Poster Session

Convolutional Neural Network (CNN) based deep learning schemes are gaining acceptability and deployment in a variety of computer vision and image analysis applications and widely perceived as achieving optimal performance in detecting and classifying objects/patterns in images. Shortcomings of CNN include high computational cost, overfitting to the training data, requiring extremely large training image datasets, and above all its black-box style of decision making with no informative explanation. The latter is a major obstacle to deployments for medical image diagnostics. Conventional machine learning approaches relay on image texture analysis to achieve high, but not optimal, performances and their decisions can be justified quantitatively. The emergence of the new paradigm of Topological Data Analysis (TDA), to deal with the growing challenges of Big Data applications, has recently been adopted to design and develop innovative image analysis schemes that automatically construct filtrations of topological shapes of image texture and use the TDA tool of persistent homology (PH) to distinguish different image classes. We initiated a study of the effect of CNN convolution layers on the discriminating strength of PH features for automatically extracted sets of texture landmarks in Ultrasound scan images of human Bladder. In particular, we estimate the effect of the pre-trained AlexNet convolutional layers filters on PH of uniform LBP landmark groups for distinguishing Benign masses from Malignant ones. We shall demonstrate, that the condition number of the pre-trained filters influences the discriminatory power of PH of these texture landmarks post convolution, which can be used for filter pruning while maintaining classification accuracy and improving efficiency.

The focal point of our research lies in the intersection of persistent homology, discrete Morse theory and Delaunay complexes. We are interested in algebraic and geometric properties of Delaunay filtrations and the application to surface reconstruction problems. Moreover, we devise an algorithm for the computation of a discrete vector field consistent with the Delaunay function.

The reach of a submanifold is a crucial invariant in geometric inference. This research provides a new estimator with improved convergence rates.

M. Farber proved that the configuration space of a tree on two points is homotopically equivalent to a banana graph on \sum (d(v)-1)(d(v)-2) edges. I give a generalization of this result for configuration spaces of trees on three points using discrete Morse theory. Also, I shall talk about the homotopy type of the 1-skeleton of the configuration space of trees on n points for any n.

Low-Rank Matrix Completion (MC) techniques are utilized in many fields of Machine Learning such as Manifold Learning (e.g. Dimension Reduction), Recommendation Systems (e.g. Netflix Problem), Cyber Security (e.g. Anomaly Detection), etc. Here, we apply low-rank MC techniques to reconstruct corrupted patches of images. A correlated sample of images such as handwritten digits and images of a person's face are naturally low-rank. This property is the salient characteristic that allows us a successful implementation of MC tools. The MC tool that we apply is an extension of Robust Principal Component Analysis (RPCA). Given an input matrix RPCA generates two output matrices, one with low rank features and one with sparse features. In our application the input matrix is formed from concatenation of vectorized images into a single matrix. We analyze the low-rank component to fill corrupted patches of images. Not only applying RPCA for the whole image-set, we also experiment with condensing subsets of images of big image-sets which mostly resemble the uncorrupted portion of the corrupted images, by learning from uncorrupted images. This sample approach both increases computational efficiency and improves performance of MC over the image-sets which include images with potentially large dissimilarity to the corrupted image. The results demonstrate that our method performs with high fidelity when it was applied to real-life datasets.

Persistent homology may offer a new method of analysing patterns formed from Turing instability in Reaction Diffusion equations. We use cubical complex filtrations to anlayse typical "spot" and "stripe" patterns, and seek a topological characterisation of this difference.

We present a parameterized algorithm for solving the Homology Localization Problem (HLP). The HLP is a computational problem where the task is to find a cycle with the most concise geometric measure that represents a given homology class in a simplicial complex. This problem is known to be NP-hard and so it is unlikely that someone will come up with an exact algorithm that runs in time polynomial measured in input size any time soon (if at all). We have therefore designed a fixed parameter tractable (FPT) algorithm for the HLP so that we may circumvent the hardness issue in practice. Our algorithm is exact and it has a runtime that is linear in its dependency on the input size. This comes at the cost of there being an exponential dependency of the runtime on a structural parameter. Our particular parameter can be seen as being a measure how connected the simplicial complex is. Though the algorithm we present do not (theoretically) perform better than the brute force algorithm in the worst case, we can solve the HLP much more efficiently whenever the parameter is small.

The theory of non-trivial abstract kernels and extensions for many kinds of algebraic objects such as groups, rings and graded rings, associative algebras, Lie algebras, restricted Lie algebras, DG-algebras and DG-Lie algebras has been widely studied since 1940’s. Gerhard Hochschild writes the earliest paper that treats associative algebra as an embryonic type in the series of kernel problems. He proves the structure theorem of kernels by presenting lots of sophisticated relations that may be least-understandable to the readers today. In this presentation we shall review the problem and elaborate the involved treatment and describe the Lie algebra case. Furthermore, we rely on the universal enveloping algebra of Lie algebra to reduce the difficulty of a direct construction for the derivation algebras.

Title: Continuous and Discrete Functions on Tessellations and Mosaics

The Voronoi tessellation in R^d is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise quadratic functions is piecewise linear, and that all three functions have the same critical points and values. The proof relies on tools from discrete and continuous mathematics, which we develop in parallel and combine to present a geometric view of Alexander duality.

This is a joint work with Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. The preprint can be found at http://pub.ist.ac.at/~edels/Papers/2020-P-02-AlexanderDuality.pdf

We introduce higher simplicial complexity of a simplicial complex K and higher combinatorial complexity of a finite space P. We relate higher simplicial complexity with higher topological complexity of | K | and higher combinatorial complexity with higher simplicial complexity of the order complex of P.

The research focus on understanding the (super) level sets of conditional random fields. Gaussian Process regression is used extensively in Statistics and Machine Learning; understanding the topological and geometric structure of excursion or level sets are of use in quantifying uncertainty.

In this poster, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, we restrict simplicial data to local matrices, while merging homological knowledge by means of the higher spectral sequence differentials. This approach has the added advantage that one can recover local information relative to the used cover. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data. To conclude, we will present PerMaViss, a python3 library that implements these ideas, together with some experimental results.