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We propose a computational framework to quantify (measure) and to optimize the reliability of complex systems. The approach uses a graph representation of the system that is subject to random failures of its components (nodes and edges).…
Multifractal analysis of stochastic processes deals with the fine scale properties of the sample paths and seeks for some global scaling property that would enable extracting the so-called spectrum of singularities. In this paper we…
A common approach to statistical learning with big-data is to randomly split it among $m$ machines and learn the parameter of interest by averaging the $m$ individual estimates. In this paper, focusing on empirical risk minimization, or…
We propose Partition Tree, a novel tree-based framework for conditional density estimation over general outcome spaces that supports both continuous and categorical variables within a unified formulation. Our approach models conditional…
Clustering algorithms remain valuable tools for grouping and summarizing the most important aspects of data. Example areas where this is the case include image segmentation, dimension reduction, signals analysis, model order reduction,…
This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second order Taylor expansions and polynomial bounding techniques is first…
An alternate definition of the box-counting dimension is proposed, to provide a better approximation for fractals involving rotation such as the 'Bradley Spiral' structure. A curve fitting comparison of this definition with the box-counting…
We consider a class of "box-like" statistically self-affine functions, and compute the almost-sure box-counting dimension of their graphs. Furthermore, we consider the differentiability of our functions, and prove that, depending on an…
Even though a train/test split of the dataset randomly performed is a common practice, could not always be the best approach for estimating performance generalization under some scenarios. The fact is that the usual machine learning…
Through a combination of rigorous analytical derivations and extensive numerical simulations, this work reports an exotic multifractal behavior, dubbed "logarithmic multifractality", in effectively infinite-dimensional systems undergoing…
Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting…
Quantifying uncertainty in model predictions is a common goal for practitioners seeking more than just point predictions. One tool for uncertainty quantification that requires minimal assumptions is conformal inference, which can help…
In quantum physics, the theoretical study of unbound many-body systems is typically quite complex -- owing to the combination of their large spatial extension and the so-called {\it curse of dimensionality}. Often, such systems are studied…
Representing sorted integer sequences in small space is a central problem for large-scale retrieval systems such as Web search engines. Efficient query resolution, e.g., intersection or random access, is achieved by carefully partitioning…
We use a bootstrap argument to enhance the eigensystem multiscale analysis, introduced by Elgart and Klein for proving localization for the Anderson model at high disorder. The eigensystem multiscale analysis studies finite volume…
We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of…
In big data analysis, a simple task such as linear regression can become very challenging as the variable dimension $p$ grows. As a result, variable screening is inevitable in many scientific studies. In recent years, randomized algorithms…
We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual…
We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND -- sparsified Nested Dissection. It is based on nested dissection, sparsification and low-rank compression. After…
Efficient and accurate estimation of multivariate empirical probability distributions is fundamental to the calculation of information-theoretic measures such as mutual information and transfer entropy. Common techniques include variations…