ข้อมูลเชิงลึก - Tensor analysis - # Overcomplete tensor decomposition

Efficient Tensor Decomposition Algorithm for Overcomplete Tensors

Q: How can the decomposition algorithm be extended to handle the fully rectangular case (m × n × p tensors)

To extend the decomposition algorithm to handle the fully rectangular case (m × n × p tensors), we can follow a similar approach as in the provided context. Adjust Slices: Modify the decomposition algorithm to accommodate tensors of format m × n × p by adjusting the slicing process to consider matrices of different dimensions. Update Matrices: Update the matrices used in the algorithm to reflect the new dimensions of the fully rectangular tensors, ensuring that the computations and transformations are applied correctly. Incorporate Additional Parameters: Integrate additional parameters or steps in the algorithm to account for the varying dimensions in the fully rectangular tensors, ensuring that the decomposition process is accurate and efficient. By adapting the algorithm to handle tensors of different formats, including fully rectangular tensors, we can ensure a comprehensive and versatile decomposition process.

Q: Can the complexity of the decomposition algorithm be analyzed in more detail, taking into account the precision required for matrix diagonalization

Analyzing the complexity of the decomposition algorithm in more detail, especially concerning the precision required for matrix diagonalization, involves considering the intricacies of the diagonalization process and its impact on the overall computational complexity. Precision Consideration: Evaluate the effect of precision requirements on the diagonalization process, as higher precision levels may increase the computational complexity of the algorithm. Algorithmic Efficiency: Assess the algorithm's efficiency in handling varying precision levels, ensuring that the diagonalization computations are accurate while maintaining reasonable time complexity. Complexity Analysis: Conduct a thorough analysis of the algorithm's complexity with respect to precision, considering factors such as the size of the input tensors, the desired precision level, and the computational resources required for matrix diagonalization. By delving into the specifics of precision requirements and their impact on the algorithm's complexity, we can gain a deeper understanding of the computational intricacies involved in matrix diagonalization within the decomposition process.

Q: Is computing the size of the smallest diagonalizable commuting extension NP-hard, as suggested by the connection to tensor rank

The suggestion that computing the size of the smallest diagonalizable commuting extension may be NP-hard stems from the intricate relationship between tensor rank and commuting extensions. Complexity Analysis: Investigate the computational complexity of determining the size of the smallest diagonalizable commuting extension, considering factors such as the dimensionality of the matrices, the constraints on diagonalizability, and the interplay with tensor rank. Reduction Techniques: Explore reduction techniques from tensor rank problems to commuting extension computations to establish the potential NP-hardness of the task, leveraging insights from existing complexity results in related areas. Theoretical Examination: Conduct a theoretical examination of the problem, drawing parallels to known NP-hard problems and utilizing formal complexity theory to ascertain the computational hardness of determining the size of diagonalizable commuting extensions. By delving into the theoretical foundations and computational intricacies of the problem, we can shed light on the potential NP-hardness of computing the size of the smallest diagonalizable commuting extension, further enriching our understanding of tensor decomposition complexities.

แนวคิดหลัก

The author presents a new, constructive uniqueness theorem for tensor decomposition that applies to order 3 tensors of format n×n×p and can prove uniqueness of decomposition for generic tensors up to rank r = 4n/3 as soon as p ≥4. This leads to the first efficient algorithm for overcomplete decomposition of generic tensors.

บทคัดย่อ

The paper presents a new uniqueness theorem and an efficient decomposition algorithm for overcomplete tensor decomposition.

Key highlights:

The uniqueness theorem applies to order 3 tensors of format n×n×p and can prove uniqueness of decomposition for generic tensors up to rank r = 4n/3 as soon as p ≥4.
This is an improvement over Kruskal's uniqueness theorem, which can only prove uniqueness up to rank n+1 for tensors of format n×n×4.
The uniqueness theorem has an algorithmic proof, leading to an efficient decomposition algorithm.
The algorithm can efficiently decompose generic tensors in the overcomplete regime (n ≤r ≤4n/3), which was not possible prior to this work.
The algorithm relies on the method of commuting extensions pioneered by Strassen, as well as the classical Jennrich algorithm for undercomplete tensor decomposition.

The author also provides a genericity result showing that the conditions of the uniqueness theorem are generically satisfied for tensors in the range n ≤r ≤4n/3.

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ข้อมูลเชิงลึกที่สำคัญจาก

An efficient uniqueness theorem for overcomplete tensor decomposition

by Pascal Koira... ที่ arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07801.pdf

An efficient uniqueness theorem for overcomplete tensor decomposition

สอบถามเพิ่มเติม

How can the decomposition algorithm be extended to handle the fully rectangular case (m × n × p tensors)

To extend the decomposition algorithm to handle the fully rectangular case (m × n × p tensors), we can follow a similar approach as in the provided context.

Adjust Slices: Modify the decomposition algorithm to accommodate tensors of format m × n × p by adjusting the slicing process to consider matrices of different dimensions.

Update Matrices: Update the matrices used in the algorithm to reflect the new dimensions of the fully rectangular tensors, ensuring that the computations and transformations are applied correctly.

Incorporate Additional Parameters: Integrate additional parameters or steps in the algorithm to account for the varying dimensions in the fully rectangular tensors, ensuring that the decomposition process is accurate and efficient.

By adapting the algorithm to handle tensors of different formats, including fully rectangular tensors, we can ensure a comprehensive and versatile decomposition process.

Can the complexity of the decomposition algorithm be analyzed in more detail, taking into account the precision required for matrix diagonalization

Analyzing the complexity of the decomposition algorithm in more detail, especially concerning the precision required for matrix diagonalization, involves considering the intricacies of the diagonalization process and its impact on the overall computational complexity.

Precision Consideration: Evaluate the effect of precision requirements on the diagonalization process, as higher precision levels may increase the computational complexity of the algorithm.

Algorithmic Efficiency: Assess the algorithm's efficiency in handling varying precision levels, ensuring that the diagonalization computations are accurate while maintaining reasonable time complexity.

Complexity Analysis: Conduct a thorough analysis of the algorithm's complexity with respect to precision, considering factors such as the size of the input tensors, the desired precision level, and the computational resources required for matrix diagonalization.

By delving into the specifics of precision requirements and their impact on the algorithm's complexity, we can gain a deeper understanding of the computational intricacies involved in matrix diagonalization within the decomposition process.

Is computing the size of the smallest diagonalizable commuting extension NP-hard, as suggested by the connection to tensor rank

The suggestion that computing the size of the smallest diagonalizable commuting extension may be NP-hard stems from the intricate relationship between tensor rank and commuting extensions.

Complexity Analysis: Investigate the computational complexity of determining the size of the smallest diagonalizable commuting extension, considering factors such as the dimensionality of the matrices, the constraints on diagonalizability, and the interplay with tensor rank.

Reduction Techniques: Explore reduction techniques from tensor rank problems to commuting extension computations to establish the potential NP-hardness of the task, leveraging insights from existing complexity results in related areas.

Theoretical Examination: Conduct a theoretical examination of the problem, drawing parallels to known NP-hard problems and utilizing formal complexity theory to ascertain the computational hardness of determining the size of diagonalizable commuting extensions.

By delving into the theoretical foundations and computational intricacies of the problem, we can shed light on the potential NP-hardness of computing the size of the smallest diagonalizable commuting extension, further enriching our understanding of tensor decomposition complexities.