Mapper is an unsupervised machine learning algorithm generalising the notion of clustering to obtain a geometric description of a dataset. The procedure splits the data into possibly overlapping bins which are then clustered. The output of the algorithm is a graph where nodes represent clusters and edges represent the sharing of data points between two clusters. However, several parameters must be selected before applying Mapper and the resulting graph may vary dramatically with the choice of parameters. We define an intrinsic notion of Mapper instability that measures the variability of the output as a function of the choice of parameters required to construct a Mapper output. Our results and discussion are general and apply to all Mapper-type algorithms. We derive theoretical results that provide estimates for the instability and suggest practical ways to control it. We provide also experiments to illustrate our results and in particular we demonstrate that a reliable candidate Mapper output can be identified as a local minimum of instability regarded as a function of Mapper input parameters.
This paper is organised as follows. In , we discuss some related work and its connections to the current paper. In , we give background on clustering stability required for the remainder of the paper. This allows us in to set out how the ideas of Ben-David and von Luxburg @cite_43 can be generalised to the Mapper setting. In particular, we introduce Mapper functions in Definition , which provide a new way of expressing Mapper outputs. Crucially, this is used to define a similarity metric between Mapper functions, @math in Definition . The Distance @math captures the structure of the whole Mapper output and leads to the definition of our notion of instability of Mapper (Definition ) with respect to a large class of clustering procedures. In we present an algorithm allowing us to experimentally obtain values of instability. This leads in section in to interesting experimental results which suggest that regions of relatively high instability correspond to structural changes in the Mapper output. Hence local minima of the instability function with respect to parameter choices are good candidates for parameter selection allowing us to study Mapper through variations of all the parameters.
@cite_43
2091344462 2963771441 2964237352 2964179816
Recent work [1], [2] has shown promising results in enabling robotic manipulation of deformable objects through learning from demonstrations. Their method computes a registration from training scene to test scene, and then applies an extrapolation of this registration to the training scene gripper motion to obtain the gripper motion for the test scene. The warping cost of scene-to-scene registrations is used to determine the nearest neighbor from a set of training demonstrations. Then once the gripper motion has been generalized to the test situation, they apply trajectory optimization [3] to plan for the robot motions that will track the predicted gripper motions. In many situations, however, the predicted gripper motions cannot be followed perfectly due to, for example, joint limits or obstacles. In this case the past work finds a path that minimizes deviation from the predicted gripper trajectory as measured by its Euclidean distance for position and angular distance for orientation. Measuring the error this way during the motion planning phase, however, ignores the underlying structure of the problem—namely the idea that rigid registrations are preferred to generalize from training scene to test scene. Deviating from the gripper trajectory predicted by the extrapolated registration effectively changes the warp induced by the registration in the part of the space where the gripper trajectories are. The main contribution of this paper is an algorithm that considers this effective final warp as the criterion to optimize for in a unified optimization that simultaneously considers the scene-to-scene warping and the robot trajectory (which were separated into two sequential steps by the past work). This results in an approach that adjusts to infeasibility in a way that adapts directly to the geometry of the scene and minimizes the introduction of additional warping cost. In addition, this paper proposes to learn the motion of the gripper pads, whereas past work considered the motion of a coordinate frame attached to the gripper as a whole. This enables learning more precise grasping motions. Our experiments, which consider the task of knot tying, show that both unified optimization and explicit consideration of gripper pad motion result in improved performance. Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^ -1 (I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero. Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs. This paper establishes the consistency of a family of graph-cut-based algorithms for clustering of data clouds. We consider point clouds obtained as samples of a ground-truth measure. We investigate approaches to clustering based on minimizing objective functionals defined on proximity graphs of the given sample. Our focus is on functionals based on graph cuts like the Cheeger and ratio cuts. We show that minimizers of these cuts converge as the sample size increases to a minimizer of a corresponding continuum cut (which partitions the ground truth measure). Moreover, we obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the consistency to hold. We provide results for two-way and for multiway cuts. Furthermore we provide numerical experiments that illustrate the results and explore the optimality of scaling in dimension two.