Hamilton-Jacobi (HJ) reachability analysis is a fundamental tool for the safety verification and control synthesis of nonlinear control systems. Classical HJ reachability analysis methods compute value functions over grids which discretize the continuous state space. Such approaches do not account for discretization errors and thus do not guarantee that the sets represented by the computed value functions over-approximate the backward reachable sets (BRS) when given avoid specifications or under-approximate the reach-avoid sets (RAS) when given reach-avoid specifications. We address this issue by presenting an algorithm for computing sound upper and lower bounds on the HJ value functions that guarantee the sound over-approximation of BRS and under-approximation of RAS. Additionally, we develop a refinement algorithm that splits the grid cells which could not be classified as within or outside the BRS or RAS given the computed bounds to obtain corresponding tighter bounds. We validate the effectiveness of our algorithm in two case studies.
We propose two novel approaches to address a critical problem of reach measurement across multiple media -- how to estimate the reach of an unobserved subset of buying groups (BGs) based on the observed reach of other subsets of BGs. Specifically, we propose a model-free approach and a model-based approach. The former provides a coarse estimate for the reach of any subset by leveraging the consistency among the reach of different subsets. Linear programming is used to capture the constraints of the reach consistency. This produces an upper and a lower bound for the reach of any subset. The latter provides a point estimate for the reach of any subset. The key idea behind the latter is to exploit the conditional independence model. In particular, the groups of the model are created by assuming each BG has either high or low reach probability in a group, and the weights of each group are determined through solving a non-negative least squares (NNLS) problem. In addition, we also provide a framework to give both confidence interval and point estimates by integrating these two approaches with training points selection and parameter fine-tuning through cross-validation. Finally, we evalu
In this paper, we propose an approach for synthesizing provable reach-avoid controllers, which drive a deterministic system operating in an unknown environment to safely reach a desired target set. The approach falls within the reachability analysis framework and is based on the computation of inner-approximations of controlled reach-avoid sets(CRSs). Given a target set and a safe set, the controlled reach-avoid set is the set of states such that starting from each of them, there exists at least one controller to ensure that the system can enter the target set while staying inside the safe set before the target hitting time. Therefore, the boundary of the controlled reach-avoid set acts as a barrier, which separating states capable of achieving the reach-avoid objective from those that are not, and thus the computed inner-approximation provides a viable space for the system to achieve the reach-avoid objective. Our approach for synthesizing reach-avoid controllers mainly consists of three steps. We first learn a safe set of states in the unknown environment from sensor measurements based on a support vector machine approach. Then, based on the learned safe set and target set, we co
Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or reach. However, these assumptions are often coupled with the requirement that the manifold be smooth, typically at least C^2 .In this paper, we prove that any manifold with positive reach can be approximated arbitrarily well by a C^$\infty$ manifold without significantly reducing the reach, by employing techniques from differential topology -partitions of unity and smoothing using convolution kernels. This result implies that nearly all theorems established for C^2 manifolds with a certain reach naturally extend to manifolds with the same reach, even if they are not C^2 , for free!
Autonomous mobile robots must maintain safety, but should not sacrifice performance, leading to the classical reach-avoid problem: find a trajectory that is guaranteed to reach a goal and avoid obstacles. This paper addresses the near danger case, also known as a narrow gap, where the agent starts near the goal, but must navigate through tight obstacles that block its path. The proposed method builds off the common approach of using a simplified planning model to generate plans, which are then tracked using a high-fidelity tracking model and controller. Existing approaches use reachability analysis to overapproximate the error between these models and ensure safety, but doing so introduces numerical approximation error conservativeness that prevents goal-reaching. The present work instead proposes a Piecewise Affine Reach-avoid Computation (PARC) method to tightly approximate the reachable set of the planning model. PARC significantly reduces conservativeness through a careful choice of the planning model and set representation, along with an effective approach to handling time-varying tracking errors. The utility of this method is demonstrated through extensive numerical experimen
Human motion generation is a challenging task that aims to create realistic motion imitating natural human behaviour. We focus on the well-studied behaviour of priming an object/location for pick up or put down - that is, the spotting of an object/location from a distance, known as gaze priming, followed by the motion of approaching and reaching the target location. To that end, we curate, for the first time, 23.7K gaze-primed human motion sequences for reaching target object locations from five publicly available datasets, i.e., HD-EPIC, MoGaze, HOT3D, ADT, and GIMO. We pre-train a text-conditioned diffusion-based motion generation model, then fine-tune it conditioned on goal pose or location, on our curated sequences. Importantly, we evaluate the ability of the generated motion to imitate natural human movement through several metrics, including the 'Reach Success' and a newly introduced 'Prime Success' metric. Tested on 5 datasets, our model generates diverse full-body motion, exhibiting both priming and reaching behaviour, and outperforming baselines and recent methods.
The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach explicitly is notoriously difficult. In this paper, we introduce a rigorous and practical method to compute a guaranteed lower bound for the reach of a submanifold described as the common zero-set of finitely many smooth functions, not necessarily polynomials. Our algorithm uses techniques from numerically verified proofs and is particularly suitable for high-performance parallel implementations. We illustrate the utility of this method through several applications. Of special note is a novel algorithm for computing the homology groups of planar curves, achieved by constructing a cubical complex that deformation retracts onto the curve--an approach potentially extendable to higher-dimensional manifolds. Additional applications include an improved comparison inequality between intrinsic and extrinsic distances for submanifolds of $\mathbb{R}^N$, lower bounds for the first eigenvalue of the Laplacian on algebraic varieties and explicit bounds on how
This paper contains three main results. Firstly, we give an elementary proof of the following statement: Let $M$ be a (closed, in both the geometrical and topological sense of the word) topological manifold embedded in $\mathbb{R}^d$. If $M$ has positive reach, then M can locally be written as the graph of a $C^{1,1}$ from the tangent space to the normal space. Conversely if $M$ can locally written as the graph of a $C^{1,1}$ function from the tangent space to the normal space, then $M$ has positive reach. The result was hinted at by Federer when he introduced the reach, and proved by Lytchak. Lytchak's proof relies heavily CAT(k)-theory. The proof presented here uses only basic results on homology. Secondly, we give optimal Lipschitz-constants for the derivative, in other words we give an optimal bound for the angle between tangent spaces in term of the distance between the points. This improves earlier results, that were either suboptimal or assumed that the manifold was $C^2$. Thirdly, we generalize a result by Aamari et al. which explains the how the reach is attained from the smooth setting to general sets of positive reach.
We study the reach (in the sense of Federer) of the natural isometric embedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space, where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if a point $x\in X$ can be joined to another point $y\in X$ by two minimizing geodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includes the cases where $X$ is a compact manifold or a non-simply connected one. On the other hand, we show that $\operatorname{reach}(X\subset W_p(X)) = \infty$ when $X$ is a CAT(0) space. The infinite reach enables us to examine the regularity of the projection map. Furthermore, we replicate these findings by considering the isometric embedding $X\hookrightarrow W_\vartheta(X)$ into an Orlicz--Wasserstein space, a generalization by Sturm of the classical Wasserstein space. Lastly, we establish the nullity of the reach for the isometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space of persistence diagrams equipped with the bottleneck distance.
We present a semi-analytical method for exact computation of the boundary of the reach set of a single-input controllable linear time invariant (LTI) system with given bounds on its input range. In doing so, we deduce a parametric formula for the boundary of the reach set of an integrator linear system with time-varying bounded input. This formula generalizes recent results on the geometry of an integrator reach set with time-invariant bounded input. We show that the same ideas allow for computing the volume of the LTI reach set.
Offline reinforcement learning aims to train agents from pre-collected datasets. However, this comes with the added challenge of estimating the value of behaviors not covered in the dataset. Model-based methods offer a potential solution by training an approximate dynamics model, which then allows collection of additional synthetic data via rollouts in this model. The prevailing theory treats this approach as online RL in an approximate dynamics model, and any remaining performance gap is therefore understood as being due to dynamics model errors. In this paper, we analyze this assumption and investigate how popular algorithms perform as the learned dynamics model is improved. In contrast to both intuition and theory, if the learned dynamics model is replaced by the true error-free dynamics, existing model-based methods completely fail. This reveals a key oversight: The theoretical foundations assume sampling of full horizon rollouts in the learned dynamics model; however, in practice, the number of model-rollout steps is aggressively reduced to prevent accumulating errors. We show that this truncation of rollouts results in a set of edge-of-reach states at which we are effectively
We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $\mathcal{C}^k$-smooth submanifold of $\mathbb{R}^D$, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand, and geodesic metric distortion on the other hand. The derived rates are adaptive, with rates depending on whether the reach of $M$ arises from curvature or from a bottleneck structure. In the process, we derive optimal geodesic metric estimation bounds.
The encoder network of an autoencoder is an approximation of the nearest point projection onto the manifold spanned by the decoder. A concern with this approximation is that, while the output of the encoder is always unique, the projection can possibly have infinitely many values. This implies that the latent representations learned by the autoencoder can be misleading. Borrowing from geometric measure theory, we introduce the idea of using the reach of the manifold spanned by the decoder to determine if an optimal encoder exists for a given dataset and decoder. We develop a local generalization of this reach and propose a numerical estimator thereof. We demonstrate that this allows us to determine which observations can be expected to have a unique, and thereby trustworthy, latent representation. As our local reach estimator is differentiable, we investigate its usage as a regularizer and show that this leads to learned manifolds for which projections are more often unique than without regularization.
We compute the reach, extremal curvature and volume of a tubular neighborhood for the Segre-Veronese variety intersected with the unit sphere.
Various problems in manifold estimation make use of a quantity called the reach, denoted by $τ\_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $\hatτ$ of $τ\_{M}$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $\mathbb{X}\_{n}$, $\hatτ(\mathbb{X}\_{n})$ is showed to achieve uniform expected loss bounds over a $\mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.
The reach of a set $M \subset \mathbb R^d$, also known as condition number when $M$ is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when $M$ is a manifold.
Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of $M$. Roughly speaking, the reach is a measure of a manifold's departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.
Signed distance fields (SDFs) are a widely used implicit surface representation, with broad applications in computer graphics, computer vision, and applied mathematics. To reconstruct an explicit triangle mesh surface corresponding to an SDF, traditional isosurfacing methods, such as Marching Cubes and and its variants, are typically used. However, these methods overlook fundamental properties of SDFs, resulting in reconstructions that exhibit severe oversmoothing and feature loss. To address this shortcoming, we propose a novel method based on a key insight: each SDF sample corresponds to a spherical region that must lie fully inside or outside the surface, depending on its sign, and that must be tangent to the surface at some point. Leveraging this understanding, we formulate an energy that gauges the degree of violation of tangency constraints by a proposed surface. We then employ a gradient flow that minimizes our energy, starting from an initial triangle mesh that encapsulates the surface. This algorithm yields superior reconstructions to previous methods, even with sparsely sampled SDFs. Our approach provides a more nuanced understanding of SDFs and offers significant improve
A mobile agent has to reach a target in the Euclidean plane. Both the agent and the target are modeled as points. In the beginning, the agent is at distance at most $D>0$ from the target. Reaching the target means that the agent gets at a {\em sensing distance} at most $r>0$ from it. The agent has a measure of length and a compass. We consider two scenarios: in the {\em static} scenario the target is inert, and in the {\em dynamic} scenario it may move arbitrarily at any (possibly varying) speed bounded by $v$. The agent has no information about the parameters of the problem, in particular it does not know $D$, $r$ or $v$. The goal is to reach the target at lowest possible cost, measured by the total length of the trajectory of the agent. Our main result is establishing the minimum cost (up to multiplicative constants) of reaching the target under both scenarios, and providing the optimal algorithm for the agent. For the static scenario the minimum cost is $Θ((\log D + \log \frac{1}{r}) D^2/r)$, and for the dynamic scenario it is $Θ((\log M + \log \frac{1}{r}) M^2/r)$, where $M=\max(D,v)$. Under the latter scenario, the speed of the agent in our algorithm grows exponentially
We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset eq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset eq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces.