We show that every Mori dream space of globally Fregular type is of Fano type. As an application, we give a characterization of varieties of Fano type in terms of the singularities of their Cox rings.
Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M -objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis. PreliminariesLet us introduce notations for defining simplicial problems and review an existing method of Bézier curve fitting. Simplicial ProblemA multi-objective optimization problem is denoted by its objective map f = (f 1 , . . . , f M ) : X → R M . Let I := { 1, . . . , M } be the index set of objective functions and ∆ M −1 := (t 1 , . . . , t M ) ∈ R M 0 ≤ t m , m∈M t m = 1 be the standard simplex in R M . For each non-empty subset J ⊆ I, we call∆ J := (t 1 , . . . , t M ) ∈ ∆ M −1 t m = 0 (m ∈ J) the J-face of ∆ M −1 andThe problem class we are interested in is as follows:We call such φ and f • φ a triangulation of the Pareto set X * (f ) and the Pareto front f X * (f ), respectively. For each non-empty subset J ⊆ I, we call X * (f J ) the J-face of X * (f ) and f X * (f J ) the J-face of f X * (f ). For each 0 ≤ m ≤ M − 1, we callthe m-skeleton of X * (f ) and f X * (f ), respectively.By definition, any subproblem of a simplicial problem is again simplicial. As shown in Figure 1b, the Pareto sets forms a simplex. The second condition asserts that f | X * (f ) : X * (f ) → R M is a C 0 -embedding. This means that the Pareto front of each subproblem is homeomorphic to its Pareto set as shown in Figure 1c. Therefore, the Pareto set/front of an M -objective simplicial problem can be identified with a curved (M − 1)-simplex. We can find its J-face by solving the J-subproblem.
In this paper, we develop a theory about the relationship between Ginvariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine transformations are G-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.
The B'ezier simplex fitting is a novel data modeling technique which utilizes geometric structures of data to approximate the Pareto set of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from skeletons of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a single object. In this paper, we analyze the asymptotic risks of those B'ezier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a B'ezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a B'ezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a B'ezier simplex of degree three where the all-at-once fitting gets an advantage.
Given a local domain (R, m) of prime characteristic that is a homomorphic image of a Gorenstein ring, Huneke and Lyubeznik proved that there exists a module-finite extension domain S such that the induced map on local cohomology modules H i m (R) −→ H i m (S) is zero for each i < dim R. We prove that the extension S may be chosen to be generically Galois, and analyze the Galois groups that arise.
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