We consider the conjecture proposed in Matsumoto, Zhang and Schiebinger (2022) suggesting that optimal transport with quadratic regularisation can be used to construct a graph whose discrete Laplace operator converges to the Laplace--Beltrami operator. We derive first order optimal potentials for the problem under consideration and find that the resulting solutions exhibit a surprising resemblance to the well-known Barenblatt--Prattle solution of the porous medium equation. Then, relying on these first order optimal potentials, we derive the pointwise $L^2$-limit of such discrete operators built from an i.i.d. random sample on a smooth compact manifold. Simulation results complementing the limiting distribution results are also presented.

Were you unable to attend Transform 2022? Check out all of the summit sessions in our on-demand library now! A few years ago, a New Jersey man was arrested for shoplifting and spent ten days in jail. He was actually 30 miles away during the time of the incident; police facial recognition software wrongfully identified him. Facial recognition's race and gender failings are well known.

Clinicians and surgeons are increasingly using medical devices based on artificial intelligence. These AI devices, which rely on data-driven algorithms to inform health care decisions, presently aid in diagnosing cancers, heart conditions and diseases of the eye, with many more applications on the way. In a new study, Stanford faculty discuss sex, gender and race bias in medical technologies. Pulse oximeters, for example, are more likely to incorrectly report blood gas levels in dark-skinned individuals and in women. Given this surge in AI, two Stanford University faculty members are calling for efforts to ensure that this technology does not exacerbate existing heath care disparities.

Clustering of data sets is a standard problem in many areas of science and engineering. The method of spectral clustering is based on embedding the data set using a kernel function, and using the top eigenvectors of the normalized Laplacian to recover the connected components. We study the performance of spectral clustering in recovering the latent labels of i.i.d. samples from a finite mixture of nonparametric distributions. The difficulty of this label recovery problem depends on the overlap between mixture components and how easily a mixture component is divided into two nonoverlapping components. When the overlap is small compared to the indivisibility of the mixture components, the principal eigenspace of the population-level normalized Laplacian operator is approximately spanned by the square-root kernelized component densities. In the finite sample setting, and under the same assumption, embedded samples from different components are approximately orthogonal with high probability when the sample size is large. As a corollary we control the fraction of samples mislabeled by spectral clustering under finite mixtures with nonparametric components.