Despite the tremendous progress in the estimation of generative models, the development of tools for diagnosing their failures and assessing their performance has advanced at a much slower pace. Recent developments have investigated metrics that quantify which parts of the true distribution are modeled well, and, on the contrary, what the model fails to capture, akin to precision and recall in information retrieval. In this paper, we present a general evaluation framework for generative models that measures the trade-off between precision and recall using Renyi divergences. Our framework provides a novel perspective on existing techniques and extends them to more general domains. As a key advantage, it allows for efficient algorithms that are directly applicable to continuous distributions directly without discretization. We further showcase the proposed techniques on a set of image synthesis models.
Exact geometric characterization of @cite_5 . @math which lie on the geodesic between @math and @math , shown as the red curve. Figure (b) shows a case where the metric defined by @cite_3 results in essentially perfect precision and recall. Arguably, in this case the precision and recall should both be low. We will now show that the two most prominent definitions of precision and recall are special cases of the presented framework.
@cite_5 @cite_3
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A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space @math , we define a sequence @math of non-negative real numbers by @math By construction, and the above result, this is a non-increasing sequence with limit @math . We study this sequence and its rates of decay with @math . We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of @math . Furthermore, we specifically analyze @math and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point. A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space X carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of X may be obtained by an appropriate finite sample (subset of X), and that persistence of any subset of X is well interleaved with the persistence of X. It follows that the persistence of X is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each (r>0 ) a density (s>0 ), so that for each s-dense sample (S X ) the corresponding fundamental group (and the first homology) of the Rips complex of S is isomorphic to the one of X, leading to an improved reconstruction result. We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle, acting as reparameterizations. In particular we investigate the L^2 inner product with respect to 1 plus curvature squared times arclength as the measure along a curve, applied to normal vector field to the curve. The curvature squared term acts as a sort of geometric Tikhonov regularization because, without it, the geodesic distance between any 2 distinct curves is 0, while in our case the distance is always positive. We give some lower bounds for the distance function, derive the geodesic equation and the sectional curvature, solve the geodesic equation with simple endpoints numerically, and pose some open questions. The space has an interesting split personality: among large smooth curves, all its sectional curvatures are positive or 0, while for curves with high curvature or perturbations of high frequency, the curvatures are negative. We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to Bourgain (1985) and Rao (1999). We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion (O ( _ X n ) , ) where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an (O ( ( _ X ) n ) ) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “low-dimensional,” improved bounds are achieved.