Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms generally lack clear relation to the initial data dimensions. Therefore, interpreting and reasoning about dimensionality reductions can be difficult. In this work, we introduce two interaction techniques, and , for reasoning dynamically about scatter plots of dimensionally reduced data. We also contribute two related visualization techniques, and to facilitate and enrich the effective use of the proposed interactions, which we integrate in a new tool called . To evaluate our techniques, we first analyze their time and accuracy performance across varying sample and dimension sizes. We then conduct a user study in which twelve data scientists use so as to assess the usefulness of the techniques in performing exploratory data analysis tasks. Results suggest that our visual interactions are intuitive and effective for exploring dimensionality reductions and generating hypotheses about the underlying data.
We compute forward projections using out-of-sample extension (or extrapolation) @cite_16 . Out-of-sample extension is the projection of a new data point into an existing DR (e.g. learned manifold model) using only the properties of the already computed DR. It is conceptually equivalent to testing a trained machine-learning model with data that was not part of the training set. For linear DR methods, out-of-sample extension is often performed by applying the learned linear transformation to the new data point. For autoencoders, the trained network defines the transformation from the high-dimensional to low-dimensional data representation @cite_41 . Back or backward projection maps a low-dimensional data point back into the original high-dimensional data space. For linear DRs, back projection is typically done by applying the inverse of the learned linear DR mapping. For nonlinear DRs, earlier research proposed DR-specific backward-projection techniques. For example, iLAMP @cite_26 introduces a back-projection method for LAMP @cite_23 using local neighborhoods and demonstrates its viability over synthetic datasets @cite_26 . Researchers also investigated general backward projection methods using radial basis functions @cite_22 @cite_27 , treating backward projection as an interpolation problem.
@cite_26 @cite_22 @cite_41 @cite_27 @cite_23 @cite_16
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Ever improving computing power and technological advances are greatly augmenting data collection and scientific observation. This has directly contributed to increased data complexity and dimensionality, motivating research of exploration techniques for multidimensional data. Consequently, a recent influx of work dedicated to techniques and tools that aid in understanding multidimensional datasets can be observed in many research fields, including biology, engineering, physics and scientific computing. While the effectiveness of existing techniques to analyze the structure and relationships of multidimensional data varies greatly, few techniques provide flexible mechanisms to simultaneously visualize and actively explore high-dimensional spaces. In this paper, we present an inverse linear affine multidimensional projection, coined iLAMP, that enables a novel interactive exploration technique for multidimensional data. iLAMP operates in reverse to traditional projection methods by mapping low-dimensional information into a high-dimensional space. This allows users to extrapolate instances of a multidimensional dataset while exploring a projection of the data to the planar domain. We present experimental results that validate iLAMP, measuring the quality and coherence of the extrapolated data; as well as demonstrate the utility of iLAMP to hypothesize the unexplored regions of a high-dimensional space. Abstract Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale. The proposed construction is shown to be similar to the Nystrom extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nystrom extension with suggestions for improvement. Several unsupervised learning algorithms based on an eigendecomposition provide either an embedding or a clustering only for given training points, with no straightforward extension for out-of-sample examples short of recomputing eigenvectors. This paper provides a unified framework for extending Local Linear Embedding (LLE), Isomap, Laplacian Eigenmaps, Multi-Dimensional Scaling (for dimensionality reduction) as well as for Spectral Clustering. This framework is based on seeing these algorithms as learning eigenfunctions of a data-dependent kernel. Numerical experiments show that the generalizations performed have a level of error comparable to the variability of the embedding algorithms due to the choice of training data. A standard approach for visualizing multivariate networks is to use one or more multidimensional views (for example, scatterplots) for selecting nodes by various metrics, possibly coordinated with a node-link view of the network. In this paper, we present three novel approaches for achieving a tighter integration of these views through hybrid techniques for multidimensional visualization, graph selection and layout. First, we present the FlowVizMenu, a radial menu containing a scatterplot that can be popped up transiently and manipulated with rapid, fluid gestures to select and modify the axes of its scatterplot. Second, the FlowVizMenu can be used to steer an attribute-driven layout of the network, causing certain nodes of a node-link diagram to move toward their corresponding positions in a scatterplot while others can be positioned manually or by force-directed layout. Third, we describe a novel hybrid approach that combines a scatterplot matrix (SPLOM) and parallel coordinates called the Parallel Scatterplot Matrix (P-SPLOM), which can be used to visualize and select features within the network. We also describe a novel arrangement of scatterplots called the Scatterplot Staircase (SPLOS) that requires less space than a traditional scatterplot matrix. Initial user feedback is reported. In recent years, a variety of nonlinear dimensionality reduction techniques have been proposed that aim to address the limitations of traditional techniques such as PCA and classical scaling. The paper presents a review and systematic comparison of these techniques. The performances of the nonlinear techniques are investigated on artificial and natural tasks. The results of the experiments reveal that nonlinear techniques perform well on selected artificial tasks, but that this strong performance does not necessarily extend to real-world tasks. The paper explains these results by identifying weaknesses of current nonlinear techniques, and suggests how the performance of nonlinear dimensionality reduction techniques may be improved.