Related papers: Geometry-Aware Maximum Likelihood Estimation of In…
Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by…
In the last decades the estimation of the intrinsic dimensionality of a dataset has gained considerable importance. Despite the great deal of research work devoted to this task, most of the proposed solutions prove to be unreliable when the…
A fundamental task in data exploration is to extract simplified low dimensional representations that capture intrinsic geometry in data, especially for faithfully visualizing data in two or three dimensions. Common approaches to this task…
We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution $\rho$ in $\mathbb{R}^D$, that is nearly supported on a $d$-dimensional set $\mathcal{M}$ - for example…
Quantification of the number of variables needed to locally explain complex data is often the first step to better understanding it. Existing techniques from intrinsic dimension estimation leverage statistical models to glean this…
Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a…
Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a…
To gain insight into the mechanisms behind machine learning methods, it is crucial to establish connections among the features describing data points. However, these correlations often exhibit a high-dimensional and strongly nonlinear…
We consider distributed estimation of the inverse covariance matrix, also called the concentration or precision matrix, in Gaussian graphical models. Traditional centralized estimation often requires global inference of the covariance…
Nonlinear manifold learning from unorganized data points is a very challenging unsupervised learning and data visualization problem with a great variety of applications. In this paper we present a new algorithm for manifold learning and…
Local intrinsic dimension (LID) estimation methods have received a lot of attention in recent years thanks to the progress in deep neural networks and generative modeling. In opposition to old non-parametric methods, new methods use…
We study mixture of linear regression (random coefficient) models, which capture population heterogeneity by allowing the regression coefficients to follow an unknown distribution $G^*$. In contrast to common parametric methods that fix the…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
Timely detection of abrupt anomalies is crucial for real-time monitoring and security of modern systems producing high-dimensional data. With this goal, we propose effective and scalable algorithms. Proposed algorithms are nonparametric as…
Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…
For many machine learning tasks, the input data lie on a low-dimensional manifold embedded in a high dimensional space and, because of this high-dimensional structure, most algorithms are inefficient. The typical solution is to reduce the…
Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to…
In this paper we study the computation of the nonparametric maximum likelihood estimator (NPMLE) in multivariate mixture models. Our first approach discretizes this infinite dimensional convex optimization problem by fixing the support…
We present Low Distortion Local Eigenmaps (LDLE), a manifold learning technique which constructs a set of low distortion local views of a dataset in lower dimension and registers them to obtain a global embedding. The local views are…
In the manifold learning problem one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper we…