8,536 research outputs found
Doctor of Philosophy
dissertationWith the ever-increasing amount of available computing resources and sensing devices, a wide variety of high-dimensional datasets are being produced in numerous fields. The complexity and increasing popularity of these data have led to new challenges and opportunities in visualization. Since most display devices are limited to communication through two-dimensional (2D) images, many visualization methods rely on 2D projections to express high-dimensional information. Such a reduction of dimension leads to an explosion in the number of 2D representations required to visualize high-dimensional spaces, each giving a glimpse of the high-dimensional information. As a result, one of the most important challenges in visualizing high-dimensional datasets is the automatic filtration and summarization of the large exploration space consisting of all 2D projections. In this dissertation, a new type of algorithm is introduced to reduce the exploration space that identifies a small set of projections that capture the intrinsic structure of high-dimensional data. In addition, a general framework for summarizing the structure of quality measures in the space of all linear 2D projections is presented. However, identifying the representative or informative projections is only part of the challenge. Due to the high-dimensional nature of these datasets, obtaining insights and arriving at conclusions based solely on 2D representations are limited and prone to error. How to interpret the inaccuracies and resolve the ambiguity in the 2D projections is the other half of the puzzle. This dissertation introduces projection distortion error measures and interactive manipulation schemes that allow the understanding of high-dimensional structures via data manipulation in 2D projections
Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion
A spectrally sparse signal of order is a mixture of damped or
undamped complex sinusoids. This paper investigates the problem of
reconstructing spectrally sparse signals from a random subset of regular
time domain samples, which can be reformulated as a low rank Hankel matrix
completion problem. We introduce an iterative hard thresholding (IHT) algorithm
and a fast iterative hard thresholding (FIHT) algorithm for efficient
reconstruction of spectrally sparse signals via low rank Hankel matrix
completion. Theoretical recovery guarantees have been established for FIHT,
showing that number of samples are sufficient for exact
recovery with high probability. Empirical performance comparisons establish
significant computational advantages for IHT and FIHT. In particular, numerical
simulations on D arrays demonstrate the capability of FIHT on handling large
and high-dimensional real data
High-dimensional Bayesian optimization with intrinsically low-dimensional response surfaces
Bayesian optimization is a powerful technique for the optimization of expensive black-box functions. It is used in a wide range of applications such as in drug and material design and training of machine learning models, e.g. large deep networks. We propose to extend this approach to high-dimensional settings, that is where the number of parameters to be optimized exceeds 10--20. In this thesis, we scale Bayesian optimization by exploiting different types of projections and the intrinsic low-dimensionality assumption of the objective function. We reformulate the problem in a low-dimensional subspace and learn a response surface and maximize an acquisition function in this low-dimensional projection. Contributions include i) a probabilistic model for axis-aligned projections, such as the quantile-Gaussian process and ii) a probabilistic model for learning a feature space by means of manifold Gaussian processes. In the latter contribution, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Finally, we present empirical results against well-known baselines in high-dimensional Bayesian optimization and provide possible directions for future research in this field.Open Acces
How Does the Low-Rank Matrix Decomposition Help Internal and External Learnings for Super-Resolution
Wisely utilizing the internal and external learning methods is a new
challenge in super-resolution problem. To address this issue, we analyze the
attributes of two methodologies and find two observations of their recovered
details: 1) they are complementary in both feature space and image plane, 2)
they distribute sparsely in the spatial space. These inspire us to propose a
low-rank solution which effectively integrates two learning methods and then
achieves a superior result. To fit this solution, the internal learning method
and the external learning method are tailored to produce multiple preliminary
results. Our theoretical analysis and experiment prove that the proposed
low-rank solution does not require massive inputs to guarantee the performance,
and thereby simplifying the design of two learning methods for the solution.
Intensive experiments show the proposed solution improves the single learning
method in both qualitative and quantitative assessments. Surprisingly, it shows
more superior capability on noisy images and outperforms state-of-the-art
methods
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