Tensors express multi-dimensional data, extending the concept of vectors and matrices to encompass higher dimensions. These structures find widespread application in fields such as machine learning, physics, more info and computer graphics. Rings, with their well-defined operations of addition and multiplication, provide a powerful framework for manipulating tensors. By leveraging the properties of rings, we can perform tensor operations like adjustment and joining efficiently and accurately.

Tensor representations within linear spaces over rings allow us to analyze complex data patterns. This framework enables the development of sophisticated algorithms for tasks like pattern recognition, feature extraction, and tensor resolution.

• Furthermore, tensors over rings provide a unified structure for representing diverse data types, including numerical, symbolic, and categorical information.
• Investigating the interplay between tensors and rings opens up new avenues for research in areas such as cryptography.

Efficient Tensor Ring Decomposition for Large-Scale Data Processing

Tensor ring decomposition (TRD) presents a powerful technique for efficiently decomposing large-scale data tensors. It involves factorizing the tensor into a set of smaller matrices, known as rings, which can be managed more efficiently than the original tensor. TRD has {foundextensive applications in various domains, including machine learning, data processing, and scientific computing. Its efficiency stems from its ability to exploit the inherent pattern of tensors and reduce the memory footprint required for storage and calculation.

• TRD algorithms typically utilize a combination of iterative optimization techniques and tensor operations to identify the optimal ring structure.
• Furthermore recent advancements in TRD have focused on developing accelerated algorithms and techniques that can handle tensors with even extensive dimensions.
• The increasing availability of large-scale datasets has emphasized the need for efficient tensor decomposition methods like TRD, which they provide a effective solution for data processing.

Adaptive Tensor Ring Factorization for Non-Euclidean Data

Tensor factorization techniques have emerged as powerful tools for interpreting high-order data. Traditional tensor factorization methods often rely on Euclidean distance metrics, which may not be suitable for non-Euclidean data, such as graphs or manifolds. Adaptive Tensor Ring Factorization (ATRF) addresses this challenge by adaptively learning a representation that models the underlying structure of non-Euclidean data. ATRF employs a hierarchical tensor ring structure, allowing for efficient representation and processing of complex relationships within the data. By optimizing the factorization parameters based on the specific properties of the non-Euclidean domain, ATRF achieves superior performance in tasks such as graph classification, embedding learning, and link prediction.

Theoretical Foundations of Tensor Rings and Their Applications

Tensor rings have emerged as a powerful framework for representing manipulating multidimensional data. Grounded in the principles of tensor algebra and linear algebra, these structures provide a rich tool for encoding complex dependencies across multiple dimensions. This theoretical foundation enables diverse applications in fields such as machine learning, where tensor rings can be used to model complex relationships and perform tasks.

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li Tensor rings offer a unified framework for representing multidimensional data.

li They leverage principles of tensor algebra and ring theory.

li Applications span machine learning, data mining, and pattern recognition.

Tensor Rings in Machine Learning: A Survey

Tensor rings have emerged as a novel paradigm in machine learning, offering effective methods for analyzing high-order data tensors. This survey provides a comprehensive exploration of tensor ring representations, delving into their mathematical foundations and extensive applications. We survey key algorithms for tensor ring construction, focusing on their capabilities over traditional tensor representation methods. Furthermore, we explore the relevance of tensor rings in a variety of machine learning tasks, including {imageanalysis, natural language understanding, and recommender systems.

• The overview also discusses the current trends in tensor ring theory, identifying promising paths for future research.
• By providing a comprehensive overview of tensor rings, this survey aims to encourage further research in this evolving field of machine learning.

Applications of Tensor Rings to Signal Processing and Image Analysis

Tensor rings have gained traction as a powerful tool in signal processing and image analysis. Leveraging their inherent multi-dimensional structure, tensor rings provide an effective framework for representing complex data structures. In signal processing, they can be applied for tasks such as signal compression, achieving remarkable performance. Similarly, in image analysis, tensor rings have shown effectiveness in applications like object detection. Their ability to capture complex dependencies within data makes them particularly applicable for these tasks.

The {rigorous{ mathematical framework and computational efficiency of tensor rings make them an attractive complement to traditional methods in both signal processing and image analysis. As research in this area develops, we can expect to see even more creative applications of tensor rings in these fields.