Anomaly detection in multivariate signals is a task of paramount importance
in many disciplines (epidemiology, finance, cognitive sciences and
neurosciences, oncology, etc.). In this perspective, Topological Data Analysis
(TDA) offers a battery of "shape" invariants that can be exploited for the
implementation of an effective detection scheme. Our contribution consists of
extending the constructions presented in \cite{chretienleveraging} on the
construction of simplicial complexes from the Signatures of signals and their
predictive capacities, rather than the use of a generic distance as in
\cite{petri2014homological}. Signature theory is a new theme in Machine
Learning arXiv:1603.03788 stemming from recent work on the notions of Rough
Paths developed by Terry Lyons and his team \cite{lyons2002system} based on the
formalism introduced by Chen \cite{chen1957integration}. We explore in
particular the detection of changes in topology, based on tracking the
evolution of homological persistence and the Betti numbers associated with the
complex introduced in \cite{chretienleveraging}. We apply our tools for the
analysis of brain signals such as EEG to detect precursor phenomena to
epileptic seizures.

多元信号的异常检测使用拓扑数据分析的方法构建了具有预测能力的单纯复合体，来跟踪拓扑的变化并探测与脑信号相关的癫痫发作前兆现象。

应用签名理论对脑电图进行时间拓扑分析

Time topological analysis of EEG using signature theory

The online prediction of multivariate signals, existing simultaneously in
space and time, from noisy partial observations is a fundamental task in
numerous applications. We propose an efficient Neural Network architecture for
the online estimation of time-varying graph signals named the Adaptive Least
Mean Squares Graph Neural Networks (LMS-GNN). LMS-GNN aims to capture the time
variation and bridge the cross-space-time interactions under the condition that
signals are corrupted by noise and missing values. The LMS-GNN is a combination
of adaptive graph filters and Graph Neural Networks (GNN). At each time step,
the forward propagation of LMS-GNN is similar to adaptive graph filters where
the output is based on the error between the observation and the prediction
similar to GNN. The filter coefficients are updated via backpropagation as in
GNN. Experimenting on real-world temperature data reveals that our LMS-GNN
achieves more accurate online predictions compared to graph-based methods like
adaptive graph filters and graph convolutional neural networks.

我们提出了一种用于在线估计时变图信号的高效神经网络架构，名为自适应最小均方图神经网络（LMS-GNN），它旨在捕捉时间变化和填补受噪声和缺失值影响的信号的空间和时间跨度相互作用，通过自适应图滤波器和图神经网络的组合实现。实验表明，与自适应图滤波器和图卷积神经网络等基于图的方法相比，我们的 LMS-GNN 实现了更准确的在线预测。

自适应最小均方图神经网络与在线图信号估计

Adaptive Least Mean Squares Graph Neural Networks and Online Graph  Signal Estimation

Multivariate signals are prevalent in various domains, such as healthcare,
transportation systems, and space sciences. Modeling spatiotemporal
dependencies in multivariate signals is challenging due to (1) long-range
temporal dependencies and (2) complex spatial correlations between sensors. To
address these challenges, we propose representing multivariate signals as
graphs and introduce GraphS4mer, a general graph neural network (GNN)
architecture that captures both spatial and temporal dependencies in
multivariate signals. Specifically, (1) we leverage Structured State Spaces
model (S4), a state-of-the-art sequence model, to capture long-term temporal
dependencies and (2) we propose a graph structure learning layer in GraphS4mer
to learn dynamically evolving graph structures in the data. We evaluate our
proposed model on three distinct tasks and show that GraphS4mer consistently
improves over existing models, including (1) seizure detection from
electroencephalography signals, outperforming a previous GNN with
self-supervised pretraining by 3.1 points in AUROC; (2) sleep staging from
polysomnography signals, a 4.1 points improvement in macro-F1 score compared to
existing sleep staging models; and (3) traffic forecasting, reducing MAE by
8.8% compared to existing GNNs and by 1.4% compared to Transformer-based
models.

提出了一种基于图的神经网络模型 GraphS4mer，用于对多变量信号进行时空建模，通过学习动态演化的图结构来捕捉数据中的空间和时间相关性，并在三个不同的任务中测试，发现相较于其它模型，GraphS4mer 有着更优的性能表现。