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A recent research trend involves treating database index structures as Machine Learning (ML) models. In this domain, single or multiple ML models are trained to learn the mapping from keys to positions inside a data set. This class of indexes is known as "Learned Indexes." Learned indexes have demonstrated improved search performance and reduced space requirements for one-dimensional data. The concept of one-dimensional learned indexes has naturally been extended to multi-dimensional (e.g., spatial) data, leading to the development of "Learned Multi-dimensional Indexes". This survey focuses on learned multi-dimensional index structures. Specifically, it reviews the current state of this research area, explains the core concepts behind each proposed method, and classifies these methods based on several well-defined criteria. We present a taxonomy that classifies and categorizes each learned multi-dimensional index, and survey the existing literature on learned multi-dimensional indexes according to this taxonomy. Additionally, we present a timeline to illustrate the evolution of research on learned indexes. Finally, we highlight several open challenges and future research directions in this emerging and highly active field.

The Center for Disease Control estimates that over 37 million US adults suffer from chronic kidney disease (CKD), yet 9 out of 10 of these individuals are unaware of their condition due to the absence of symptoms in the early stages. It has a significant impact on patients' quality of life, particularly when it progresses to the need for dialysis. Early prediction of dialysis is crucial as it can significantly improve patient outcomes and assist healthcare providers in making timely and informed decisions. However, developing an effective machine learning (ML)-based Clinical Decision Support System (CDSS) for early dialysis prediction poses a key challenge due to the imbalanced nature of data. To address this challenge, this study evaluates various data augmentation techniques to understand their effectiveness on real-world datasets. We propose a new approach named Binary Gaussian Copula Synthesis (BGCS). BGCS is tailored for binary medical datasets and excels in generating synthetic minority data that mirrors the distribution of the original data. BGCS enhances early dialysis prediction by outperforming traditional methods in detecting dialysis patients. For the best ML model, Random Forest, BCGS achieved a 72% improvement, surpassing the state-of-the-art augmentation approaches. Also, we present a ML-based CDSS, designed to aid clinicians in making informed decisions. CDSS, which utilizes decision tree models, is developed to improve patient outcomes, identify critical variables, and thereby enable clinicians to make proactive decisions, and strategize treatment plans effectively for CKD patients who are more likely to require dialysis in the near future. Through comprehensive feature analysis and meticulous data preparation, we ensure that the CDSS's dialysis predictions are not only accurate but also actionable, providing a valuable tool in the management and treatment of CKD.

We are concerned with a novel Bayesian statistical framework for the characterization of natural subsurface formations, a very challenging task. Because of the large dimension of the stochastic space of the prior distribution in the framework, typically a dimensional reduction method, such as a Karhunen-Leove expansion (KLE), needs to be applied to the prior distribution to make the characterization computationally tractable. Due to the large variability of properties of subsurface formations (such as permeability and porosity) it may be of value to localize the sampling strategy so that it can better adapt to large local variability of rock properties. In this paper, we introduce the concept of multiscale sampling to localize the search in the stochastic space. We combine the simplicity of a preconditioned Markov Chain Monte Carlo method with a new algorithm to decompose the stochastic space into orthogonal subspaces, through a one-to-one mapping of the subspaces to subdomains of a non-overlapping domain decomposition of the region of interest. The localization of the search is performed by a multiscale blocking strategy within Gibbs sampling: we apply a KL expansion locally, at the subdomain level. Within each subdomain, blocking is applied again, for the sampling of the KLE random coefficients. The effectiveness of the proposed framework is tested in the solution of inverse problems related to elliptic partial differential equations arising in porous media flows. We use multi-chain studies in a multi-GPU cluster to show that the new algorithm clearly improves the convergence rate of the preconditioned MCMC method. Moreover, we illustrate the importance of a few conditioning points to further improve the convergence of the proposed method.

In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian statistical framework with a preconditioned Markov Chain Monte Carlo (MCMC) algorithm for the solution of the inverse problem. Reduction of uncertainty can be accomplished by incorporating measurements at sparse locations (static data) in the prior distribution. We present a new method to condition Gaussian fields (the log of permeability fields) to available sparse measurements. A truncated Karhunen-Lo\`eve expansion (KLE) is used for dimension reduction. In the proposed method the imposition of static data is made through the projection of a sample (expressed as a vector of independent, identically distributed normal random variables) onto the nullspace of a data matrix, that is defined in terms of the KLE. The numerical implementation of the proposed method is straightforward. Through numerical experiments for a model of second-order elliptic equation, we show that the proposed method in multi-chain studies converges much faster than the MCMC method without conditioning. These studies indicate the importance of conditioning in accelerating the MCMC convergence., Comment: 16 pages, 6 figures, conference paper