Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor and generalizes to tensors with any number of modes. The algorithm is based on a new random walk model which we call the super-spacey random surfer. We show that our method out-performs state-of-the-art co-clustering methods on several synthetic datasets with ground truth clusters and then use the algorithm to analyze several real-world datasets.

With the rapid development of web services, large amounts of time series data are generated and accumulated across various domains such as finance, healthcare, and online platforms. As such data often co-evolves with multiple variables interacting with each other, estimating the time-varying dependencies between variables (i.e., the dynamic network structure) has become crucial for accurate modeling. However, real-world data is often represented as tensor time series with multiple modes, resulting in large, entangled networks that are hard to interpret and computationally intensive to estimate. In this paper, we propose Kronecker Time-Varying Graphical Lasso (KTVGL), a method designed for modeling tensor time series. Our approach estimates mode-specific dynamic networks in a Kronecker product form, thereby avoiding overly complex entangled structures and producing interpretable modeling results. Moreover, the partitioned network structure prevents the exponential growth of computational time with data dimension. In addition, our method can be extended to stream algorithms, making the computational time independent of the sequence length. Experiments on synthetic data show that the proposed method achieves higher edge estimation accuracy than existing methods while requiring less computation time. To further demonstrate its practical value, we also present a case study using real-world data. Our source code and datasets are available at https://github.com/Higashiguchi-Shingo/KTVGL.

Weshowtheperformance ofGETF onhigh order tensor data inFigure 6. Noted, when theorder increases, the tensor size would increase exponentially.

We show the performance of GETF on high order tensor data in Figure 6. Noted, when the order increases, the tensor size would increase exponentially.

Abstract--The Large Margin Distribution Machine (LMDM) is a recent advancement in classifier design that optimizes not just the minimum margin (as in SVM) but the entire margin distribution, thereby improving generalization. However, existing LMDM formulations are limited to vectorized inputs and struggle with high-dimensional tensor data due to the need for flattening, which destroys the data's inherent multi-mode structure and increases computational burden. In this paper, we propose a Structure-Preserving Margin Distribution Learning for High-Order T ensor Data with Low-Rank Decomposition (SPMD-LRT) that operates directly on tensor representations without vectorization. The SPMD-LRT preserves multi-dimensional spatial structure by incorporating first-order and second-order tensor statistics (margin mean and variance) into the objective, and it leverages low-rank tensor decomposition techniques including rank-1(CP), higher-rank CP, and T ucker decomposition to parameterize the weight tensor . An alternating optimization (double-gradient descent) algorithm is developed to efficiently solve the SPMD-LRT, iteratively updating factor matrices and core tensor . This approach enables SPMD-LRT to maintain the structural information of high-order data while optimizing margin distribution for improved classification. Extensive experiments on diverse datasets (including MNIST, images and fMRI neuroimaging) demonstrate that SPMD-LRT achieves superior classification accuracy compared to conventional SVM, vector-based LMDM, and prior tensor-based SVM extensions (Support T ensor Machines and Support T ucker Machines). These results confirm the effectiveness and robustness of SPMD-LRT in handling high-dimensional tensor data for classification. Dvances in data acquisition have led to an abundance of high-order tensor data (multi-dimensional arrays) across various domains, such as video sequences, medical imaging, and spatiotemporal sensor readings. Effectively learning from such tensor-structured data has become a pressing research focus [1] [2]. The multi-dimensional structure of tensors offers rich information (e.g.

High-dimensional data, particularly in the form of high-order tensors, presents a major challenge in self-supervised learning. While MLP-based autoencoders (AE) are commonly employed, their dependence on flattening operations exacerbates the curse of dimensionality, leading to excessively large model sizes, high computational overhead, and challenging optimization for deep structural feature capture. Although existing tensor networks alleviate computational burdens through tensor decomposition techniques, most exhibit limited capability in learning non-linear relationships. To overcome these limitations, we introduce the Mode-Aware Non-linear Tucker Autoencoder (MA-NTAE). MA-NTAE generalized classical Tucker decomposition to a non-linear framework and employs a Pick-and-Unfold strategy, facilitating flexible per-mode encoding of high-order tensors via recursive unfold-encode-fold operations, effectively integrating tensor structural priors. Notably, MA-NTAE exhibits linear growth in computational complexity with tensor order and proportional growth with mode dimensions. Extensive experiments demonstrate MA-NTAE's performance advantages over standard AE and current tensor networks in compression and clustering tasks, which become increasingly pronounced for higher-order, higher-dimensional tensors.

Currently, existing tensor recovery methods fail to recognize the impact of tensor scale variations on their structural characteristics. Furthermore, existing studies face prohibitive computational costs when dealing with large-scale high-order tensor data. To alleviate these issue, assisted by the Krylov subspace iteration, block Lanczos bidiagonalization process, and random projection strategies, this article first devises two fast and accurate randomized algorithms for low-rank tensor approximation (LRTA) problem. Theoretical bounds on the accuracy of the approximation error estimate are established. Next, we develop a novel generalized nonconvex modeling framework tailored to large-scale tensor recovery, in which a new regularization paradigm is exploited to achieve insightful prior representation for large-scale tensors. On the basis of the above, we further investigate new unified nonconvex models and efficient optimization algorithms, respectively, for several typical high-order tensor recovery tasks in unquantized and quantized situations. To render the proposed algorithms practical and efficient for large-scale tensor data, the proposed randomized LRTA schemes are integrated into their central and time-intensive computations. Finally, we conduct extensive experiments on various large-scale tensors, whose results demonstrate the practicability, effectiveness and superiority of the proposed method in comparison with some state-of-the-art approaches.

During the training of Large Language Models (LLMs), tensor data is periodically "checkpointed" to persistent storage to allow recovery of work done in the event of failure. The volume of data that must be copied during each checkpoint, even when using reduced-precision representations such as bfloat16, often reaches hundreds of gigabytes. Furthermore, the data must be moved across a network and written to a storage system before the next epoch occurs. With a view to ultimately building an optimized checkpointing solution, this paper presents experimental analysis of checkpoint data used to derive a design that maximizes the use of lossless compression to reduce the volume of data. We examine how tensor data and its compressibility evolve during model training and evaluate the efficacy of existing common off-the-shelf general purpose compression engines combined with known data optimization techniques such as byte-grouping and incremental delta compression. Leveraging our analysis we have built an effective compression solution, known as Language Model Compressor (LMC), which is based on byte-grouping and Huffman encoding. LMC offers more compression performance than the best alternative (BZ2) but with an order-of-magnitude reduction in the time needed to perform the compression. We show that a 16-core parallel implementation of LMC can attain compression and decompression throughput of 2.78 GiB/s and 3.76 GiB/s respectively. This increase in performance ultimately reduces the CPU resources needed and provides more time to copy the data to the storage system before the next epoch thus allowing for higher-frequency checkpoints.

Tensor decomposition is a fundamental tool for analyzing multi-dimensional data by learning low-rank factors to represent high-order interactions. While recent works on temporal tensor decomposition have made significant progress by incorporating continuous timestamps in latent factors, they still struggle with general tensor data with continuous indexes not only in the temporal mode but also in other modes, such as spatial coordinates in climate data. Additionally, the problem of determining the tensor rank remains largely unexplored in temporal tensor models. To address these limitations, we propose \underline{G}eneralized temporal tensor decomposition with \underline{R}ank-r\underline{E}vealing laten\underline{T}-ODE (GRET). Our approach encodes continuous spatial indexes as learnable Fourier features and employs neural ODEs in latent space to learn the temporal trajectories of factors. To automatically reveal the rank of temporal tensors, we introduce a rank-revealing Gaussian-Gamma prior over the factor trajectories. We develop an efficient variational inference scheme with an analytical evidence lower bound, enabling sampling-free optimization. Through extensive experiments on both synthetic and real-world datasets, we demonstrate that GRET not only reveals the underlying ranks of temporal tensors but also significantly outperforms existing methods in prediction performance and robustness against noise.