Conditional mean independence (CMI) testing is crucial for statistical tasks including model determination and variable importance evaluation. In this work, we introduce a novel population CMI measure and a bootstrap-based testing procedure that utilizes deep generative neural networks to estimate the conditional mean functions involved in the population measure. The test statistic is thoughtfully constructed to ensure that even slowly decaying nonparametric estimation errors do not affect the asymptotic accuracy of the test. Our approach demonstrates strong empirical performance in scenarios with high-dimensional covariates and response variable, can handle multivariate responses, and maintains nontrivial power against local alternatives outside an $n^{-1/2}$ neighborhood of the null hypothesis. We also use numerical simulations and real-world imaging data applications to highlight the efficacy and versatility of our testing procedure.

Motivated by the computation of the non-parametric maximum likelihood estimator (NPMLE) and the Bayesian posterior in statistics, this paper explores the problem of convex optimization over the space of all probability distributions. We introduce an implicit scheme, called the implicit KL proximal descent (IKLPD) algorithm, for discretizing a continuous-time gradient flow relative to the Kullback–Leibler (KL) divergence for minimizing a convex target functional. We show that IKLPD converges to a global optimum at a polynomial rate from any initialization; moreover, if the objective functional is strongly convex relative to the KL divergence, for example, when the target functional itself is a KL divergence as in the context of Bayesian posterior computation, IKLPD exhibits globally exponential convergence. Computationally, we propose a numerical method based on normalizing flow to realize IKLPD. Conversely, our numerical method can also be viewed as a new approach that sequentially trains a normalizing flow for minimizing a convex functional with a strong theoretical guarantee.

When the target magnetic structures have clear interfaces and significant petrophysical contrasts to the surrounding strata, the focusing (or sharp-boundary) inversion methods are preferred over the smooth inversion methods. The result of focusing inversions depends on the types of the focusing model constraint (stabilizer). In this study, we propose an adjustable exponential minimum support (AEMS) stabilizer, which is an extension of the previously introduced exponential minimum support (EMS) stabilizer. AEMS is similar to EMS, but it includes an adjustable focusing scheme to control the sharpness of the resulting structure. Using synthetic models, we compared the inversion results obtained with the AEMS method to those obtained by smoothing stabilizers and other focusing stabilizers. AEMS inversion results have a more accurate physical-property distribution and are closer to the true models, indicating an improved model recovery from the other inversion methods. The improvement is mainly attributed to the adjustable scheme of the focusing parameter in the AEMS method. Then, we inverted a field dataset using the AEMS inversion method to obtain a new reference model for copper-gold porphyry deposit detection and interpretation.