Polypols are natural generalizations of polytopes, with boundaries given by non-linear algebraic hypersurfaces.We describe polypols in the plane and in 3-space that admit a unique adjoint hypersurface and study them from an algebro-geometric perspective. We relate planar polypols to positive geometries introduced originally in particle physics, and identify the adjoint curve of a planar polypol with the numerator of the canonical differential form associated with the positive geometry.We settle several cases of a conjecture by Wachspress claiming that the adjoint curve of a regular planar polypol does not intersect its interior. In particular, we provide a complete characterization of the real topology of the adjoint curve for arbitrary convex polygons. Finally, we determine all types of planar polypols such that the rational map sending a polypol to its adjoint is finite, and explore connections of our topic with algebraic statistics.

We study convolutional neural networks with monomial activation functions. Specifically, we prove that their parameterization map is regular and is an isomorphism almost everywhere, up to rescaling the filters. By leveraging on tools from algebraic geometry, we explore the geometric properties of the image in function space of this map - typically referred to as neuromanifold. In particular, we compute the dimension and the degree of the neuromanifold, which measure the expressivity of the model, and describe its singularities. Moreover, for a generic large dataset, we derive an explicit formula that quantifies the number of critical points arising in the optimization of a regression loss.

Rolling shutter (RS) cameras dominate consumer and smartphone markets. Several methods for computing the absolute pose of RS cameras have appeared in the last 20 years, but the relative pose problem has not been fully solved yet. We provide a unified theory for the important class of order-one rolling shutter (RS<inf>1</inf>) cameras. These cameras generalize the perspective projection to RS cameras, projecting a generic space point to exactly one image point via a rational map. We introduce a new back-projection RS camera model, characterize RS<inf>1</inf> cameras, construct explicit parameterizations of such cameras, and determine the image of a space line. We classify all minimal problems for solving the relative camera pose problem with linear RS<inf>1</inf> cameras and discover new practical cases. Finally, we show how the theory can be used to explain RS models previously used for absolute pose computation.

We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We present a dictionary that relates notions of stability from geometric invariant theory to the existence and uniqueness of a maximum likelihood estimate. Our dictionary holds for both discrete and continuous statistical models: we discuss log-linear models and Gaussian models, including matrix normal models and directed Gaussian graphical models. Our approach reveals promising consequences of the interplay between invariant theory and statistics. For instance, algorithms from statistics can be used in invariant theory, and vice versa.

We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear first-order partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials.

We study the geometry of linear networks with one-dimensional convolutional layers. The function spaces of these networks can be identified with semialgebraic families of polynomials admitting sparse factorizations. We analyze the impact of the network's architecture on the function space's dimension, boundary, and singular points. We also describe the critical points of the network's parameterization map. Furthermore, we study the optimization problem of training a network with the squared error loss. We prove that for architectures where all strides are larger than one and generic data, the nonzero critical points of that optimization problem are smooth interior points of the function space. This property is known to be false for dense linear networks and linear convolutional networks with stride one.

We present a complete classification of minimal problems for generic arrangements of points and lines in space observed partially by three calibrated perspective cameras when each line is incident to at most one point. This is a large class of interesting minimal problems that allows missing observations in images due to occlusions and missed detections. There is an infinite number of such minimal problems; however, we show that they can be reduced to 140,616 equivalence classes by removing superfluous features and relabeling the cameras. We also introduce camera-minimal problems, which are practical for designing minimal solvers, and show how to pick a simplest camera-minimal problem for each minimal problem. This simplification results in 74,575 equivalence classes. Only 76 of these were known; the rest are new. To identify problems having potential for practical solving of image matching and 3D reconstruction, we present several natural subfamilies of camera-minimal problems as well as compute solution counts for all camera-minimal problems which have fewer than 300 solutions for generic data.