Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces

Transfer operators such as the Perron--Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis

Kernel methods for detecting coherent structures in dynamical data

We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes (VAMP) score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD). Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods

GraphKKE: graph Kernel Koopman embedding for human microbiome analysis

More and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, or some cancer types. Thanks to modern high-throughput omics technologies, it becomes possible to directly analyze human microbiome and its influence on the health status. Microbial communities are monitored over long periods of time and the associations between their members are explored. These relationships can be described by a time-evolving graph. In order to understand responses of the microbial community members to a distinct range of perturbations such as antibiotics exposure or diseases and general dynamical properties, the time-evolving graph of the human microbial communities has to be analyzed. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to solving this problem is the representation of the time-evolving graphs as fixed-length feature vectors preserving the original dynamics. We propose a method for learning the embedding of the time-evolving graph that is based on the spectral analysis of transfer operators and graph kernels. We demonstrate that our method can capture temporary changes in the time-evolving graph on both synthetic data and real-world data. Our experiments demonstrate the efficacy of the method. Furthermore, we show that our method can be applied to human microbiome data to study dynamic processes

Unsupervised approaches for time-evolving graph embeddings with application to human microbiome

More and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, and even some types of cancer. Advances in high-throughput omics technologies have made it possible to directly analyze the human microbiome and its impact on human health and physiology. Microbial composition is usually observed over long periods of time and the interactions between their members are explored. Numerous studies have used microbiome data to accurately differentiate disease states and understand the differences in microbiome profiles between healthy and ill individuals. However, most of them mainly focus on various statistical approaches, omitting microbe-microbe interactions among a large number of microbiome species that, in principle, drive microbiome dynamics. Constructing and analyzing time-evolving graphs is needed to understand how microbial ecosystems respond to a range of distinct perturbations, such as antibiotic exposure, diseases, or other general dynamic properties. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to addressing this challenge lies in representing time-evolving graphs constructed from microbiome data as fixed-length, low-dimensional feature vectors that preserve the original dynamics. Therefore, we propose two unsupervised approaches that map the time-evolving graph constructed from microbiome data into a low-dimensional space where the initial dynamic, such as the number of metastable states and their locations, is preserved. The first method relies on the spectral analysis of transfer operators, such as the Perron--Frobenius or Koopman operator, and graph kernels. These components enable us to extract topological information such as complex interactions of species from the time-evolving graph and take into account the dynamic changes in the human microbiome composition. Further, we study how deep learning techniques can contribute to the study of a complex network of microbial species. The method consists of two key components: 1) the Transformer, the state-of-the-art architecture used in the sequential data, that learns both structural patterns of the time-evolving graph and temporal changes of the microbiome system and 2) contrastive learning that allows the model to learn the low-dimensional representation while maintaining metastability in a low-dimensional space. Finally, this thesis will address an important challenge in microbiome data, specifically identifying which species or interactions of species are responsible for or affected by the changes that the microbiome undergoes from one state (healthy) to another state (diseased or antibiotic exposure). Using interpretability techniques of deep learning models, which, at the outset, have been used as methods to prove the trustworthiness of a deep learning model, we can extract structural information of the time-evolving graph pertaining to particular metastable states

Understanding microbiome dynamics via interpretable graph representation learning

Large-scale perturbations in the microbiome constitution are strongly correlated, whether as a driver or a consequence, with the health and functioning of human physiology. However, understanding the difference in the microbiome profiles of healthy and ill individuals can be complicated due to the large number of complex interactions among microbes. We propose to model these interactions as a time-evolving graph where nodes represent microbes and edges are interactions among them. Motivated by the need to analyse such complex interactions, we develop a method that can learn a low-dimensional representation of the time-evolving graph while maintaining the dynamics occurring in the high-dimensional space. Through our experiments, we show that we can extract graph features such as clusters of nodes or edges that have the highest impact on the model to learn the low-dimensional representation. This information is crucial for identifying microbes and interactions among them that are strongly correlated with clinical diseases. We conduct our experiments on both synthetic and real-world microbiome datasets

Manifold Learning in Atomistic Simulations: A Conceptual Review

Analyzing large volumes of high-dimensional data requires dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. Such practice is needed in atomistic simulations of complex systems where even thousands of degrees of freedom are sampled. An abundance of such data makes gaining insight into a specific physical problem strenuous. Our primary aim in this review is to focus on unsupervised machine learning methods that can be used on simulation data to find a low-dimensional manifold providing a collective and informative characterization of the studied process. Such manifolds can be used for sampling long-timescale processes and free-energy estimation. We describe methods that can work on datasets from standard and enhanced sampling atomistic simulations. Unlike recent reviews on manifold learning for atomistic simulations, we consider only methods that construct low-dimensional manifolds based on Markov transition probabilities between high-dimensional samples. We discuss these techniques from a conceptual point of view, including their underlying theoretical frameworks and possible limitations