Related papers: A Novel Approach for Intrinsic Dimension Estimatio…
We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we…
Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to…
Linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data, due to their simple geometric interpretations and typically attractive computational properties. These methods capture many data features of…
Latent variable models represent a useful tool for the analysis of complex data when the constructs of interest are not observable. A problem related to these models is that the integrals involved in the likelihood function cannot be solved…
We develop an efficient estimation procedure for identifying and estimating the central subspace. Using a new way of parameterization, we convert the problem of identifying the central subspace to the problem of estimating a finite…
A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system…
In this note we discuss a common misconception, namely that embeddings are always used to reduce the dimensionality of the item space. We show that when we measure dimensionality in terms of information entropy then the embedding of sparse…
High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these…
The problem of finding a reduced dimensionality representation of categorical variables while preserving their most relevant characteristics is fundamental for the analysis of complex data. Specifically, given a co-occurrence matrix of two…
A basic problem in machine learning is to find a mapping $f$ from a low dimensional latent space $\mathcal{Y}$ to a high dimensional observation space $\mathcal{X}$. Modern tools such as deep neural networks are capable to represent general…
Dimension reduction is often an important step in the analysis of high-dimensional data. PCA is a popular technique to find the best low-dimensional approximation of high-dimensional data. However, classical PCA is very sensitive to…
State-of-the-art models rely on massive widths despite exhibiting low Intrinsic Dimension (ID). We posit that this redundancy serves the non-convex optimization search rather than the final representation. We validate this hypothesis by…
We present a Bayesian model selection approach to estimate the intrinsic dimensionality of a high-dimensional dataset. To this end, we introduce a novel formulation of the probabilisitic principal component analysis model based on a…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
This paper proposes a supervised dimension reduction methodology for tensor data which has two advantages over most image-based prognostic models. First, the model does not require tensor data to be complete which expands its application to…
The monitoring and management of high-volume feature-rich traffic in large networks offers significant challenges in storage, transmission and computational costs. The predominant approach to reducing these costs is based on performing a…
One of the fundamental problems within the field of machine learning is dimensionality reduction. Dimensionality reduction methods make it possible to combat the so-called curse of dimensionality, visualize high-dimensional data and, in…
The vast majority of Dimensionality Reduction (DR) techniques rely on second-order statistics to define their optimization objective. Even though this provides adequate results in most cases, it comes with several shortcomings. The methods…
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…
Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower…