We consider noisy Euclidean traveling salesman problems in the plane, which are random combinatorial problems with underlying structure. Gibbs sampling is used to compute average trajectories, which estimate the underlying structure common to all instances. This procedure requires identifying the exact relationship between permutations and tours. In a learning setting, the average trajec(cid:173) tory is used as a model to construct solutions to new instances sampled from the same source. Experimental results show that the average trajectory can in fact estimate the underlying structure and that overfitting effects occur if the trajectory adapts too closely to a single instance.

We present a modified version of the perceptron learning algorithm (PLA) which solves semidefinite programs (SDPs) in polynomial time. The algorithm is based on the following three observations: (i) Semidefinite programs are linear programs with infinitely many (linear) constraints; (ii) every linear program can be solved by a sequence of constraint satisfaction problems with linear constraints; (iii) in general, the perceptron learning algorithm solves a constraint satisfaction problem with linear constraints in finitely many updates. Combining the PLA with a probabilistic rescaling algorithm (which, on average, increases the size of the feasable region) results in a prob- abilistic algorithm for solving SDPs that runs in polynomial time. We present preliminary results which demonstrate that the algo- rithm works, but is not competitive with state-of-the-art interior point methods.

This paper proposes constraint propagation relaxation (CPR), a probabilistic approach to classical constraint propagation that provides another view on the whole parametric family of survey propagation algorithms SP(ρ), ranging from belief propagation (ρ 0) to (pure) survey propagation(ρ 1).

We propose a novel inference framework for finding maximal cliques in a weighted graph that satisfy hard constraints. The constraints specify the graph nodes that must belong to the solution as well as mutual exclusions of graph nodes, i.e., sets of nodes that cannot belong to the same solution. The proposed inference is based on a novel particle filter algorithm with state permeations. We apply the inference framework to a challenging problem of learning part-based, deformable object models. Two core problems in the learning framework, matching of image patches and finding salient parts, are formulated as two instances of the problem of finding maximal cliques with hard constraints.

We present a recurrent neuronal network, modeled as a continuous-time dynamical system, that can solve constraint satisfaction problems. Discrete variables are represented by coupled Winner-Take-All (WTA) networks, and their values are encoded in localized patterns of oscillations that are learned by the recurrent weights in these networks. Constraints over the variables are encoded in the network connectivity. Although there are no sources of noise, the network can escape from local optima in its search for solutions that satisfy all constraints by modifying the effective network connectivity through oscillations. If there is no solution that satisfies all constraints, the network state changes in a pseudo-random manner and its trajectory approximates a sampling procedure that selects a variable assignment with a probability that increases with the fraction of constraints satisfied by this assignment.

We study a class of generalized linear programs (GLP) in a large-scale setting, which includes simple, possibly nonsmooth convex regularizer and simple convex set constraints. By reformulating (GLP) as an equivalent convex-concave min-max problem, we show that the linear structure in the problem can be used to design an efficient, scalable first-order algorithm, to which we give the name \emph{Coordinate Linear Variance Reduction} (\textsc{clvr}; pronounced "clever"). \textsc{clvr} yields improved complexity results for (GLP) that depend on the max row norm of the linear constraint matrix in (GLP) rather than the spectral norm. When the regularization terms and constraints are separable, \textsc{clvr} admits an efficient lazy update strategy that makes its complexity bounds scale with the number of nonzero elements of the linear constraint matrix in (GLP) rather than the matrix dimensions. On the other hand, for the special case of linear programs, by exploiting sharpness, we propose a restart scheme for \textsc{clvr} to obtain empirical linear convergence. Then we show that Distributionally Robust Optimization (DRO) problems with ambiguity sets based on both $f$-divergence and Wasserstein metrics can be reformulated as (GLPs) by introducing sparsely connected auxiliary variables. We complement our theoretical guarantees with numerical experiments that verify our algorithm's practical effectiveness, in terms of wall-clock time and number of data passes.

The importance of ground Mobile Robots (MRs) and Unmanned Aerial Vehicles (UAVs) within the research community, industry, and society is growing fast. Many of these agents are nowadays equipped with communication systems that are, in some cases, essential to successfully achieve certain tasks. In this context, we have begun to witness the development of a new interdisciplinary research field at the intersection of robotics and communications. This research field has been boosted by the intention of integrating UAVs within the 5G and 6G communication networks. This research will undoubtedly lead to many important applications in the near future. Nevertheless, one of the main obstacles to the development of this research area is that most researchers address these problems by oversimplifying either the robotics or the communications aspect. This impedes the ability of reaching the full potential of this new interdisciplinary research area. In this tutorial, we present some of the modelling tools necessary to address problems involving both robotics and communication from an interdisciplinary perspective. As an illustrative example of such problems, we focus in this tutorial on the issue of communication-aware trajectory planning.

Cyber-physical systems (CPS) like autonomous vehicles, that utilize learning components, are often sensitive to noise and out-of-distribution (OOD) instances encountered during runtime. As such, safety critical tasks depend upon OOD detection subsystems in order to restore the CPS to a known state or interrupt execution to prevent safety from being compromised. However, it is difficult to guarantee the performance of OOD detectors as it is difficult to characterize the OOD aspect of an instance, especially in high-dimensional unstructured data. To distinguish between OOD data and data known to the learning component through the training process, an emerging technique is to incorporate variational autoencoders (VAE) within systems and apply classification or anomaly detection techniques on their latent spaces. The rationale for doing so is the reduction of the data domain size through the encoding process, which benefits real-time systems through decreased processing requirements, facilitates feature analysis for unstructured data and allows more explainable techniques to be implemented. This study places probably approximately correct (PAC) based guarantees on OOD detection using the encoding process within VAEs to quantify image features and apply conformal constraints over them. This is used to bound the detection error on unfamiliar instances with user-defined confidence. The approach used in this study is to empirically establish these bounds by sampling the latent probability distribution and evaluating the error with respect to the constraint violations that are encountered. The guarantee is then verified using data generated from CARLA, an open-source driving simulator.

Synthesizing 3D human avatars interacting realistically with a scene is an important problem with applications in AR/VR, video games and robotics. Towards this goal, we address the task of generating a virtual human -- hands and full body -- grasping everyday objects. Existing methods approach this problem by collecting a 3D dataset of humans interacting with objects and training on this data. However, 1) these methods do not generalize to different object positions and orientations, or to the presence of furniture in the scene, and 2) the diversity of their generated full-body poses is very limited. In this work, we address all the above challenges to generate realistic, diverse full-body grasps in everyday scenes without requiring any 3D full-body grasping data. Our key insight is to leverage the existence of both full-body pose and hand grasping priors, composing them using 3D geometrical constraints to obtain full-body grasps. We empirically validate that these constraints can generate a variety of feasible human grasps that are superior to baselines both quantitatively and qualitatively. See our webpage for more details: https://flex.cs.columbia.edu/.

In this work, we introduce a collision model specifically tailored for the simulation of inextensible textiles. The model considers friction, contacts, and inextensibility constraints all at the same time without any decoupling. Self-collisions are modeled in a natural way that allows considering the thickness of cloth without introducing unwanted oscillations. The discretization of the equations of motion leads naturally to a sequence of quadratic problems with inequality and equality constraints. In order to solve these problems efficiently, we develop a novel active-set algorithm that takes into account past active constraints to accelerate the resolution of unresolved contacts. We put to a test the developed collision procedure with diverse scenarios involving static and dynamic friction, sharp objects, and complex-topology folding sequences. Finally, we perform an experimental validation of the collision model by comparing simulations with recordings of real textiles as given by a $\textit{Motion Capture System}$. The results are very accurate, having errors around 1 cm for DIN A2 textiles (42 x 59.4 cm) even in difficult scenarios involving fast and strong hits with a rigid object.