to explain the neighborhood relations displayed in some linear projection. Furthermore, we introduce a new approach to discover a diverse set of high quality linear projections and show that in practice the information of k linear projections is often jointly encoded in ∼ k axis-aligned plots. We have integrated these ideas into an interactive visualization system that allows users to jointly browse both linear projections and their axis-aligned representatives. Using a number of case studies we show how the resulting plots lead to more intuitive visualizations and new insights.

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Section: Introductionmentioning

confidence: 87%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exploring High‐Dimensional Structure via Axis‐Aligned Decomposition of Linear Projections

Thiagarajan

Liu

Ramamurthy

et al. 2018

Computer Graphics Forum

Self Cite

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show abstract

“…The Star Coordinates [15] visualization technique, for example, provides interactive linear projections of high-dimensional data. Recently measure-driven approaches for exploration have gained interest, e.g., by Liu et al [18] as well as by Lehmann and Theisel [17]. Visualizing the projection matrix of linear dimensionality reduction techniques (instead of projections of the data) can be done with factor maps or Hinton diagrams [2,12].…”

Section: Related Workmentioning

confidence: 99%

Uncertainty-Aware Principal Component Analysis

Görtler

Spinner

Streeb

et al. 2020

IEEE Trans. Visual. Comput. Graphics

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0 0 0 (a) Regular PCA 0 0 0 (b) s = 0.5 0 0 0 (c) s = 0.8 0 0 0 (d) s = 1.0 Fig. 1. Data uncertainty can have a significant influence on the outcome of dimensionality reduction techniques. We propose a generalization of principal component analysis (PCA) that takes into account the uncertainty in the input. The top row shows the dataset with varying degrees of uncertainty and the corresponding principal components, whereas the bottom row shows the projection of the dataset, using our method, onto the first principal component. In Figures (a) and (b), with relatively low uncertainty, the blue and the orange distributions are comprised by the red and the green distributions. In Figures (c) and (d), with a larger amount of uncertainty, the projection changes drastically: now the orange and blue distributions encompass the red and the green distributions.Abstract-We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach.

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“…Many researchers write small scripts to generate data. In a few approaches the underlying model is well described and defined, e.g., by intersecting planes [72] or variations of the SwissRoll [97]. Other approaches [55,56] use rules and statistics to encode relationships between data instances (e.g., older person implies higher income) and allow one to insert anomalies for different applications.…”

Section: Multi-dimensional Data Visualizationmentioning

confidence: 99%

Generative Data Models for Validation and Evaluation of Visualization Techniques

Schulz

Nocaj

El‐Assady

et al. 2016

Proceedings of the Sixth Workshop on Beyond Time and Errors on Novel Evaluation Methods for Visualization

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We argue that there is a need for substantially more research on the use of generative data models in the validation and evaluation of visualization techniques. For example, user studies will require the display of representative and unconfounded visual stimuli, while algorithms will need functional coverage and assessable benchmarks. However, data is often collected in a semi-automatic fashion or entirely hand-picked, which obscures the view of generality, impairs availability, and potentially violates privacy. There are some sub-domains of visualization that use synthetic data in the sense of generative data models, whereas others work with real-world-based data sets and simulations. Depending on the visualization domain, many generative data models are "side projects" as part of an ad-hoc validation of a techniques paper and thus neither reusable nor general-purpose. We review existing work on popular data collections and generative data models in visualization to discuss the opportunities and consequences for technique validation, evaluation, and experiment design. We distill handling and future directions, and discuss how we can engineer generative data models and how visualization research could benefit from more and better use of generative data models.

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The Grassmannian Atlas: A General Framework for Exploring Linear Projections of High‐Dimensional Data

Cited by 13 publications

References 25 publications

Exploring High‐Dimensional Structure via Axis‐Aligned Decomposition of Linear Projections

Exploring High‐Dimensional Structure via Axis‐Aligned Decomposition of Linear Projections

Uncertainty-Aware Principal Component Analysis

Generative Data Models for Validation and Evaluation of Visualization Techniques

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