Mean ergodic theorems on norming dual pairs

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair (C_b(E), M(E)) all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.Comment: 18 pages, 1 figur

HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inference

A large proportion of recent invertible neural architectures is based on a coupling block design. It operates by dividing incoming variables into two sub-spaces, one of which parameterizes an easily invertible (usually affine) transformation that is applied to the other. While the Jacobian of such a transformation is triangular, it is very sparse and thus may lack expressiveness. This work presents a simple remedy by noting that (affine) coupling can be repeated recursively within the resulting sub-spaces, leading to an efficiently invertible block with dense triangular Jacobian. By formulating our recursive coupling scheme via a hierarchical architecture, HINT allows sampling from a joint distribution p(y,x) and the corresponding posterior p(x|y) using a single invertible network. We demonstrate the power of our method for density estimation and Bayesian inference on a novel data set of 2D shapes in Fourier parameterization, which enables consistent visualization of samples for different dimensionalities

Using an n-zone TDI camera for acquisition of multiple images with different illuminations in a single scan

For fast scanning of large surfaces with microscopic resolution or for scanning of roll-fed material, TDI line scan cameras are typically used. TDI cameras sum up the light collected in adjacent lines of the image sensor synchronous to the motion of the object. Therefore TDI cameras have much higher sensitivity than standard line cameras. For many applications in the field of optical inspection more than one image of the object under test are needed with different illumination situations. For this task we need either more than one TDI camera or we have to scan the object several times in different illumination situations. Both solutions are often not entirely satisfying. In this paper we present a solution of this task using a modified TDI sensor consisting of three or more separate TDI zones. With this n-zone TDI camera it is possible to acquire multiple images with different illuminations in a single scan. In a simulation we demonstrate the principle of operation of the camera and the necessary image preprocessing which can be implemented in the frame grabber hardware

On some normability conditions

Various normability conditions of locally convex spaces (including Vogt interpolation classes DN and as well as quasi- and asymptotic normability) are investigated. In particular, it is shown that on the class of Schwartz spaces the property of asymptotic normability coincides with the property GS , which is a natural generalization of Gelfand-Shilov countable normability (cf. [9, 25], where the metrizable case was treated). It is observed also that there are certain natural duality relationships among some of normability conditions

Adaptive least-squares space-time finite element methods

We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations

ProDAS: Probabilistic Dataset of Abstract Shapes

We introduce a novel and comprehensive dataset, named ProDAS, which enables the generation of diverse objects with varying shape, size, rotation, and texture/color through a latent factor model. ProDAS offers complete access and control over the data generation process, serving as an ideal environment for investigating disentanglement, causal discovery, out-of-distribution detection, and numerous other research questions. We provide pre-defined functions for the important cases of creating distinct and interconnected distributions, allowing the investigation of distribution shifts and other intriguing applications. The library can be found at https://github.com/XarwinM/ProDAS

A topological sampling theorem for Robust boundary reconstruction and image segmentation

AbstractExisting theories on shape digitization impose strong constraints on admissible shapes, and require error-free data. Consequently, these theories are not applicable to most real-world situations. In this paper, we propose a new approach that overcomes many of these limitations. It assumes that segmentation algorithms represent the detected boundary by a set of points whose deviation from the true contours is bounded. Given these error bounds, we reconstruct boundary connectivity by means of Delaunay triangulation and α-shapes. We prove that this procedure is guaranteed to result in topologically correct image segmentations under certain realistic conditions. Experiments on real and synthetic images demonstrate the good performance of the new method and confirm the predictions of our theory

Positive Difference Distribution for Image Outlier Detection using Normalizing Flows and Contrastive Data

Detecting test data deviating from training data is a central problem for safe and robust machine learning. Likelihoods learned by a generative model, e.g., a normalizing flow via standard log-likelihood training, perform poorly as an outlier score. We propose to use an unlabelled auxiliary dataset and a probabilistic outlier score for outlier detection. We use a self-supervised feature extractor trained on the auxiliary dataset and train a normalizing flow on the extracted features by maximizing the likelihood on in-distribution data and minimizing the likelihood on the contrastive dataset. We show that this is equivalent to learning the normalized positive difference between the in-distribution and the contrastive feature density. We conduct experiments on benchmark datasets and compare to the likelihood, the likelihood ratio and state-of-the-art anomaly detection methods