In this note, we propose polynomial-time algorithms solving the Monge and Kantorovich formulations of the $\infty$-optimal transport problem in the discrete and finite setting. It is the first time, to the best of our knowledge, that efficient numerical methods for these problems have been proposed.
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We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.
We introduce the local information cost (LIC), which quantifies the amount of information that nodes in a network need to learn when solving a graph problem. We show that the local information cost presents a natural lower bound on the communication complexity of distributed algorithms. For the synchronous CONGEST $KT_1$ model, where each node has initial knowledge of its neighbors' IDs, we prove that $\Omega(\frac{\text{LIC}_\gamma(P)}{\log\tau \log n})$ bits are required for solving a graph problem $P$ with a $\tau$-round algorithm that errs with probability at most $\gamma$. Our result is the first lower bound that yields a general trade-off between communication and time for graph problems in the CONGEST $KT_1$ model. We demonstrate how to apply the local information cost by deriving a lower bound on the communication complexity of computing routing tables for all-pairs-shortest-paths (APSP) routing, as well as for computing a spanner with multiplicative stretch $2t-1$ that consists of at most $O(n^{1+\frac{1}{t} + \epsilon})$ edges, where $\epsilon = O( {1}/{t^2} )$. More concretely, we derive the following lower bounds in the CONGEST model under the $KT_1$ assumption: For constructing routing tables, we show that any $O(\text{poly}(n))$-time algorithm has a communication complexity of $\Omega( {n^2}/{\log^2 n} )$ bits. Our main result is for constructing graph spanners: We show that any $O(\text{poly}(n))$-time algorithm must send at least $\tilde\Omega(\tfrac{1}{t^2} n^{1+{1}/{2t}})$ bits. Previously, only a trivial lower bound of $\tilde \Omega(n)$ bits was known for these problems.
Line coverage is the task of servicing a given set of one-dimensional features in an environment. It is important for the inspection of linear infrastructure such as road networks, power lines, and oil and gas pipelines. This paper addresses the single robot line coverage problem for aerial and ground robots by modeling it as an optimization problem on a graph. The problem belongs to the broad class of arc routing problems and is closely related to the rural postman problem (RPP) on asymmetric graphs. The paper presents an integer linear programming formulation with proofs of correctness. Using the minimum cost flow problem, we develop approximation algorithms with guarantees on the solution quality. These guarantees also improve the existing results for the asymmetric RPP. The main algorithm partitions the problem into three cases based on the structure of the required graph, i.e., the graph induced by the features that require servicing. We evaluate our algorithms on road networks from the 50 most populous cities in the world, consisting of up to 730 road segments. The algorithms, augmented with improvement heuristics, run within 3s and generate solutions that are within 10% of the optimum. We experimentally demonstrate our algorithms with commercial UAVs.
Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the \emph{data separation function}. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to $1$ in the general case of arbitrary normalized dataset consisting of $n$ samples in $d$ dimension as $n/\sqrt{d}\rightarrow 0$. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes \emph{kinetic optimal} as $n/\sqrt{d}\rightarrow 0$. We further support this theory with empirical experiments on ImageNet.
Real-time computation of optimal control is a challenging problem and, to solve this difficulty, many frameworks proposed to use learning techniques to learn (possibly sub-optimal) controllers and enable their usage in an online fashion. Among these techniques, the optimal motion framework is a simple, yet powerful technique, that obtained success in many complex real-world applications. The main idea of this approach is to take advantage of dynamic motion primitives, a widely used tool in robotics to learn trajectories from demonstrations. While usually these demonstrations come from humans, the optimal motion framework is based on demonstrations coming from optimal solutions, such as the ones obtained by numeric solvers. As usual in many learning techniques, a drawback of this approach is that it is hard to estimate the suboptimality of learned solutions, since finding easily computable and non-trivial upper bounds to the error between an optimal solution and a learned solution is, in general, unfeasible. However, we show in this paper that it is possible to estimate this error for a broad class of problems. Furthermore, we apply this estimation technique to achieve a novel and more efficient sampling scheme to be used within the optimal motion framework, enabling the usage of this framework in some scenarios where the computational resources are limited.
We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework overcomes the challenges related to the imposition of boundary conditions, the choice of collocation points in physics-informed neural networks, and the integration of variational physics-informed neural networks. A numerical experiment set confirms the framework's capability of handling various forward and inverse problems. In particular, the trained neural network generalises well for smooth problems, beating finite element solutions by some orders of magnitude. We finally propose an effective one-loop solver with an initial data fitting step (to obtain a cheap initialisation) to solve inverse problems.
We design and compute a class of optimal control problems for reaction-diffusion systems. They form mean field control problems related to multi-density reaction-diffusion systems. To solve proposed optimal control problems numerically, we first apply high-order finite element methods to discretize the space-time domain and then solve the optimal control problem using augmented Lagrangian methods (ALG2). Numerical examples, including generalized optimal transport and mean field control problems between Gaussian distributions and image densities, demonstrate the effectiveness of the proposed modeling and computational methods for mean field control problems involving reaction-diffusion equations/systems.
We consider the problem of estimating the causal effect of a treatment on an outcome in linear structural causal models (SCM) with latent confounders when we have access to a single proxy variable. Several methods (such as difference-in-difference (DiD) estimator or negative outcome control) have been proposed in this setting in the literature. However, these approaches require either restrictive assumptions on the data generating model or having access to at least two proxy variables. We propose a method to estimate the causal effect using cross moments between the treatment, the outcome, and the proxy variable. In particular, we show that the causal effect can be identified with simple arithmetic operations on the cross moments if the latent confounder in linear SCM is non-Gaussian. In this setting, DiD estimator provides an unbiased estimate only in the special case where the latent confounder has exactly the same direct causal effects on the outcomes in the pre-treatment and post-treatment phases. This translates to the common trend assumption in DiD, which we effectively relax. Additionally, we provide an impossibility result that shows the causal effect cannot be identified if the observational distribution over the treatment, the outcome, and the proxy is jointly Gaussian. Our experiments on both synthetic and real-world datasets showcase the effectiveness of the proposed approach in estimating the causal effect.
A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential) bound are essentially optimized to perform well on such worst case graphs. However, such graphs are often either unlikely or actively avoided in practice, where benign graph instances usually admit much faster solutions. To circumnavigate these drawbacks, the concept of universal complexity analysis in the distributed setting was suggested by [Kutten and Peleg, PODC'95] and actively pursued by [Haeupler et al., STOC'21]. Here, the aim is to gauge the complexity of a distributed graph problem depending on the given graph instance. The challenge is to identify and understand the graph property that allows to accurately quantify the complexity of a distributed problem on a given graph. In the present work, we consider distributed shortest paths problems in the HYBRID model of distributed computing, where nodes have simultaneous access to two different modes of communication: one is restricted by locality and the other is restricted by congestion. We identify the graph parameter of neighborhood quality and show that it accurately describes a universal bound for the complexity of certain class of shortest paths problems in the HYBRID model.
In this work, we prove rigorous error estimates for a hybrid method introduced in [15] for solving the time-dependent radiation transport equation (RTE). The method relies on a splitting of the kinetic distribution function for the radiation into uncollided and collided components. A high-resolution method (in angle) is used to approximate the uncollided components and a low-resolution method is used to approximate the the collided component. After each time step, the kinetic distribution is reinitialized to be entirely uncollided. For this analysis, we consider a mono-energetic problem on a periodic domains, with constant material cross-sections of arbitrary size. To focus the analysis, we assume the uncollided equation is solved exactly and the collided part is approximated in angle via a spherical harmonic expansion ($\text{P}_N$ method). Using a non-standard set of semi-norms, we obtain estimates of the form $C(\varepsilon,\sigma,\Delta t)N^{-s}$ where $s\geq 1$ denotes the regularity of the solution in angle, $\varepsilon$ and $\sigma$ are scattering parameters, $\Delta t$ is the time-step before reinitialization, and $C$ is a complicated function of $\varepsilon$, $\sigma$, and $\Delta t$. These estimates involve analysis of the multiscale RTE that includes, but necessarily goes beyond, usual spectral analysis. We also compute error estimates for the monolithic $\text{P}_N$ method with the same resolution as the collided part in the hybrid. Our results highlight the benefits of the hybrid approach over the monolithic discretization in both highly scattering and streaming regimes.
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