There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

Data Mining - Concepts and Techniques.

“Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks”の学習報告

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

Fast Graph Representation Learning with PyTorch Geometric

Popular graph neural networks implement convolution operations on graphs based on polynomial spectral filters. In this paper, we propose a novel graph convolutional layer inspired by the auto-regressive moving average (ARMA) filter that, compared to polynomial ones, provides a more flexible frequency response, is more robust to noise, and better captures the global graph structure. We propose a graph neural network implementation of the ARMA filter with a recursive and distributed formulation, obtaining a convolutional layer that is efficient to train, localized in the node space, and can be transferred to new graphs at test time. We perform a spectral analysis to study the filtering effect of the proposed ARMA layer and report experiments on four downstream tasks: semi-supervised node classification, graph signal classification, graph classification, and graph regression. Results show that the proposed ARMA layer brings significant improvements over graph neural networks based on polynomial filters.

Graph Neural Networks with convolutional ARMA filters

The key component in forecasting demand and consumption of resources in a supply network is an accurate prediction of real-valued time series. Indeed, both service interruptions and resource waste can be reduced with the implementation of an effective forecasting system. Significant research has thus been devoted to the design and development of methodologies for short term load forecasting over the past decades. A class of mathematical models, called Recurrent Neural Networks, are nowadays gaining renewed interest among researchers and they are replacing many practical implementation of the forecasting systems, previously based on static methods. Despite the undeniable expressive power of these architectures, their recurrent nature complicates their understanding and poses challenges in the training procedures. Recently, new important families of recurrent architectures have emerged and their applicability in the context of load forecasting has not been investigated completely yet. In this paper we perform a comparative study on the problem of Short-Term Load Forecast, by using different classes of state-of-the-art Recurrent Neural Networks. We test the reviewed models first on controlled synthetic tasks and then on different real datasets, covering important practical cases of study. We provide a general overview of the most important architectures and we define guidelines for configuring the recurrent networks to predict real-valued time series.

An overview and comparative analysis of Recurrent Neural Networks for Short Term Load Forecasting

Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem and train a GNN to compute cluster assignments that minimize this objective. Our GNN-based implementation is differentiable, does not require to compute the spectral decomposition, and learns a clustering function that can be quickly evaluated on out-of-sample graphs. From the proposed clustering method, we design a graph pooling operator that overcomes some important limitations of state-of-the-art graph pooling techniques and achieves the best performance in several supervised and unsupervised tasks.

Spectral Clustering with Graph Neural Networks for Graph Pooling

In this paper, we approach the problem of forecasting a time series (TS) of an electrical load measured on the Azienda Comunale Energia e Ambiente (ACEA) power grid, the company managing the electricity distribution in Rome, Italy, with an echo state network (ESN) considering two different leading times of 10 min and 1 day. We use a standard approach for predicting the load in the next 10 min, while, for a forecast horizon of one day, we represent the data with a high-dimensional multi-variate TS, where the number of variables is equivalent to the quantity of measurements registered in a day. Through the orthogonal transformation returned by PCA decomposition, we reduce the dimensionality of the TS to a lower number $k$ of distinct variables; this allows us to cast the original prediction problem in $k$ different one-step ahead predictions. The overall forecast can be effectively managed by $k$ distinct prediction models, whose outputs are combined together to obtain the final result. We employ a genetic algorithm for tuning the parameters of the ESN and compare its prediction accuracy with a standard autoregressive integrated moving average model.

Short-Term Electric Load Forecasting Using Echo State Networks and PCA Decomposition

In this paper, we elaborate over the well-known interpretability issue in echo-state networks (ESNs). The idea is to investigate the dynamics of reservoir neurons with time-series analysis techniques developed in complex systems research. Notably, we analyze time series of neuron activations with recurrence plots (RPs) and recurrence quantification analysis (RQA), which permit to visualize and characterize high-dimensional dynamical systems. We show that this approach is useful in a number of ways. First, the 2-D representation offered by RPs provides a visualization of the high-dimensional reservoir dynamics. Our results suggest that, if the network is stable, reservoir and input generate similar line patterns in the respective RPs. Conversely, as the ESN becomes unstable, the patterns in the RP of the reservoir change. As a second result, we show that an RQA measure, called $L_\mathrm {max}$ , is highly correlated with the well-established maximal local Lyapunov exponent. This suggests that complexity measures based on RP diagonal lines distribution can quantify network stability. Finally, our analysis shows that all RQA measures fluctuate on the proximity of the so-called edge of stability, where an ESN typically achieves maximum computational capability. We leverage on this property to determine the edge of stability and show that our criterion is more accurate than two well-known counterparts, both based on the Jacobian matrix of the reservoir. Therefore, we claim that RPs and RQA-based analyses are valuable tools to design an ESN, given a specific problem.

/pdf/investigating-echo-state-networks-dynamics-by-means-of-4kb05nuj7s.pdf

Filippo Maria Bianchi

Papers

Graph Neural Networks with convolutional ARMA filters

An overview and comparative analysis of Recurrent Neural Networks for Short Term Load Forecasting

Spectral Clustering with Graph Neural Networks for Graph Pooling

Short-Term Electric Load Forecasting Using Echo State Networks and PCA Decomposition

Investigating Echo-State Networks Dynamics by Means of Recurrence Analysis