Discrepancy and integration in function spaces with dominating mixed smoothness

Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey of discrepancy results and the calculation of the best known constant in Roth's Theorem. We give a larger class of point sets satisfying the optimal upper bounds than already known from Hinrichs for the plane and solve the problem in arbitrary dimension for certain parameters considering a celebrated constructions by Chen and Skriganov which are known to achieve optimal $L_2$-norm of the discrepancy function. Since those constructions are $b$-adic, we give $b$-adic characterizations of the spaces. Finally results for Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and for the integration error are concluded

L_p- and S_{p,q}^rB-discrepancy of (order 2) digital nets

Dick proved that all order $2$ digital nets satisfy optimal upper bounds of the $L_2$-discrepancy. We give an alternative proof for this fact using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds of the $S_{p,q}^r B$-discrepancy for a certain parameter range and enlarge that range for order $2$ digitals nets. $L_p$-, $S_{p,q}^r F$- and $S_p^r H$-discrepancy is considered as well

Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible $L_2$-norm of the discrepancy function. We consider the discrepancy function of the Chen-Skriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a $b$-adic generalization of the Haar system and corresponding characterizations of the Besov space norm. Results for further function spaces and integration errors are concluded.Comment: arXiv admin note: text overlap with arXiv:1109.454

Adversarial Learning of Mappings Onto Regularized Spaces for Biometric Authentication

We present AuthNet: a novel framework for generic biometric authentication which, by learning a regularized mapping instead of a classification boundary, leads to higher performance and improved robustness. The biometric traits are mapped onto a latent space in which authorized and unauthorized users follow simple and well-behaved distributions. In turn, this enables simple and tunable decision boundaries to be employed in order to make a decision. We show that, differently from the deep learning and traditional template-based authentication systems, regularizing the latent space to simple target distributions leads to improved performance as measured in terms of Equal Error Rate (EER), accuracy, False Acceptance Rate (FAR) and Genuine Acceptance Rate (GAR). Extensive experiments on publicly available datasets of faces and fingerprints confirm the superiority of AuthNet over existing methods

DiffuScene: Denoising Diffusion Models for Generative Indoor Scene Synthesis

We present DiffuScene for indoor 3D scene synthesis based on a novel scene configuration denoising diffusion model. It generates 3D instance properties stored in an unordered object set and retrieves the most similar geometry for each object configuration, which is characterized as a concatenation of different attributes, including location, size, orientation, semantics, and geometry features. We introduce a diffusion network to synthesize a collection of 3D indoor objects by denoising a set of unordered object attributes. Unordered parametrization simplifies and eases the joint distribution approximation. The shape feature diffusion facilitates natural object placements, including symmetries. Our method enables many downstream applications, including scene completion, scene arrangement, and text-conditioned scene synthesis. Experiments on the 3D-FRONT dataset show that our method can synthesize more physically plausible and diverse indoor scenes than state-of-the-art methods. Extensive ablation studies verify the effectiveness of our design choice in scene diffusion models.Comment: CVPR 202