Updating truncated Singular Value Decomposition (SVD) has extensive applications in representation learning. The continuous evolution of massive-scaled data matrices in practical scenarios highlights the importance of aligning SVD-based models with fast-paced updates. Recent methods for updating truncated SVD can be recognized as Rayleigh-Ritz projection procedures where their projection matrices are augmented based on the original singular vectors. However, the updating process in these methods densifies the update matrix and applies the projection to all singular vectors, resulting in inefficiency. This paper presents a novel method for dynamically approximating the truncated SVD of a sparse and temporally evolving matrix. The proposed method takes advantage of sparsity in the orthogonalization process of the augment matrices and employs an extended decomposition to store projections in the column space of singular vectors independently. Numerical experimental results on updating truncated SVD for evolving sparse matrices show an order of magnitude improvement in the efficiency of our proposed method while maintaining precision comparing to previous methods.

Time Series Classification (TSC) encompasses two settings: classifying entire sequences or classifying segmented subsequences. The raw time series for segmented TSC usually contain Multiple classes with Varying Duration of each class (MVD). Therefore, the characteristics of MVD pose unique challenges for segmented TSC, yet have been largely overlooked by existing works. Specifically, there exists a natural temporal dependency between consecutive instances (segments) to be classified within MVD. However, mainstream TSC models rely on the assumption of independent and identically distributed (i.i.d.), focusing on independently modeling each segment. Additionally, annotators with varying expertise may provide inconsistent boundary labels, leading to unstable performance of noise-free TSC models. To address these challenges, we first formally demonstrate that valuable contextual information enhances the discriminative power of classification instances. Leveraging the contextual priors of MVD at both the data and label levels, we propose a novel consistency learning framework Con4m, which effectively utilizes contextual information more conducive to discriminating consecutive segments in segmented TSC tasks, while harmonizing inconsistent boundary labels for training. Extensive experiments across multiple datasets validate the effectiveness of Con4m in handling segmented TSC tasks on MVD. The source code is available at https://github.com/MrNobodyCali/Con4m.

Network embedding, a graph representation learning method illustrating network topology by mapping nodes into lower-dimension vectors, is challenging to accommodate the ever-changing dynamic graphs in practice. Existing research is mainly based on node-by-node embedding modifications, which falls into the dilemma of efficient calculation and accuracy. Observing that the embedding dimensions are usually much smaller than the number of nodes, we break this dilemma with a novel dynamic network embedding paradigm that rotates and scales the axes of embedding space instead of a node-by-node update. Specifically, we propose the Dynamic Adjacency Matrix Factorization (DAMF) algorithm, which achieves an efficient and accurate dynamic network embedding by rotating and scaling the coordinate system where the network embedding resides with no more than the number of edge modifications changes of node embeddings. Moreover, a dynamic Personalized PageRank is applied to the obtained network embeddings to enhance node embeddings and capture higher-order neighbor information dynamically. Experiments of node classification, link prediction, and graph reconstruction on different-sized dynamic graphs suggest that DAMF advances dynamic network embedding. Further, we unprecedentedly expand dynamic network embedding experiments to billion-edge graphs, where DAMF updates billion-level parameters in less than 10ms.