Interpolating between measures supported by polygonal or polyhedral domains is a problem that has been recently addressed by the semi-discrete optimal transport framework. Within this framework, one of the domains is discretized with a set of samples, while the other one remains continuous. In this paper we present a method to introduce some symmetry into the solution using coupled power diagrams. This symmetry is key to capturing the discontinuities of the transport map reflected in the geometry of the power cells. We design our method as a fixed-point algorithm alternating between computations of semi-discrete transport maps and recentering of the sites. The resulting objects are coupled power diagrams with identical geometry, allowing us to approximate displacement interpolation through linear interpolation of the meshes vertices. Through these coupled power diagrams, we have a natural way of jointly sampling measures.
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The convex rope problem is to find a counterclockwise or clockwise convex rope starting at the vertex a and ending at the vertex b of a simple polygon P, where a is a vertex of the convex hull of P and b is visible from infinity. The convex rope mentioned is the shortest path joining a and b that does not enter the interior of P. In this paper, the problem is reconstructed as the one of finding such shortest path in a simple polygon and solved by the method of multiple shooting. We then show that if the collinear condition of the method holds at all shooting points, then these shooting points form the shortest path. Otherwise, the sequence of paths obtained by the update of the method converges to the shortest path. The algorithm is implemented in C++ for numerical experiments.
Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms. The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related $\mathbb{F}_p$-Matrix Factorization. Here, given a matrix $A$ over a finite field GF($p$) where $p$ is a prime, and an integer $r$, our objective is to find a matrix $B$ over the same field with GF($p$)-rank at most $r$ minimizing some norm of $A-B$. Our empirical research on synthetic and real-world data demonstrates the advantage of the new algorithms over previous works on BMF and $\mathbb{F}_p$-Matrix Factorization.
In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic L\'evy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, we introduce new tools of discrete functional analysis, such as discrete nonlocal Poincar\'e and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.
LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation of roundoff errors, which can lead solvers to return feasibility and optimality claims that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are established through the derivation of theoretical insights. Their advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the methodology is that the length of any coefficient calculated via the proposed algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches.
The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.
Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and $\lambda\leq0$ or differentiable and $\lambda\leq1$, where $\lambda$ is the parameter in Gegenbauer projections. If the underlying function is analytic and $\lambda>0$ or differentiable and $\lambda>1$, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of $n^{\lambda}$ and $n^{\lambda-1}$, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all $\lambda>-1/2$. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods.
In recent years, electricity generation has been responsible for more than a quarter of the greenhouse gas emissions in the US. Integrating a significant amount of renewables into a power grid is probably the most accessible way to reduce carbon emissions from power grids and slow down climate change. Unfortunately, the most accessible renewable power sources, such as wind and solar, are highly fluctuating and thus bring a lot of uncertainty to power grid operations and challenge existing optimization and control policies. The chance-constrained alternating current (AC) optimal power flow (OPF) framework finds the minimum cost generation dispatch maintaining the power grid operations within security limits with a prescribed probability. Unfortunately, the AC-OPF problem's chance-constrained extension is non-convex, computationally challenging, and requires knowledge of system parameters and additional assumptions on the behavior of renewable distribution. Known linear and convex approximations to the above problems, though tractable, are too conservative for operational practice and do not consider uncertainty in system parameters. This paper presents an alternative data-driven approach based on Gaussian process (GP) regression to close this gap. The GP approach learns a simple yet non-convex data-driven approximation to the AC power flow equations that can incorporate uncertainty inputs. The latter is then used to determine the solution of CC-OPF efficiently, by accounting for both input and parameter uncertainty. The practical efficiency of the proposed approach using different approximations for GP-uncertainty propagation is illustrated over numerous IEEE test cases.
Implicit Processes (IPs) represent a flexible framework that can be used to describe a wide variety of models, from Bayesian neural networks, neural samplers and data generators to many others. IPs also allow for approximate inference in function-space. This change of formulation solves intrinsic degenerate problems of parameter-space approximate inference concerning the high number of parameters and their strong dependencies in large models. For this, previous works in the literature have attempted to employ IPs both to set up the prior and to approximate the resulting posterior. However, this has proven to be a challenging task. Existing methods that can tune the prior IP result in a Gaussian predictive distribution, which fails to capture important data patterns. By contrast, methods producing flexible predictive distributions by using another IP to approximate the posterior process cannot tune the prior IP to the observed data. We propose here the first method that can accomplish both goals. For this, we rely on an inducing-point representation of the prior IP, as often done in the context of sparse Gaussian processes. The result is a scalable method for approximate inference with IPs that can tune the prior IP parameters to the data, and that provides accurate non-Gaussian predictive distributions.
When learning disconnected distributions, Generative adversarial networks (GANs) are known to face model misspecification. Indeed, a continuous mapping from a unimodal latent distribution to a disconnected one is impossible, so GANs necessarily generate samples outside of the support of the target distribution. This raises a fundamental question: what is the latent space partition that minimizes the measure of these areas? Building on a recent result of geometric measure theory, we prove that an optimal GANs must structure its latent space as a 'simplicial cluster' - a Voronoi partition where cells are convex cones - when the dimension of the latent space is larger than the number of modes. In this configuration, each Voronoi cell maps to a distinct mode of the data. We derive both an upper and a lower bound on the optimal precision of GANs learning disconnected manifolds. Interestingly, these two bounds have the same order of decrease: $\sqrt{\log m}$, $m$ being the number of modes. Finally, we perform several experiments to exhibit the geometry of the latent space and experimentally show that GANs have a geometry with similar properties to the theoretical one.
The conventional room geometry blind inference techniques with acoustic signals are conducted based on the prior knowledge of the environment, such as the room impulse response (RIR) or the sound source position, which will limit its application under known scenarios. To solve this problem, we have proposed a room geometry reconstruction method in this paper by using the geometric relation between the direct signal and first-order reflections. In addition to the information of the compact microphone array itself, this method does not need any precognition of the environmental parameters. Besides, the learning-based DNN models are designed and used to improve the accuracy and integrity of the localization results of the direct source and first-order reflections. The direction of arrival (DOA) and time difference of arrival (TDOA) information of the direct and reflected signals are firstly estimated using the proposed DCNN and TD-CNN models, which have higher sensitivity and accuracy than the conventional methods. Then the position of the sound source is inferred by integrating the DOA, TDOA and array height using the proposed DNN model. After that, the positions of image sources and corresponding boundaries are derived based on the geometric relation. Experimental results of both simulations and real measurements verify the effectiveness and accuracy of the proposed techniques compared with the conventional methods under different reverberant environments.
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