We prove that the image of period map is algebraic, as conjectured by Griffiths.

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In a Data-Generating Experiment, the observed sample, x, has intractable or unavailable c.d.f., F_θ, and θ’s statistical nature is unknown; θ is element of metric space (Θ, dΘ). Matching estimates of θ are introduced, learning from the “best” x-matches with samples X^∗ from F_{θ^∗}, θ^∗ ∈ Θ. Under mild conditions, these nonparametric estimates are uniformly consistent and the upper bounds on their rates of convergence in probability have the same rate and depend on the Kolmogorov entropies of an increasing sequence of sets covering Θ. When Θ ⊆ Rm and the observations are i.i.d. the upper bounds can be, \sqrt{log n}/\sqrt{n} when m is known, and \sqrt{m_n·log n}/\sqrt{n} when m is unknown; m ≥ 1, m_n ↑ ∞ at a desired rate. Upper bounds can also be obtained for dependent observations. These rates hold for observations in R^d, complementing recent results obtained for real, i.i.d. observations, under stronger assumptions and using weak probability distances; d ≥ 1. In simulations, the Matching estimates are successful for the mixture of 2 normals and for Tukey’s (a, b, g, h) and the (a, b, g, k) models. Computers’ evolution will allow for more and faster comparisons, resulting in improved Matching estimates for universal use in Machine Learning.

In shape analysis, finding an optimal 1-1 correspondence between 3D surfaces within a large class of admissible bijective mappings is of great importance. Such a process is called surface registration. The difficulty lies in the fact that the space of all surface diffeomorphisms is a complicated functional space, making it challenging to exhaustively search for the best mapping. To tackle this problem, we propose a simple representation of bijective surface maps using Beltrami coefficients (BCs)—complex-valued functions defined on surfaces with supremum norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of BCs. Hence, every bijective surface map may be represented by a unique BC. Conversely, given a BC, we can reconstruct the unique surface map associated with it using the Beltrami Holomorphic flow (BHF) method. Using BCs to represent surface maps is advantageous because it is a much simpler functional space, which captures many essential features of a surface map. By adjusting BCs, we equivalently adjust surface diffeomorphisms to obtain the optimal map with desired properties. More specifically, BHF gives us the variation of the associated map under the variation of BC. Using this, a variational problem over the space of surface diffeomorphisms can be easily reformulated into a variational problem over the space of BCs. This makes the minimization procedure much easier. More importantly, the diffeomorphic property is always preserved. We test our method on synthetic examples and real medical applications. Experimental results demonstrate the effectiveness of our proposed algorithm for surface registration.

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