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Dejan Slepčev

Publications and source records attributed to Dejan Slepčev.

28 records · Page 2Linked to original sources
arXiv2016

Transport-based analysis, modeling, and learning from signal and data distributions

Transport-based techniques for signal and data analysis have received increased attention recently. Given their abilities to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications including content-based retrieval, cancer detection, image super-resolution, and statistical machine learning, to name a few, and shown to produce state of the art in several applications. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here we provide an overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications.

cs.CV↗
arXiv2016

Multiple penalized principal curves: analysis and computation

We study the problem of finding the one-dimensional structure in a given data set. In other words we consider ways to approximate a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of principal curves and introduce a new functional which allows for multiple curves. We prove the existence of minimizers and establish their basic properties. We develop an efficient algorithm for obtaining (near) minimizers of the functional. While both of the functionals used are nonconvex, we argue that enlarging the configuration space to allow for multiple curves leads to a simpler energy landscape with fewer undesirable (high-energy) local minima. Furthermore we note that the approach proposed is able to find the one-dimensional structure even for data with considerable amount of noise.

math.AP↗
arXiv2016

Estimating perimeter using graph cuts

We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For $Ω\subset D = (0,1)^d$, with $d \geq 2$, we are given $n$ random i.i.d. points on $D$ whose membership in $Ω$ is known. We consider the sample as a random geometric graph with connection distance $\varepsilon>0$. We estimate the perimeter of $Ω$ (relative to $D$) by the, appropriately rescaled, graph cut between the vertices in $Ω$ and the vertices in $D \backslash Ω$. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to $n$ and $\varepsilon$. We consider two scaling regimes: the dense (when the average degree of the vertices goes to $\infty$) and the sparse one (when the degree goes to $0$). In the dense regime there is a crossover in the nature of approximation at dimension $d=5$: we show that in low dimensions $d=2,3,4$ one can obtain confidence intervals for the approximation error, while in higher dimensions one can only obtain error estimates for testing the hypothesis that the perimeter is less than a given number.

math.ST↗
arXiv2015

A variational approach to the consistency of spectral clustering

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the spectral convergence of both unnormalized and normalized graph Laplacians towards the appropriate operators in the continuum domain. We obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the spectral convergence to hold. We also show that the discrete clusters obtained via spectral clustering converge towards a continuum partition of the ground truth measure. Such continuum partition minimizes a functional describing the continuum analogue of the graph-based spectral partitioning. Our approach, based on variational convergence, is general and flexible.

math.ST↗
arXiv2015

Existence of Ground States of Nonlocal-Interaction Energies

We investigate which nonlocal-interaction energies have a ground state (global minimizer). We consider this question over the space of probability measures and establish a sharp condition for the existence of ground states. We show that this condition is closely related to the notion of stability (i.e. $H$-stability) of pairwise interaction potentials. Our approach uses the direct method of the calculus of variations.

math.AP↗
arXiv2014

Average-distance problem for parameterized curves

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure $μ$, for $p \geq 1$ and $λ>0$ we consider the functional \[ E(γ) = \int_{\mathbb{R}^d} d(x, Γ_γ)^p dμ(x) + λ\,\textrm{Length}(γ) \] where $γ:I \to \mathbb{R}^d$, $I$ is an interval in $\mathbb{R}$, $Γ_γ= γ(I)$, and $d(x, Γ_γ)$ is the distance of $x$ to $Γ_γ$. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure $\mathcal H^1$, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures $μ$ supported in two dimensions the minimizing curve is injective if $p \geq 2$ or if $μ$ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.

math.AP↗
arXiv2014

Continuum limit of total variation on point clouds

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of available data points increases, of graph-based machine learning algorithms for tasks such as clustering. In particular, we study when is the cut capacity, and more generally total variation, on these graphs a good approximation of the perimeter (total variation) in the continuum setting. We address this question in the setting of $Γ$-convergence. We obtain almost optimal conditions on the scaling, as number of points increases, of the size of the neighborhood over which the points are connected by an edge for the $Γ$-convergence to hold. Taking the limit is enabled by a transportation based metric which allows to suitably compare functionals defined on different point clouds.

math.ST↗
arXiv2014

Nonlocal-interaction equations on uniformly prox-regular sets

We study the well-posedness of a class of nonlocal-interaction equations on general domains $Ω\subset \mathbb{R}^d$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $λ$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity. Our approach is based on the theory of gradient flows in spaces of probability measures.

math.AP↗
arXiv2009

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a non-rigorous argument indicating that they are not displacement semiconvex.

math.AP↗