Related papers: Dimensionally Consistent Learning with Buckingham …
Symmetry in differential equations reveals invariances and offers a powerful means to reduce model complexity. Lie group analysis characterizes these symmetries through infinitesimal generators, which provide a local, linear criterion for…
Principal component analysis continues to be a powerful tool in dimension reduction of high dimensional data. We assume a variance-diverging model and use the high-dimension, low-sample-size asymptotics to show that even though the…
In addition to being extremely non-linear, modern problems require millions if not billions of parameters to solve or at least to get a good approximation of the solution, and neural networks are known to assimilate that complexity by…
Deep learning systems achieve remarkable empirical performance, yet the stability of the training process itself remains poorly understood. Training unfolds as a high-dimensional dynamical system in which small perturbations to…
This paper introduces a new clustering technique, called {\em dimensional clustering}, which clusters each data point by its latent {\em pointwise dimension}, which is a measure of the dimensionality of the data set local to that point.…
Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC…
Recently, covariance descriptors have received much attention as powerful representations of set of points. In this research, we present a new metric learning algorithm for covariance descriptors based on the Dykstra algorithm, in which the…
Data sets tend to live in low-dimensional non-linear subspaces. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The symmetric Riemannian geometry setting can be suitable for a variety of…
Structural breaks have been commonly seen in applications. Specifically for detection of change points in time, research gap still remains on the setting in ultra high dimension, where the covariates may bear spurious correlations. In this…
Symmetries are key properties of physical models and of experimental designs, but any proposed symmetry may or may not be realized in nature. In this paper, we introduce a practical and general method to test such suspected symmetries in…
High-dimensional systems that have a low-dimensional dominant behavior allow for model reduction and simplified analysis. We use differential analysis to formalize this important concept in a nonlinear setting. We show that dominance can be…
Dimensionality reduction is crucial both for visualization and preprocessing high dimensional data for machine learning. We introduce a novel method based on a hierarchy built on 1-nearest neighbor graphs in the original space which is used…
Two-sample hypothesis testing is a fundamental problem with various applications, which faces new challenges in the high-dimensional context. To mitigate the issue of the curse of dimensionality, high-dimensional data are typically assumed…
Popularly used eigendecomposition-based criteria such as BIC type, ratio estimation and principal component-based criterion often underdetermine model dimensionality for regressions or the number of factors for factor models. This…
We propose a novel data-driven method called QENDy (Quadratic Embedding of Nonlinear Dynamics) that not only allows us to learn quadratic representations of highly nonlinear dynamical systems, but also to identify the governing equations.…
Aligning beamlines at synchrotron light sources is a high-dimensional, expensive-to-sample optimization problem, as beams are focused using a series of dynamic optical components. Bayesian Optimization is an efficient machine learning…
Data are not only ubiquitous in society, but are increasingly complex both in size and dimensionality. Dimension reduction offers researchers and scholars the ability to make such complex, high dimensional data spaces simpler and more…
Most linear dimension reduction methods proposed in the literature can be formulated using an appropriate pair of scatter matrices, see e.g. Ye and Weiss (2003), Tyler et al. (2009), Bura and Yang (2011), Liski et al. (2014) and Luo and Li…
Physics-informed neural networks have emerged as a powerful tool in the scientific machine learning community, with applications to both forward and inverse problems. While they have shown considerable empirical success, significant…
We make two theoretical contributions to disentanglement learning by (a) defining precise semantics of disentangled representations, and (b) establishing robust metrics for evaluation. First, we characterize the concept "disentangled…