We address two-dimensional shape-based classification, considering shapes described by arbitrary sets of unlabeled points, or landmarks. This is relevant in practice because, in many applications, the points describing the shapes come from automatic processes, e.g., edge detection, thus without labels. Rather than attempting to compute point correspondences (a quagmire, when dealing with nontrivial shapes), we use what we call the analytic signature (ANSIG) of the shapes, a representation that has the key feature of being invariant to point labeling. Geometric transformations, such as translation, rotation, and scale, and different cardinality of point sets, are also dealt with by this representation. We demonstrate the capabilities of our representation with several shape classification experiments.