Randomized Low-Rank Approximation of Tensors in Tucker Format using Multilinear Nyström Method

Temel Kavramlar

The multilinear Nyström (MLN) algorithm provides an efficient and stable method for computing low-rank approximations of tensors in Tucker format using only tensor mode-j products.

Özet

The content discusses the development of a randomized algorithm called the multilinear Nyström (MLN) method for computing low-rank approximations of tensors in Tucker format.
Key highlights:

Tensors offer a natural way to model higher-order structures, but suffer from the curse of dimensionality, making them computationally intractable. Low-rank tensor approximations can help mitigate this issue.
The authors extend the generalized Nyström method, originally developed for matrices, to the tensor setting, creating the MLN algorithm. This allows for efficient and stable low-rank tensor approximations.
The MLN algorithm avoids expensive orthogonalization steps and can be implemented in a numerically stable fashion, unlike previous tensor decomposition methods.
The authors provide a detailed theoretical analysis of the accuracy and stability of the MLN algorithm, showing that it achieves near-optimal approximation quality and can be implemented reliably even in the presence of floating-point errors.
Numerical experiments demonstrate that MLN outperforms state-of-the-art methods in terms of memory requirements, computational cost, and number of accesses to the original tensor data.

İstatistikler

The tensor A has dimensions n1 x ... x nd.
The multilinear rank of the tensor A is r = (r1, ..., rd).
The oversampling parameter is ℓ = (ℓ1, ..., ℓd).

Alıntılar

"The Nyström method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nyström method)."
"We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nyström method, positioning it as a viable substitute for the randomized higher-order SVD algorithm."

Önemli Bilgiler Şuradan Elde Edildi

A multilinear Nyström algorithm for low-rank approximation of tensors in Tucker format

by Alberto Bucc... : arxiv.org 04-24-2024

https://arxiv.org/pdf/2309.02877.pdf

Daha Derin Sorular

How can the MLN algorithm be extended or adapted to handle tensors with special structures, such as sparsity or symmetry

The MLN algorithm can be extended or adapted to handle tensors with special structures, such as sparsity or symmetry, by incorporating specific constraints or modifications into the algorithm.

Handling Sparsity:

For sparse tensors, the MLN algorithm can be enhanced by incorporating techniques from sparse tensor decomposition methods. This may involve adjusting the sketching matrices Xk and Yk to account for the sparsity pattern in the tensor. Additionally, the regularization parameter in the stabilized version of MLN can be optimized to preserve the sparsity structure during approximation.

Symmetry Constraints:

To handle symmetric tensors, constraints can be imposed on the sketching matrices to preserve the symmetry properties. By ensuring that the sketching matrices respect the symmetry of the tensor, the MLN algorithm can provide more accurate approximations for symmetric tensors.

Tensor-Specific Modifications:

Tailoring the MLN algorithm to exploit known structures in the tensor, such as block-diagonal patterns or hierarchical relationships, can improve the efficiency and accuracy of the approximation. By customizing the sketching and projection steps based on the tensor's specific structure, MLN can be adapted to handle a wide range of tensor characteristics.

What are the potential limitations or drawbacks of the MLN approach compared to other tensor decomposition methods, and how can they be addressed

While the MLN approach offers several advantages for low-rank tensor approximation, there are potential limitations and drawbacks compared to other tensor decomposition methods. These limitations can be addressed through various strategies:

Computational Complexity:

MLN may have higher computational complexity compared to some tensor decomposition methods, especially for large tensors or high-dimensional data. To address this, optimizing the sketching matrices and projection steps can help reduce computational costs and improve efficiency.

Stability Concerns:

The stability of MLN in the presence of floating-point errors or numerical inaccuracies can be a drawback. By incorporating regularization techniques, such as the use of stabilized pseudoinverses, the stability of the algorithm can be enhanced to mitigate the impact of numerical errors.

Memory Requirements:

MLN may require significant memory resources, especially for storing intermediate results during the approximation process. Implementing memory-efficient data structures and algorithms can help alleviate this limitation and improve the scalability of the MLN approach.

What are the implications of the MLN algorithm for applications that rely on efficient tensor representations, such as high-dimensional PDEs, multivariate functions, or machine learning models

The implications of the MLN algorithm for applications relying on efficient tensor representations, such as high-dimensional PDEs, multivariate functions, or machine learning models, are significant:

Improved Computational Efficiency:

MLN offers a computationally efficient way to approximate tensors in the Tucker format, making it suitable for applications with high-dimensional data or complex tensor structures. By providing near-optimal low-rank approximations, MLN can enhance the computational efficiency of tensor-based algorithms.

Enhanced Model Performance:

In machine learning models that utilize tensors for data representation, the use of MLN can lead to improved model performance by reducing the computational burden associated with high-dimensional tensor operations. This can result in faster training times and more accurate predictions.

Scalability and Flexibility:

MLN's ability to handle tensors with varying structures and dimensions makes it a versatile tool for applications such as high-dimensional PDEs and multivariate functions. Its streamable nature and stability properties make it suitable for large-scale data processing and analysis, contributing to the scalability of tensor-based algorithms.