Related papers: Learning polytopes with fixed facet directions
Fast and robust three-dimensional reconstruction of facial geometric structure from a single image is a challenging task with numerous applications. Here, we introduce a learning-based approach for reconstructing a three-dimensional face…
We propose a novel camera pose estimation or perspective-n-point (PnP) algorithm, based on the idea of consistency regions and half-space intersections. Our algorithm has linear time-complexity and a squared reconstruction error that…
Vertex direction algorithms have been around for a few decades in the experimental design and mixture models literature. We briefly review this type of algorithm and describe a new member of the family: the support reduction algorithm. The…
We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…
While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex…
Oberman gave a stochastic control formulation of the problem of estimating the convex envelope of a non-convex function. Based on this, we develop a reinforcement learning scheme to approximate the convex envelope, using a variant of…
Soft extrapolation refers to the problem of recovering a function from its samples, multiplied by a fast-decaying window and perturbed by an additive noise, over an interval which is potentially larger than the essential support of the…
Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and…
In this work, we investigate a particular class of shape optimization problems under uncertainties on the input parameters. More precisely, we are interested in the minimization of the expectation of a quadratic objective in a situation…
Structural learning, a method to estimate the parameters for discrete energy minimization, has been proven to be effective in solving computer vision problems, especially in 3D scene parsing. As the complexity of the models increases,…
We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a…
The estimation of direction of arrivals with help of $TV$-minimization is studied. Contrary to prior work in this direction, which has only considered certain antenna placement designs, we consider general antenna geometries. Applying the…
We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this…
By using elementary yet interesting observations and refining techniques used in a recent work by Fei Xue et al., we present new upper bounds for covering functionals of convex polytopes in $\mathbb{R}^n$ with few vertices. In these…
Classifiers and rating scores are prone to implicitly codifying biases, which may be present in the training data, against protected classes (i.e., age, gender, or race). So it is important to understand how to design classifiers and scores…
We show the existence of an FPTAS for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.
Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model. In this paper we blend…
These are lecture notes supporting a minicourse taught at the Summer School in Total Positivity and Quantum Field Theory at CMSA Harvard in June 2025. We give an introduction to positive geometries and their canonical forms. We present the…
We study the convergence rate of the proximal-gradient homotopy algorithm applied to norm-regularized linear least squares problems, for a general class of norms. The homotopy algorithm reduces the regularization parameter in a series of…
Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable…