Related papers: Persistent homology for low-complexity models
This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant…
An intriguing phenomenon in many instances of compressed sensing is that the reconstruction quality is governed not just by the overall sparsity of the signal, but also on its structure. This paper is about understanding this phenomenon,…
Dimensionality reduction techniques are widely used for visualizing high-dimensional data in two dimensions. Existing methods are typically designed to preserve either local (e.g., $t$-SNE, UMAP) or global (e.g., MDS, PCA) structure of the…
Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger…
Sufficient dimension reduction aims for reduction of dimensionality of a regression without loss of information by replacing the original predictor with its lower-dimensional subspace. Partial (sufficient) dimension reduction arises when…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
The challenge of taking many variables into account in optimization problems may be overcome under the hypothesis of low effective dimensionality. Then, the search of solutions can be reduced to the random embedding of a low dimensional…
In this work we show that the classification performance of high-dimensional structural MRI data with only a small set of training examples is improved by the usage of dimension reduction methods. We assessed two different dimension…
Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the…
Text documents are complex high dimensional objects. To effectively visualize such data it is important to reduce its dimensionality and visualize the low dimensional embedding as a 2-D or 3-D scatter plot. In this paper we explore…
Gaussian process-based latent variable models are flexible and theoretically grounded tools for nonlinear dimension reduction, but generalizing to non-Gaussian data likelihoods within this nonlinear framework is statistically challenging.…
Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful…
The quest for simplification in physics drives the exploration of concise mathematical representations for complex systems. This Dissertation focuses on the concept of dimensionality reduction as a means to obtain low-dimensional…
Real world-datasets characterized by discrete features are ubiquitous: from categorical surveys to clinical questionnaires, from unweighted networks to DNA sequences. Nevertheless, the most common unsupervised dimensional reduction methods…
The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input…
Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative…
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…
In 'An asymptotic result on compressed sensing matrices', a new construction for compressed sensing matrices using combinatorial design theory was introduced. In this paper, we use deterministic and probabilistic methods to analyse the…