Sleep Staging from Airflow Signals Using Fourier Approximations of Persistence Curves

Sleep staging is a challenging task, typically manually performed by sleep technologists based on electroencephalogram and other biosignals of patients taken during overnight sleep studies. Recent work aims to leverage automated algorithms to perform sleep staging not based on electroencephalogram signals, but rather based on the airflow signals of subjects. Prior work uses ideas from topological data analysis (TDA), specifically Hermite function expansions of persistence curves (HEPC) to featurize airflow signals. However, finite order HEPC captures only partial information. In this work, we propose Fourier approximations of persistence curves (FAPC), and use this technique to perform sleep staging based on airflow signals. We analyze performance using an XGBoost model on 1155 pediatric sleep studies taken from the Nationwide Children's Hospital Sleep DataBank (NCHSDB), and find that FAPC methods provide complimentary information to HEPC methods alone, leading to a 4.9% increase in performance over baseline methods.

Is Graph Convolution Always Beneficial For Every Feature?

Graph Neural Networks (GNNs) have demonstrated strong capabilities in processing structured data. While traditional GNNs typically treat each feature dimension equally during graph convolution, we raise an important question: Is the graph convolution operation equally beneficial for each feature? If not, the convolution operation on certain feature dimensions can possibly lead to harmful effects, even worse than the convolution-free models. In prior studies, to assess the impacts of graph convolution on features, people proposed metrics based on feature homophily to measure feature consistency with the graph topology. However, these metrics have shown unsatisfactory alignment with GNN performance and have not been effectively employed to guide feature selection in GNNs. To address these limitations, we introduce a novel metric, Topological Feature Informativeness (TFI), to distinguish between GNN-favored and GNN-disfavored features, where its effectiveness is validated through both theoretical analysis and empirical observations. Based on TFI, we propose a simple yet effective Graph Feature Selection (GFS) method, which processes GNN-favored and GNN-disfavored features separately, using GNNs and non-GNN models. Compared to original GNNs, GFS significantly improves the extraction of useful topological information from each feature with comparable computational costs. Extensive experiments show that after applying GFS to 8 baseline and state-of-the-art (SOTA) GNN architectures across 10 datasets, 83.75% of the GFS-augmented cases show significant performance boosts. Furthermore, our proposed TFI metric outperforms other feature selection methods. These results validate the effectiveness of both GFS and TFI. Additionally, we demonstrate that GFS's improvements are robust to hyperparameter tuning, highlighting its potential as a universal method for enhancing various GNN architectures.

ClusterGraph: a new tool for visualization and compression of multidimensional data

Understanding the global organization of complicated and high dimensional data is of primary interest for many branches of applied sciences. It is typically achieved by applying dimensionality reduction techniques mapping the considered data into lower dimensional space. This family of methods, while preserving local structures and features, often misses the global structure of the dataset. Clustering techniques are another class of methods operating on the data in the ambient space. They group together points that are similar according to a fixed similarity criteria, however unlike dimensionality reduction techniques, they do not provide information about the global organization of the data. Leveraging ideas from Topological Data Analysis, in this paper we provide an additional layer on the output of any clustering algorithm. Such data structure, ClusterGraph, provides information about the global layout of clusters, obtained from the considered clustering algorithm. Appropriate measures are provided to assess the quality and usefulness of the obtained representation. Subsequently the ClusterGraph, possibly with an appropriate structure--preserving simplification, can be visualized and used in synergy with state of the art exploratory data analysis techniques.

Quantum computing and persistence in topological data analysis

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales -- is $\mathsf{BQP}_1$-hard and contained in $\mathsf{BQP}$. This result implies an exponential quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

Memetic collaborative approaches for finding balanced incomplete block designs

The balanced incomplete block design (BIBD) problem is a difficult combinatorial problem with a large number of symmetries, which add complexity to its resolution. In this paper, we propose a dual (integer) problem representation that serves as an alternative to the classical binary formulation of the problem. We attack this problem incrementally: firstly, we propose basic algorithms (i.e. local search techniques and genetic algorithms) intended to work separately on the two different search spaces (i.e. binary and integer); secondly, we propose two hybrid schemes: an integrative approach (i.e. a memetic algorithm) and a collaborative model in which the previous methods work in parallel, occasionally exchanging information. Three distinct two-dimensional structures are proposed as communication topology among the algorithms involved in the collaborative model, as well as a number of migration and acceptance criteria for sending and receiving data. An empirical analysis comparing a large number of instances of our schemes (with algorithms possibly working on different search spaces and with/without symmetry breaking methods) shows that some of these algorithms can be considered the state of the art of the metaheuristic methods applied to finding BIBDs. Moreover, our cooperative proposal is a general scheme from which distinct algorithmic variants can be instantiated to handle symmetrical optimisation problems. For this reason, we have also analysed its key parameters, thereby providing general guidelines for the design of efficient/robust cooperative algorithms devised from our proposal.

UnSegMedGAT: Unsupervised Medical Image Segmentation using Graph Attention Networks Clustering

The data-intensive nature of supervised classification drives the interest of the researchers towards unsupervised approaches, especially for problems such as medical image segmentation, where labeled data is scarce. Building on the recent advancements of Vision transformers (ViT) in computer vision, we propose an unsupervised segmentation framework using a pre-trained Dino-ViT. In the proposed method, we leverage the inherent graph structure within the image to realize a significant performance gain for segmentation in medical images. For this, we introduce a modularity-based loss function coupled with a Graph Attention Network (GAT) to effectively capture the inherent graph topology within the image. Our method achieves state-of-the-art performance, even significantly surpassing or matching that of existing (semi)supervised technique such as MedSAM which is a Segment Anything Model in medical images. We demonstrate this using two challenging medical image datasets ISIC-2018 and CVC-ColonDB. This work underscores the potential of unsupervised approaches in advancing medical image analysis in scenarios where labeled data is scarce. The github repository of the code is available on [https://github.com/mudit-adityaja/UnSegMedGAT].

Persistent Homology for MCI Classification: A Comparative Analysis between Graph and Vietoris-Rips Filtrations

Mild cognitive impairment (MCI), often linked to early neurodegeneration, is characterized by subtle cognitive declines and disruptions in brain connectivity. The present study offers a detailed analysis of topological changes associated with MCI, focusing on two subtypes: Early MCI and Late MCI. This analysis utilizes fMRI time series data from two distinct populations: the publicly available ADNI dataset (Western cohort) and the in-house TLSA dataset (Indian Urban cohort). Persistent Homology, a topological data analysis method, is employed with two distinct filtration techniques - Vietoris-Rips and graph filtration-for classifying MCI subtypes. For Vietoris-Rips filtration, inter-ROI Wasserstein distance matrices between persistent diagrams are used for classification, while graph filtration relies on the top ten most persistent homology features. Comparative analysis shows that the Vietoris-Rips filtration significantly outperforms graph filtration, capturing subtle variations in brain connectivity with greater accuracy. The Vietoris-Rips filtration method achieved the highest classification accuracy of 85.7\% for distinguishing between age and gender matched healthy controls and MCI, whereas graph filtration reached a maximum accuracy of 71.4\% for the same task. This superior performance highlights the sensitivity of Vietoris-Rips filtration in detecting intricate topological features associated with neurodegeneration. The findings underscore the potential of persistent homology, particularly when combined with the Wasserstein distance, as a powerful tool for early diagnosis and precise classification of cognitive impairments, offering valuable insights into brain connectivity changes in MCI.

Topology-Aware Graph Augmentation for Predicting Clinical Trajectories in Neurocognitive Disorders

Brain networks/graphs derived from resting-state functional MRI (fMRI) help study underlying pathophysiology of neurocognitive disorders by measuring neuronal activities in the brain. Some studies utilize learning-based methods for brain network analysis, but typically suffer from low model generalizability caused by scarce labeled fMRI data. As a notable self-supervised strategy, graph contrastive learning helps leverage auxiliary unlabeled data. But existing methods generally arbitrarily perturb graph nodes/edges to generate augmented graphs, without considering essential topology information of brain networks. To this end, we propose a topology-aware graph augmentation (TGA) framework, comprising a pretext model to train a generalizable encoder on large-scale unlabeled fMRI cohorts and a task-specific model to perform downstream tasks on a small target dataset. In the pretext model, we design two novel topology-aware graph augmentation strategies: (1) hub-preserving node dropping that prioritizes preserving brain hub regions according to node importance, and (2) weight-dependent edge removing that focuses on keeping important functional connectivities based on edge weights. Experiments on 1, 688 fMRI scans suggest that TGA outperforms several state-of-the-art methods.

Feature Learning in Attention Mechanisms Is More Compact and Stable Than in Convolution

Attention and convolution are fundamental techniques in machine learning. While they use different approaches to learn features - attention mechanisms capture both global and local data relathionships, while convolutional layers focus on local patterns - both methods are effective for various tasks. Although the feature learning of both models is well-studied individually, there has not been a direct comparison of their feature learning dynamics. In this paper, we compare their Lipschitz continuity with respect to the Wasserstein distance and covering numbers under similar settings. We demonstrate that attention processes data in a more compact and stable manner. Compactness refers to the lower variance and intrinsic dimensionality of the activation outputs, while stability refers to the changes between inputs and outputs. We validate our findings through experiments using topological data analysis, measuring the 1-, 2-, and infinity-Wasserstein distances between the outputs of each layer from both models. Furthermore, we extend our comparison to Vision Transformers (ViTs) and ResNets, showing that while ViTs have higher output variance, their feature learning is more stable than that of ResNets.

TopoDiffusionNet: A Topology-aware Diffusion Model

Diffusion models excel at creating visually impressive images but often struggle to generate images with a specified topology. The Betti number, which represents the number of structures in an image, is a fundamental measure in topology. Yet, diffusion models fail to satisfy even this basic constraint. This limitation restricts their utility in applications requiring exact control, like robotics and environmental modeling. To address this, we propose TopoDiffusionNet (TDN), a novel approach that enforces diffusion models to maintain the desired topology. We leverage tools from topological data analysis, particularly persistent homology, to extract the topological structures within an image. We then design a topology-based objective function to guide the denoising process, preserving intended structures while suppressing noisy ones. Our experiments across four datasets demonstrate significant improvements in topological accuracy. TDN is the first to integrate topology with diffusion models, opening new avenues of research in this area.