Invited Speakers

Spectral Laplace-Beltrami Wavelets and Geometric Convolutional Neural Network for Signal Processing and Classification

Abstract: The Laplace-Beltrami operator is a generalization of the Euclidean representation of the Laplace operator to an arbitrary Riemannian manifold. It is a self-adjoint operator and its eigenfunctions form a complete set of real-valued orthonormal basis functions. In this talk, I will introduce spectral Laplace-Beltrami wavelets and its computational algorithm. I will then demonstrate its use for smoothing and classification of the data defined on smooth surfaces embedded in the 3-D Euclidean space. Furthermore, I will discuss that the spectral Laplace-Beltrami Wavelets can be used for the construction of geometric convolutional neural network (CNN) and then introduce a vertex-based geometric CNN algorithm for regular surfaces in which translation and downsampling on surfaces can be the same as those in the regular grid. I will show the use of this method for the prediction of Alzheimer’s Disease.

Higher-Order Attention Networks

Abstract: This paper introduces higher-order attention networks (HOANs), a novel class of attention-based neural networks defined on a generalized higher-order domain called a combinatorial complex (CC). Similar to hypergraphs, CCs admit arbitrary set-like relations between a collection of abstract entities. Simultaneously, CCs permit the construction of hierarchical higher-order relations analogous to those supported by cell complexes. Thus, CCs effectively generalize both hypergraphs and cell complexes and combine their desirable characteristics. By exploiting the rich combinatorial nature of CCs, HOANs define a new class of message-passing attention-based networks that unifies higher-order neural networks. Our evaluation on tasks related to mesh shape analysis and graph learning demonstrates that HOANs attain competitive, and in some examples superior, predictive performance in comparison to state-of-the-art neural networks.

Persistence Surfaces: A Novel Frequency Specific Topological Data Analysis Summary with Application to EEG Data Set

Abstract: Over the years, TDA has been applied to weighted networks with great success. In this talk, we begin by outlining the rationale for using TDA in the analysis of functional brain dependence networks, which emerge from brain imaging data such as electroencephalograms (EEGs). Following the overview, we present a novel approach to sample networks with arbitrary number of cycles and from different two-dimensional manifolds. From these manifolds, we propose a method to sample multivariate time series data with specific patterns in its dependence structure. Finally, we will present a novel TDA summary that we coin persistence surface, which is a generalization of the famous persistence landscape. We present an application where our new approach provides new insights in successfully discriminating between the topological structures derived from EEGs, of healthy subjects from those diagnosed with attention deficit hyperactivity disorder (ADHD).

3D data analysis of X-ray CT images with persistent homology and NMF

Abstract: In this talk, I will present data analysis of 3D images of iron ore sinters using persistent homology and nonnegative matrix factorization. "Concatenated persistence images" technique was used to extract coexisting structures in the persistence diagrams of different dimensions hidden behind the data, and we found three types of typical structures in the data.

Topological invariants for flows on surfaces and metric spaces

Abstract: Data of spaces and materials are topologically analyzed by roughly classifying their shapes. Similarly, data of flows can be topologically analyzed by roughly classifying them. The Morse graph for a flow is one of the most popular tools for classifying flows. Such graphs can capture hyperbolic behaviors of flows but cannot describe global recurrent behaviors in general. Therefore, we generalize such topological invariants to those that can describe both hyperbolic and recurrent structures. Roughly speaking, the topological invariants can be obtained by dividing the spaces along certain orbits and by coding the connections between the pieces of spaces. In particular, such topological invariants are complete for generic flows on surfaces. On the other hand, such topological invariants for flows on metric spaces are rough classifications, which are refinements of Morse graphs, Reeb graphs of Hamiltonian flows on surfaces, and Morse decompositions for generic gradient flows on manifolds. Furthermore, the topological invariants for flows are generalized to those for decompositions and semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems).

Topological loss functions and topological representation learning

Abstract: Persistent homology encodes the topological features of given data into persistence diagrams, which are multisets in the two-dimensional space. In connection with machine learning, many techniques have been developed to incorporate persistence diagrams into loss functions for controlling the topology of parameters. In this talk, I discuss several recent developments of TDA-based loss functions and a theoretical guarantee for the convergence of such functions for stochastic subgradient descent. I also talk about a data-driven approach to estimating (vectorization of) persistence diagrams of point clouds. Our proposed neural network architecture, RipsNet, can efficiently estimate topological descriptors after training. Moreover, we prove that RipsNet is robust to input perturbations in terms of 1-Wasserstein distance, allowing RipsNet to substantially outperform exactly-computed PDs in noisy settings.

On Topological Contraction for Multiple Functions

Abstract: Computational topology typically deals with scalar fields, or simply, a single function. Reeb graphs are a classical example of this, which contracts the level sets of a function into points. Real-world problems, however, often challenge researchers with multiple functions. What, then, can we learn if topological contraction is extended to a combination of functions? The speaker specializes himself in this extension known as the Reeb space. In this talk, the speaker will share his knowledge on topological contraction of multiple functions from the point of view of visualization studies, together with some mathematical background in the singularity theory. He will also discuss his ongoing work on understanding multi-objective optimization with topological contraction.

The unreasonably effective interaction between math and applications: A case study on persistence images

Abstract: Wigner described the unreasonable effectiveness of mathematics in the natural sciences: ideas from mathematics are unreasonably effective in advancing applications, and ideas from applications are unreasonably effective in advancing mathematics. We describe a case study on persistent images, a stable vector representation of persistent homology. If you combine topology with data, you get persistent homology. If you combine persistent homology with machine learning, you might get persistent landscapes or persistence images or a host of other options. The first attempt at persistence images were not stable (i.e. continuous), but by making them stable, their machine learning performance improves, as we will describe on examples ranging from materials science to biology. This ping-ponging behavior of injecting ideas from mathematics, then injecting ideas from applications, etc, leads to robust applied tools and new mathematical questions. Joint work with Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier.

Modeling Cycles in Brain Networks using Hodge-Laplacian and Hodge-Decomposition

Abstract: Decoding the dominant loops amidst a myriad of connections in a brain network is the key to understand the higher-order interactions and synchronization. Growing evidences suggest that complex brain functions and disorders are strongly connected to the higher-order interactions. The closed path or cycles in a brain network embeds higher order signal transmission paths, which provide fundamental insights into the functioning of the brain. In this talk, I will present an efficient algorithm for systematic identification and modeling of 1-cycles using persistent homology and the Hodge-Laplacian. Further, I will discuss the Hodge decomposition technique and demonstrate how the harmonic component of the decomposition can also be used to extract the 1-cycles from a network. Our methods are extensively validated on simulations and then applied to human brain networks obtained through the resting-state functional magnetic resonance images. New statistical inference procedures on 1-cycles are developed for discriminating male and female brain networks. The code for modeling cycles through the Hodge Laplacian is provided in https://github.com/laplcebeltrami/hodge.

keywords: Topological data analysis, Hodge Laplacian, Hodge decomposition, Cycles

Exploring the relationship between statistical physics and topological data analysis

Abstract: Conceptually, around criticality when phase transitions occur there are abrupt changes in the properties of a physical system, including topological ones. The criticality hypothesis of the brain states that brain dynamics self-organize into a critical configuration that balances excitatory and inhibitory interactions to achieve maximal adaptivity. Further, recent evidence suggests that synaptic hyperexcitation due to the loss of inhibitory interneurons in the hippocampus may represent some of the earliest changes of Alzheimer’s disease (AD). In this talk, I will discuss how our research team attempts to measure Excitation-Inhibition (E/I) balance using human neuroimaging data by fitting a mixed-spin Ising model, a classic statistical physics model that exhibits second-order phase transition, to data derived from resting-state functional and diffusion-weighted MR imaging. Then, I will discuss recent research efforts that suggest a connection between phase transitions and topological data analysis, and conjecture how these new efforts could inform other branches of natural science including computational neuroimaging.

Variable topological perspectives on time-series data via one-parameter reduction in multi-parameter persistence theory

Abstract: In various applications of data classification problems, multi-parameter analysis is effective and crucial because data are usually defined in multi-parametric space. Multi-parameter persistent homology, an extension of persistent homology of one-parameter data analysis, has been developed for topological data analysis (TDA) in multi-parametric space. Although it is conceptually attractive, multi-parameter persistent homology still has challenges in theory and practical applications. In this study, we consider time-series data and its classification problems using multi-parameter persistent homology. We developed a multi-parameter filtration method based on Fourier decomposition and showed that we could obtain variable classifications depending on the choice of curve in the multi-parameter filtration domain. That is, the chosen curves provide different classification criteria or perspectives. For this, we first consider the continuousization of time-series data based on Fourier decomposition towards the construction of the exact persistent barcode formula for the Vietoris-Rips complex of the point cloud generated by sliding window embedding. We will explain the geometric meaning of the barcode of the projected point cloud onto some specific subspaces. We will provide the proof that usual one-parameter persistent homology is equivalent to choosing diagonal line in multi-parameter filtration space for the case we consider time-series data. This approach is useful in that it provides various different topological perspectives for the given datasets.

The Effects of Topological Features on Convolutional Neural Networks – How Topological Signatures Enhance CNNs

Abstract: Topology characterizes the global structure of data based on topological invariants via e.g. topological data analysis (TDA), while Convolutional Neural Networks (CNNs) are capable of characterizing local features towards the global structure of data. A combined model of TDA and CNN, a family of multimodal networks, simultaneously takes the image and the corresponding topological features as the input of the network for classification problems and significantly improves the performance of a single CNN. Its success has recently been reported in various applications. However, there is a lack of explanation as to how and why topological signatures – when combined with a CNN – improve the discriminative power of the original CNN. In this paper, we use persistent homology to compute topological features and demonstrate visually the effects of topological signatures on a CNN, for which the Grad-CAM analysis of multimodal networks and topological inverse image map are proposed and utilized. For the numerical experiments, we used two famous datasets, i.e., the transient versus bogus image dataset and the HAM10000 dataset. With Grad-CAM analysis on multimodal networks, we show that topological features enforce the image network of a CNN to focus more on significant and meaningful regions across images, rather than task-irrelevant artifacts such as background noise and texture

Simplifying complex chemical reaction networks

Abstract: Inside living cells, chemical reactions form a large web of networks. Understanding the behavior of those complex reaction networks is an important and challenging problem. In many situations, it is hard to identify the details of the reactions, such as the reaction kinetics and parameter values. It would be good if we can clarify what we can say about the behavior of reaction systems, when we know the structure of reaction networks but reaction kinetics is unknown. In this talk, we discuss a method for the reduction of chemical reaction networks, by which important substructures can be extracted. Mathematical concepts such as homology and cohomology groups are found to be useful for characterizing the shapes of reaction networks and for tracking the changes of them under reductions. For a given chemical reaction network, we identify topological conditions on its subnetwork, reduction of which preserves the original steady state exactly. This method allows us to reduce a reaction network while preserving its original steady-state properties, and complex reaction systems can be studied efficiently.

Mathematical AI for Molecular Sciences

Abstract: With great accumulations in experimental data, computing power and learning models, artificial intelligence (AI) is making great advancements in molecular sciences. Recently, the breakthrough of AlphaFold 2 in protein folding herald a new era for AI-based molecular data analysis for materials, chemistry, and biology. A major challenge remains in AI-based molecular sciences which is to design and achieve effective molecular descriptors or fingerprints. In this talk, we propose several advanced mathematical based representations and featurizations. Molecular structures and their interactions can be represented by graphs, simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex) and hypergraphs. Molecular representations can be systematically featurized using various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models to form quantitative prediction models. Our models have demonstrated great advantage over traditional models in drug design, material informatics and chemical informatics.