Convergence of Non-Symmetric Diffusion Processes on RCD spaces

We construct non-symmetric diffusion processes associated with Dirichlet forms consisting of uniformly elliptic forms and derivation operators with killing terms on RCD spaces by aid of non-smooth differential structures introduced by Gigli '16. After constructing diffusions, we investigate conservativeness and the weak convergence of the laws of diffusions in terms of a geometric convergence of the underling spaces and convergences of the corresponding coefficients.Comment: 41 pages. To appear in Calc. Var. PDEs. In the second version, the following have been modified: Section 2.3, 2.4, 2.5, 2.6 were added. Assumption 3.3, Proposition 3.4, Remark 3.5, and Example 3.7 were deleted. Example 7.2 was replaced with Corollary 7.2. Theorem 4.4 was modifie

On the ergodicity of interacting particle systems under number rigidity

In this paper, we provide relations among the following properties:(a) the tail triviality of a probability measure μ on the configuration space ϒ; (b) the finiteness of a suitable L2-transportation-type distance d¯ϒ; (c) the irreducibility of local μ-symmetric Dirichlet forms on ϒ. As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including sine2, Airy2, Besselα,2 (α ≥ 1), and Ginibre point processes. In particular, the case of the unlabelled Dyson Brownianmotion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role

An Algorithmic Framework for Computing Validation Performance Bounds by Using Suboptimal Models

Practical model building processes are often time-consuming because many different models must be trained and validated. In this paper, we introduce a novel algorithm that can be used for computing the lower and the upper bounds of model validation errors without actually training the model itself. A key idea behind our algorithm is using a side information available from a suboptimal model. If a reasonably good suboptimal model is available, our algorithm can compute lower and upper bounds of many useful quantities for making inferences on the unknown target model. We demonstrate the advantage of our algorithm in the context of model selection for regularized learning problems