Intelligent transport systems planning is a research area that offers a variety of problems that are interesting from a mathematical point of view. In the last decades, due to technology evolution, one could argue that such planning might be enhanced and optimized by the use of satellite navigation devices that offer real-time information. However, as expressed in [9] and analyzed in [6], the use of such technologies generates the displacement of congestion from one zone to another. It is believed that such effects are a consequence of not knowing the travel choice patterns of users and making no behavior prediction. Hence, it is still very useful and valuable to study a deterministic and static version of the problem.

A novel and intuitive nearest neighbours based clustering algorithm is introduced, in which a cluster is defined in terms of an equilibrium condition which balances its size and cohesiveness. The formulation of the equilibrium condition allows for a quantification of the strength of alignment of each point to a cluster, with these cluster alignment strengths leading naturally to a model selection criterion which renders the proposed approach fully automatable. The algorithm is simple to implement and computationally efficient, and produces clustering solutions of extremely high quality in comparison with relevant benchmarks from the literature. R code to implement the approach is available from https://github.com/DavidHofmeyr/ I. Introduction Clustering, or cluster analysis, is the task of partitioning a set of data into groups, or clusters, which are seen to be relatively more homogeneous than the data as a whole. Clustering is one of the fundamental data analytic tasks, and forms an integral component of exploratory data analysis. Clustering is also of arguably increasing relevance, as data are increasingly being collected/generated from automated processes, where typically very little prior knowledge is available, making exploratory methods a necessity. In the classical clustering problem there is no explicit information about how the data should be grouped, and various interpretations of how clusters of points may be defined have led to the development of a very large number of methods for identifying them. Almost universally, however, clusters are determined from the geometric properties of the data, with pairs of points which are near to one another typically being seen as likely to be in the same cluster and pairs which are distant more likely to be in different clusters.

This problem arises in a number of mechanism design contexts, where intervening on a system constitutes changing the rules of the game. Calculating counterfactual value requires reasoning about how rule changes affect equilibrium behavior of the agents. Under strong assumptions this counterfactual value is point-identified, but these assumptions are often implausible. The authors present a scheme for relaxing these assumptions, and characterizing the set of values that are compatible with the observed data under this relaxation. The relaxation of point-identification assumptions is presented in terms of a second game, which the authors call the Revelation Game.

We present a new perspective on online learning that we refer to as gradient equilibrium: a sequence of iterates achieves gradient equilibrium if the average of gradients of losses along the sequence converges to zero. In general, this condition is not implied by nor implies sublinear regret. It turns out that gradient equilibrium is achievable by standard online learning methods such as gradient descent and mirror descent with constant step sizes (rather than decaying step sizes, as is usually required for no regret). Further, as we show through examples, gradient equilibrium translates into an interpretable and meaningful property in online prediction problems spanning regression, classification, quantile estimation, and others. Notably, we show that the gradient equilibrium framework can be used to develop a debiasing scheme for black-box predictions under arbitrary distribution shift, based on simple post hoc online descent updates. We also show that post hoc gradient updates can be used to calibrate predicted quantiles under distribution shift, and that the framework leads to unbiased Elo scores for pairwise preference prediction.

In this work, we propose a geometric framework for analyzing mechanical manipulation, for example, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. We first review and show that the natural geometric setting is represented by the cotangent bundle and its Lagrangian submanifolds. These are standard concepts in geometric mechanics but usually presented within dynamical frameworks. We review the basic definitions from a static mechanics perspective and show how Lagrangian submanifolds are naturally derived from a first order analysis. Then, via a second order analysis, we derive the Hessian of total energy. As this is not necessarily positive-definite from a control perspective, we propose the use of the squared-Hessian for optimality measures, motivated by insights {derived from both mechanics (Gauss's Principle) and biology (Separation Principle)}. We conclude by showing how such methods can be applied, for example, to the simple case of an elastically driven pendulum. The example is simple enough to allow for analytical solution. However, an extension is further derived and numerically solved, which is more realistically connected with actual robotic manipulation problems.

Wire actuation in tendon-driven continuum robots enables the transmission of force from a distance, but it is understood that tension control problems can arise when a pulley is used to actuate two cables in a push-pull mode. This paper analyzes the relationship between angle of rotation, pressure, as well as variables of a single continuum unit in a quasi-static equilibrium. The primary objective of the quasi-static analysis was to output pressure and the analysis, given the tensions applied. Static equilibrium condition was established, and the bisection method was carried out for the angle of rotation. The function for the bisection method considered pressure-induced forces, friction forces, and weight. {\theta} was 17.14{\deg}, and p was 405.6 Pa when Tl and Ts were given the values of 1 N and 2 N, respectively. The results seemed to be consistent with the preliminary design specification, calling for further simulations and experiments.

In this work, we comprehensively reveal the learning dynamics of neural network with normalization, weight decay (WD), and SGD (with momentum), named as Spherical Motion Dynamics (SMD). Most related works study SMD by focusing on "effective learning rate" in "equilibrium" condition, where weight norm remains unchanged. However, their discussions on why equilibrium condition can be reached in SMD is either absent or less convincing. Our work investigates SMD by directly exploring the cause of equilibrium condition. Specifically, 1) we introduce the assumptions that can lead to equilibrium condition in SMD, and prove that weight norm can converge at linear rate with given assumptions; 2) we propose "angular update" as a substitute for effective learning rate to measure the evolving of neural network in SMD, and prove angular update can also converge to its theoretical value at linear rate; 3) we verify our assumptions and theoretical results on various computer vision tasks including ImageNet and MSCOCO with standard settings. Experiment results show our theoretical findings agree well with empirical observations.

In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new "equilibrium" property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

In typical machine learning tasks and applications, it is necessary to obtain or create large labeled datasets in order to to achieve high performance. Unfortunately, large labeled datasets are not always available and can be expensive to source, creating a bottleneck towards more widely applicable machine learning. The paradigm of weak supervision offers an alternative that allows for integration of domain-specific knowledge by enforcing constraints that a correct solution to the learning problem will obey over the output space. In this work, we explore the application of this paradigm to 2-D physical systems governed by non-linear differential equations. We demonstrate that knowledge of the partial differential equations governing a system can be encoded into the loss function of a neural network via an appropriately chosen convolutional kernel. We demonstrate this by showing that the steady-state solution to the 2-D heat equation can be learned directly from initial conditions by a convolutional neural network, in the absence of labeled training data. We also extend recent work in the progressive growing of fully convolutional networks to achieve high accuracy (< 1.5% error) at multiple scales of the heat-flow problem, including at the very large scale (1024x1024). Finally, we demonstrate that this method can be used to speed up exact calculation of the solution to the differential equations via finite difference.