ToFU: Topology functional units for deep learning

<p style='text-indent:20px;'>We propose ToFU, a new trainable neural network unit with a persistence diagram dissimilarity function as its activation. Since persistence diagrams are topological summaries of structures, this new activation measures and learns the topology of data to leverage it in machine learning tasks. We showcase the utility of ToFU in two experiments: one involving the classification of discrete-time autoregressive signals, and another involving a variational autoencoder. In the former, ToFU yields competitive results with networks that use spectral features while outperforming CNN architectures. In the latter, ToFU produces topologically-interpretable latent space representations of inputs without sacrificing reconstruction fidelity.</p>

Topological inference from spontaneous activity structures in FMRI videos with peristence barcodes

Spatio-temporal brain activities with variable delay detectable in resting-state functional magnetic resonance imaging (rs-fMRI) give rise to highly reproducible structures, termed cortical lag threads, that can propagate from one brain region to another. Using a computational topology of data approach, we found that Betti numbers that are cycle counts and the areas of vortex cycles covering brain activation regions in triangulated rs-fMRI video frames make it possible to track persistent, recurring blood oxygen level dependent (BOLD) signals. Our findings have been codified and visualized in what are known as persistent barcodes. Importantly, a topology of data offers a practical approach in coping with and sidestepping massive noise in neuro data, such as unwanted dark (low intensity) regions in the neighbourhood of non-zero BOLD signals. A natural outcome of a topology of data approach is the tracking of persistent, non-trivial BOLD signals that appear intermittently in a sequence of rs-fMRI video frames. The end result of this tracking of changing lag structures is a persistent barcode, which is a pictograph that offers a convenient visual means of exhibiting, comparing and classifying brain activation patterns.

Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells

Single-cell RNA-seq quantifies biological heterogeneity across both discrete cell types and continuous cell transitions. Partition-based graph abstraction (PAGA) provides an interpretable graph-like map of the arising data manifold, based on estimating connectivity of manifold partitions (https://github.com/theislab/paga). PAGA maps provide interpretable discrete and continuous latent coordinates for both disconnected and continuous structure in data, preserve the global topology of data, allow analyzing data at different resolutions and result in much higher computational efficiency of the typical exploratory data analysis workflow — one million cells take on the order of a minute, a speedup of 130 times compared to UMAP. We demonstrate the method by inferring structure-rich cell maps with consistent topology across four hematopoietic datasets, confirm the reconstruction of lineage relations of adult planaria and the zebrafish embryo, benchmark computational performance on a neuronal dataset and detect a biological trajectory in one deep-learning processed image dataset.

Fuzzy Amendments of Trust Model for Power Grid with Intrusion Tolerance

When the topology of data network in power grid was destroy, the system still need to provide intrusion tolerance service. The trust values relationship of the nodes in data network need be calculated. The trust value of the upper and lower limit can be deduced and adjusted to eliminate the problem of over-reliance on threshold. And the introduction of confidence values to assess the quality of the trust value. A recursive algorithm is used to obtain the average of the trust data to compute confidence values. The experimental result proves the veracity and validity of the method.

Fuzzy Amendments of Trust Model with Intrusion Tolerance

When the topology of data network in power grid was destroy, the system still need to provide intrusion tolerance service. The trust values relationship of the nodes in data network need be calculated. The trust value of the upper and lower limit can be deduced and adjusted to eliminate the problem of over-reliance on threshold. And the introduction of confidence values to assess the quality of the trust value. A recursive algorithm is used to obtain the average of the trust data to compute confidence values. The experimental result proves the veracity and validity of the method.