Related papers: Data-driven dimensional analysis: algorithms for u…

To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem,…

Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality…

Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a…

High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…

Dimensionality effects pose major challenges in high-dimensional and non-Euclidean data analysis. Graph-based two-sample tests and change-point detection are particularly attractive in this context, as they make minimal distributional…

One of the outstanding problems in complexity science and dynamical system theory is understanding the dynamic behavior of high-dimensional networked systems and their susceptibility to transitions to undesired states. Because of varied…

We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first…

Lower-dimensional subspaces that impact estimates of uncertainty are often described by Linear combinations of input variables, leading to active variables. This paper extends the derivative-based active subspace methods and…

Dimension reduction is often the first step in statistical modeling or prediction of multivariate spatial data. However, most existing dimension reduction techniques do not account for the spatial correlation between observations and do not…

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of…

Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. Recently, much effort has been dedicated towards…

Nonlinear dynamical systems are ubiquitous in nature and they are hard to forecast. Not only they may be sensitive to small perturbations in their initial conditions, but they are often composed of processes acting at multiple scales.…

High-dimensional group inference is an essential part of statistical methods for analysing complex data sets, including hierarchical testing, tests of interaction, detection of heterogeneous treatment effects and inference for local…

The dynamics of many-body systems can often be captured in terms of only a few relevant variables. Mathematical and numerical approaches exist to identify these variables by exploiting a separation of time scales between slow relevant and…

Density estimation plays a crucial role in many data analysis tasks, as it infers a continuous probability density function (PDF) from discrete samples. Thus, it is used in tasks as diverse as analyzing population data, spatial locations in…

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are…

Two different models exhibiting self-organized criticality are analyzed by means of the dynamic renormalization group. Although the two models differ by their behavior under a parity transformation of the order parameter, it is shown that…

We propose an algorithmic framework, that employs active subspace techniques, for scalable global optimization of functions with low effective dimension (also referred to as low-rank functions). This proposal replaces the original…

We study nonlinear regression of real valued data in an individual sequence manner, where we provide results that are guaranteed to hold without any statistical assumptions. We address the convergence and undertraining issues of…