More Like this

1. Spectral Adapter: Fine-Tuning in Spectral Space

   Fangzhao, Zhang; Pilanci, Mert (December 2024, Neural Information Processing Systems (Neurips) 2024)
2. Efficient Spectral Methods for PDEs with Spectral Fractional Laplacian

   https://doi.org/10.1007/s10915-021-01491-2  

   Sheng, Changtao; Cao, Duo; Shen, Jie (July 2021, Journal of Scientific Computing)

   null (Ed.)
3. Towards Scalable Spectral Embedding and Data Visualization via Spectral Coarsening

   https://doi.org/10.1145/3437963.3441767  

   Zhao, Zhiqiang; Zhang, Ying; Feng, Zhuo (March 2021, ACM International Conference on Web Search and Data Mining)

   null (Ed.)

   This paper proposes a scalable multilevel framework for the spectral embedding of large undirected graphs. The proposed method first computes much smaller yet sparse graphs while preserving the key spectral (structural) properties of the original graph, by exploiting a nearly-linear time spectral graph coarsening approach. Then, the resultant spectrally-coarsened graphs are leveraged for the development of much faster algorithms for multilevel spectral graph embedding (clustering) as well as visualization of large data sets. We conducted extensive experiments using a variety of large graphs and datasets and obtained very promising results. For instance, we are able to coarsen the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges into a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-coarsened graphs allow us to achieve up to 1,100X speedup for multilevel spectral graph embedding (clustering) and up to 60X speedup for t-SNE visualization of large data sets. 

   more » « less
4. Spectral Exterior Calculus

   https://doi.org/10.1002/cpa.21885  

   Berry, Tyrus; Giannakis, Dimitrios (February 2020, Communications on Pure and Applied Mathematics)
5. Persistent spectral graph

   https://doi.org/10.1002/cnm.3376  

   Wang, Rui; Nguyen, Duc Duy; Wei, Guo‐Wei (June 2020, International Journal for Numerical Methods in Biomedical Engineering)

   Persistent homology is constrained to purely topological persistence while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified lowdimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high‐dimensional datasets. For a point‐cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, i.e., the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non‐harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerenes stability is predicted by using both harmonic and non‐harmonic persistent spectra. While non‐harmonic persistent spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B‐factors for which current popular biophysical models break down. 

   more » « less