Multimodal imaging has transformed neuroscience research. While it presents unprecedented opportunities, it also imposes serious challenges. Particularly, it is di cult to combine the merits of the interpretability attributed to a simple association model with the exibility achieved by a highly adaptive nonlinear model. In this article, we propose an orthogonalized kernel debiased machine learning approach, which is built upon the Neyman orthogonality and a form of decomposition orthogonality, for multimodal data analysis. We target the setting that naturally arises in almost all multimodal studies, where there is a primary modality of interest, plus additional auxiliary modalities. We establish the root-Nconsistency and asymptotic normality of the estimated primary parameter, the semi-parametric estimation e ciency, and the asymptotic validity of the con dence band of the predicted primary modality e ect. Our proposal enjoys, to a good extent, both model interpretability and model exibility. It is also considerably di erent from the existing statistical methods for multimodal data integration, as well as the orthogonality-based methods for high-dimensional inferences. We demonstrate the e cacy of our method through both simulations and an application to a multimodal neuroimaging study of Alzheimer's disease. Supplementary materials for this article are available online.
We study the statistical properties of the dynamic trajectory of stochastic gradient descent (SGD). We approximate the mini-batch SGD and the momentum SGD as stochastic differential equations (SDEs). We exploit the continuous formulation of SDE and the theory of Fokker-Planck equations to develop new results on the escaping phenomenon and the relationship with large batch and sharp minima. In particular, we find that the stochastic process solution tends to converge to flatter minima regardless of the batch size in the asymptotic regime. However, the convergence rate is rigorously proven to depend on the batch size. These results are validated empirically with various datasets and models.
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