The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch

Recent and forthcoming advances in instrumentation, and giant new surveys, are creating astronomical data sets that are not amenable to the methods of analysis familiar to astronomers. Traditional methods are often inadequate not merely because of the size in bytes of the data sets, but also because of the complexity of modern data sets. Mathematical limitations of familiar algorithms and techniques in dealing with such data sets create a critical need for new paradigms for the representation, analysis and scientific visualization (as opposed to illustrative visualization) of heterogeneous, multiresolution data across application domains. Some of the problems presented by the new data sets have been addressed by other disciplines such as applied mathematics, statistics and machine learning and have been utilized by other sciences such as space-based geosciences. Unfortunately, valuable results pertaining to these problems are mostly to be found only in publications outside of astronomy. Here we offer brief overviews of a number of concepts, techniques and developments, some "old" and some new. These are generally unknown to most of the astronomical community, but are vital to the analysis and visualization of complex datasets and images. In order for astronomers to take advantage of the richness and complexity of the new era of data, and to be able to identify, adopt, and apply new solutions, the astronomical community needs a certain degree of awareness and understanding of the new concepts. One of the goals of this paper is to help bridge the gap between applied mathematics, artificial intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in Astronomy, special issue "Robotic Astronomy

Improving dimensionality reduction projections for data visualization

In data science and visualization, dimensionality reduction techniques have been extensively employed for exploring large datasets. These techniques involve the transformation of high-dimensional data into reduced versions, typically in 2D, with the aim of preserving significant properties from the original data. Many dimensionality reduction algorithms exist, and nonlinear approaches such as the t-SNE (t-Distributed Stochastic Neighbor Embedding) and UMAP (Uniform Manifold Approximation and Projection) have gained popularity in the field of information visualization. In this paper, we introduce a simple yet powerful manipulation for vector datasets that modifies their values based on weight frequencies. This technique significantly improves the results of the dimensionality reduction algorithms across various scenarios. To demonstrate the efficacy of our methodology, we conduct an analysis on a collection of well-known labeled datasets. The results demonstrate improved clustering performance when attempting to classify the data in the reduced space. Our proposal presents a comprehensive and adaptable approach to enhance the outcomes of dimensionality reduction for visual data exploration.This research was funded by PID2021-122136OB-C21 from the Ministerio de Ciencia e Innovación, Spain, by 839 FEDER (EU) funds.Peer ReviewedPostprint (published version

Spectral methods for multimodal data analysis

Spectral methods have proven themselves as an important and versatile tool in a wide range of problems in the fields of computer graphics, machine learning, pattern recognition, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding a few of its eigenvalues and eigenfunctions. Classical examples include the computation of diffusion distances on manifolds in computer graphics, Laplacian eigenmaps, and spectral clustering in machine learning. In many cases, one has to deal with multiple data spaces simultaneously. For example, clustering multimedia data in machine learning applications involves various modalities or ``views'' (e.g., text and images), and finding correspondence between shapes in computer graphics problems is an operation performed between two or more modalities. In this thesis, we develop a generalization of spectral methods to deal with multiple data spaces and apply them to problems from the domains of computer graphics, machine learning, and image processing. Our main construction is based on simultaneous diagonalization of Laplacian operators. We present an efficient numerical technique for computing joint approximate eigenvectors of two or more Laplacians in challenging noisy scenarios, which also appears to be the first general non-smooth manifold optimization method. Finally, we use the relation between joint approximate diagonalizability and approximate commutativity of operators to define a structural similarity measure for images. We use this measure to perform structure-preserving color manipulations of a given image

Towards a virtual reality mobile App for exploratory data navigation in data mining education

Exploratory data visualization is a key component of data mining processes. It is particularly useful to gain insights from high-dimensional data and improve the interpretability of these processes as a result. Data visualization is limited to the capabilities of the viewer. These perceptual limitations in the process of data visualization can be at least partially overcome through interactive systems that allow us to “navigate” through visual displays. Virtual Reality (VR) is an adequate paradigm to achieve such interaction, but, despite its current blossoming, most efforts in that direction so far have required complex systems and specialized environments and are still far from being standardized. We are currently witnessing a quick industrial move towards wearable VR systems associated to mobile telephony. In parallel, the educational world is steadily moving towards ubiquitously-accessible e-learning environments. Acknowledging both trends, we propose in this thesis the foundations for the future development of a VR mobile app for visual data exploration in the context of educational data mining. The focus on data visualization for data mining is meant to be a proof of concept that could be extended to the teaching of many other aspects of data mining

Improved integration of information to reduce subsurface model bias

Subsurface modeling deals with data-related issues like cognitive and sampling biases, and model-related challenges including statistical assumptions, misspecification, and algorithmic biases. These challenges introduce four critical implications during subsurface modeling. Firstly, subsurface sampling is subject to sampling bias, which compromises statistical representativeness. Secondly, analog selection methodologies rely on multivariate statistics and expert judgment that overlook spatial information and data dimensionality. Thirdly, subsurface inferential workflows that utilize dimensionality reduction seldom provide repeatable frameworks that maintain model stability and are invariant to Euclidean transformations. Lastly, deep learning methods for dimensionality reduction, characterized as black-box models, lack interpretability and robust evaluation metrics, increasing susceptibility to algorithmic bias. Consequently, neglecting these challenges in subsurface modeling could lead to erroneous predictions, inconsistent inferences, diminished model reliability, and suboptimal decision-making that impacts project economics. This dissertation integrates information within subsurface models to reduce model bias and significantly improve their accuracy, robustness, and generalizability. First, I create spatial declustering methods to debias spatial datasets with single and multiscale preferential sampling in stationary populations. Second, I introduce a novel geostatistics-based machine learning method for identifying subsurface resource analogs that integrate spatial information in subsurface datasets with high dimensionality. Next, I efficiently combine machine learning and computational geometry methods to stabilize lower dimensional spaces for uncertainty quantification and interpretation. Finally, I create a methodology to assess, evaluate, and interpret the stability of deep learning latent feature spaces. These novel methodologies demonstrate the importance of improved techniques for information integration in subsurface modeling and show better results over naïve methods. This results in objective sampling debiasing in spatial stationary populations with single or multiple data scales, improving statistical representativity. Also, the results show better generalization and accurate identification of spatial analogs in high-dimensional datasets. Moreover, the methods yield Euclidean transformation-invariant lower-dimensional spaces, ensuring unique and repeatable solutions that improve model reliability and interpretability, for rational comparisons. Finally, the results indicate that deep learning models for dimensionality reduction exhibit algorithmic biases and instabilities, including sample, structural, and inferential instability, affecting their reliability and interpretability. Together, these innovations ultimately reduce model bias and significantly improve subsurface modeling.Petroleum and Geosystems Engineerin