We study routing for on-demand last-mile logistics with two crucial novel features: i) Multiple depots, optimizing where to pick-up every order, ii) Allowing vehicles to perform depot returns prior to being empty, thus adapting their routes to include new orders online. Both features result in shorter distances and more agile planning. We propose a scalable dynamic method to deliver orders as fast as possible. Following a rolling horizon approach, each time step the following is executed. First, define potential pick-up locations and identify which groups of orders can be transported together, with which vehicle and following which route. Then, decide which of these potential groups of orders will be executed and by which vehicle by solving an integer linear program. We simulate one day of service in Amsterdam that considers 10,000 requests, compare results to several strategies and test different scenarios. Results underpin the advantages of the proposed method

We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it $(\lambda,\tau)$-barycenter, where $\lambda$ is the inner regularization strength and $\tau$ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of $\lambda,\tau \geq 0$ and generalizes them. First, in spite of -- and in fact owing to -- being \emph{doubly} regularized, we show that our formulation is debiased for $\tau=\lambda/2$: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength $\lambda^2$ of entropic regularization, instead of $\max\{\lambda,\tau\}$ in general. We discuss this phenomenon for isotropic Gaussians where all $(\lambda,\tau)$-barycenters have closed form. Second, we show that for $\lambda,\tau>0$, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given $n$ samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate $n^{-1/2}$. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple \emph{noisy particle gradient descent} which, in the mean-field limit, converges globally at an exponential rate to the barycenter.

The two-stage object pose estimation paradigm first detects semantic keypoints on the image and then estimates the 6D pose by minimizing reprojection errors. Despite performing well on standard benchmarks, existing techniques offer no provable guarantees on the quality and uncertainty of the estimation. In this paper, we inject two fundamental changes, namely conformal keypoint detection and geometric uncertainty propagation, into the two-stage paradigm and propose the first pose estimator that endows an estimation with provable and computable worst-case error bounds. On one hand, conformal keypoint detection applies the statistical machinery of inductive conformal prediction to convert heuristic keypoint detections into circular or elliptical prediction sets that cover the groundtruth keypoints with a user-specified marginal probability (e.g., 90%). Geometric uncertainty propagation, on the other, propagates the geometric constraints on the keypoints to the 6D object pose, leading to a Pose UnceRtainty SEt (PURSE) that guarantees coverage of the groundtruth pose with the same probability. The PURSE, however, is a nonconvex set that does not directly lead to estimated poses and uncertainties. Therefore, we develop RANdom SAmple averaGing (RANSAG) to compute an average pose and apply semidefinite relaxation to upper bound the worst-case errors between the average pose and the groundtruth. On the LineMOD Occlusion dataset we demonstrate: (i) the PURSE covers the groundtruth with valid probabilities; (ii) the worst-case error bounds provide correct uncertainty quantification; and (iii) the average pose achieves better or similar accuracy as representative methods based on sparse keypoints.

In model-based reinforcement learning for safety-critical control systems, it is important to formally certify system properties (e.g., safety, stability) under the learned controller. However, as existing methods typically apply formal verification \emph{after} the controller has been learned, it is sometimes difficult to obtain any certificate, even after many iterations between learning and verification. To address this challenge, we propose a framework that jointly conducts reinforcement learning and formal verification by formulating and solving a novel bilevel optimization problem, which is differentiable by the gradients from the value function and certificates. Experiments on a variety of examples demonstrate the significant advantages of our framework over the model-based stochastic value gradient (SVG) method and the model-free proximal policy optimization (PPO) method in finding feasible controllers with barrier functions and Lyapunov functions that ensure system safety and stability.

When considering an invariant that gives a Bayesian generalization error, that is a real log canonical threshold, in general, papers replace a mean error function with a relatively simple polynomial whose real log canonical threshold corresponds to that of the mean error function, and obtain its real log canonical threshold by resolving its singularities through an algebraic operation called blow-up. Though it is known that the singularities of any polynomial can be resolved by a finite number of blow-up iterations, it is not clarified well whether or not it is possible to resolve singularities of a specific polynomial by applying a specific blow-up algorithm. Therefore this paper proposes a blow-up algorithm that can be applied to the polynomials called sum-of-products polynomials and proves that it halts. Furthermore, this paper considers real log canonical thresholds of sum-of-products polynomials by using the algorithm. First, this section explains the foundation of Bayesian learning theory and details the relation to a real log canonical threshold and blow-up. Then this section defines exclusive sum-of-products polynomials which is subject to previous studies and explains the novelty and utility of this paper.

Once developed for quantum theory, tensor networks have been established as a successful machine learning paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum machine learning to assess problems that classical computers are unable to solve efficiently. Their nature at the interface between physics and machine learning makes tensor networks easily deployable on quantum computers. In this review article, we shed light on one of the major architectures considered to be predestined for variational quantum machine learning. In particular, we discuss how layouts like MPS, PEPS, TTNs and MERA can be mapped to a quantum computer, how they can be used for machine learning and data encoding and which implementation techniques improve their performance.

In this work, we propose a simultaneous localization and mapping (SLAM) system using a monocular camera and Ultra-wideband (UWB) sensors. Our system, referred to as VRSLAM, is a multi-stage framework that leverages the strengths and compensates for the weaknesses of each sensor. Firstly, we introduce a UWB-aided 7 degree-of-freedom (scale factor, 3D position, and 3D orientation) global alignment module to initialize the visual odometry (VO) system in the world frame defined by the UWB anchors. This module loosely fuses up-to-scale VO and ranging data using either a quadratically constrained quadratic programming (QCQP) or nonlinear least squares (NLS) algorithm based on whether a good initial guess is available. Secondly, we provide an accompanied theoretical analysis that includes the derivation and interpretation of the Fisher Information Matrix (FIM) and its determinant. Thirdly, we present UWBaided bundle adjustment (UBA) and UWB-aided pose graph optimization (UPGO) modules to improve short-term odometry accuracy, reduce long-term drift as well as correct any alignment and scale errors. Extensive simulations and experiments show that our solution outperforms UWB/camera-only and previous approaches, can quickly recover from tracking failure without relying on visual relocalization, and can effortlessly obtain a global map even if there are no loop closures.

Predicting body fat can provide medical practitioners and users with essential information for preventing and diagnosing heart diseases. Hybrid machine learning models offer better performance than simple regression analysis methods by selecting relevant body measurements and capturing complex nonlinear relationships among selected features in modelling body fat prediction problems. There are, however, some disadvantages to them. Current machine learning. Modelling body fat prediction as a combinatorial single- and multi-objective optimisation problem often gets stuck in local optima. When multiple feature subsets produce similar or close predictions, avoiding local optima becomes more complex. Evolutionary feature selection has been used to solve several machine-learning-based optimisation problems. A fuzzy set theory determines appropriate levels of exploration and exploitation while managing parameterisation and computational costs. A weighted-sum body fat prediction approach was explored using evolutionary feature selection, fuzzy set theory, and machine learning algorithms, integrating contradictory metrics into a single composite goal optimised by fuzzy adaptive evolutionary feature selection. Hybrid fuzzy adaptive global learning local search universal diversity-based feature selection is applied to this single-objective feature selection-machine learning framework (FAGLSUD-based FS-ML). While using fewer features, this model achieved a more accurate and stable estimate of body fat percentage than other hybrid and state-of-the-art machine learning models. A multi-objective FAGLSUD-based FS-MLP is also proposed to analyse accuracy, stability, and dimensionality conflicts simultaneously. To make informed decisions about fat deposits in the most vital body parts and blood lipid levels, medical practitioners and users can use a well-distributed Pareto set of trade-off solutions.

We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliar variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and Strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.

We consider maximization of stochastic monotone continuous submodular functions (CSF) with a diminishing return property. Existing algorithms only guarantee the performance \textit{in expectation}, and do not bound the probability of getting a bad solution. This implies that for a particular run of the algorithms, the solution may be much worse than the provided guarantee in expectation. In this paper, we first empirically verify that this is indeed the case. Then, we provide the first \textit{high-probability} analysis of the existing methods for stochastic CSF maximization, namely PGA, boosted PGA, SCG, and SCG++. Finally, we provide an improved high-probability bound for SCG, under slightly stronger assumptions, with a better convergence rate than that of the expected solution. Through extensive experiments on non-concave quadratic programming (NQP) and optimal budget allocation, we confirm the validity of our bounds and show that even in the worst-case, PGA converges to $OPT/2$, and boosted PGA, SCG, SCG++ converge to $(1 - 1/e)OPT$, but at a slower rate than that of the expected solution.