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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We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of pixel images in about 0.4 s using a single graphics processing unit (GPU). In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise non-linearity) when the normal operator ( @math , where @math is the adjoint of the forward imaging operator, @math ) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a @math image on the GPU. Locally weighted projection regression (LWPR) is a new algorithm for incremental nonlinear function approximation in high-dimensional spaces with redundant and irrelevant input dimensions. At its core, it employs nonparametric regression with locally linear models. In order to stay computationally efficient and numerically robust, each local model performs the regression analysis with a small number of univariate regressions in selected directions in input space in the spirit of partial least squares regression. We discuss when and how local learning techniques can successfully work in high-dimensional spaces and review the various techniques for local dimensionality reduction before finally deriving the LWPR algorithm. The properties of LWPR are that it (1) learns rapidly with second-order learning methods based on incremental training, (2) uses statistically sound stochastic leave-one-out cross validation for learning without the need to memorize training data, (3) adjusts its weighting kernels based on only local information in order to minimize the danger of negative interference of incremental learning, (4) has a computational complexity that is linear in the number of inputs, and (5) can deal with a large number of—possibly redundant—inputs, as shown in various empirical evaluations with up to 90 dimensional data sets. For a probabilistic interpretation, predictive variance and confidence intervals are derived. To our knowledge, LWPR is the first truly incremental spatially localized learning method that can successfully and efficiently operate in very high-dimensional spaces. In this paper, we present a method that estimates reflectance and illumination information from a single image depicting a single-material specular object from a given class under natural illumination. We follow a data-driven, learning-based approach trained on a very large dataset, but in contrast to earlier work we do not assume one or more components (shape, reflectance, or illumination) to be known. We propose a two-step approach, where we first estimate the object’s reflectance map, and then further decompose it into reflectance and illumination. For the first step, we introduce a Convolutional Neural Network (CNN) that directly predicts a reflectance map from the input image itself, as well as an indirect scheme that uses additional supervision, first estimating surface orientation and afterwards inferring the reflectance map using a learning-based sparse data interpolation technique. For the second step, we suggest a CNN architecture to reconstruct both Phong reflectance parameters and high-resolution spherical illumination maps from the reflectance map. We also propose new datasets to train these CNNs. We demonstrate the effectiveness of our approach for both steps by extensive quantitative and qualitative evaluation in both synthetic and real data as well as through numerous applications, that show improvements over the state-of-the-art.