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LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
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SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors
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ACM SIGGRAPH 2004 Papers
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ACM Transactions on Mathematical Software (TOMS)
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Journal of Complexity
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Proceedings of the 2008 ACM conference on Recommender systems
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
SIAM Journal on Matrix Analysis and Applications
Symmetric Tensors and Symmetric Tensor Rank
SIAM Journal on Matrix Analysis and Applications
On the Tensor SVD and the Optimal Low Rank Orthogonal Approximation of Tensors
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Tensor Decompositions and Applications
SIAM Review
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
SIAM Journal on Scientific Computing
Hierarchical Singular Value Decomposition of Tensors
SIAM Journal on Matrix Analysis and Applications
Quasi-Newton Methods on Grassmannians and Multilinear Approximations of Tensors
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Hybrid clustering of multiple information sources via HOSVD
ISNN'10 Proceedings of the 7th international conference on Advances in Neural Networks - Volume Part II
IEEE Transactions on Signal Processing - Part II
Discrimination of speech from nonspeech based on multiscale spectro-temporal Modulations
IEEE Transactions on Audio, Speech, and Language Processing
Kronecker product approximation for preconditioning in three-dimensional imaging applications
IEEE Transactions on Image Processing
Higher Order SVD Analysis for Dynamic Texture Synthesis
IEEE Transactions on Image Processing
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We present an alternative strategy for truncating the higher-order singular value decomposition (T-HOSVD). An error expression for an approximate Tucker decomposition with orthogonal factor matrices is presented, leading us to propose a novel truncation strategy for the HOSVD, which we refer to as the sequentially truncated higher-order singular value decomposition (ST-HOSVD). This decomposition retains several favorable properties of the T-HOSVD, while reducing the number of operations required to compute the decomposition and practically always improving the approximation error. Three applications are presented, demonstrating the effectiveness of ST-HOSVD. In the first application, ST-HOSVD, T-HOSVD, and higher-order orthogonal iteration (HOOI) are employed to compress a database of images of faces. On average, the ST-HOSVD approximation was only $0.1\%$ worse than the optimum computed by HOOI, while cutting the execution time by a factor of $20$. In the second application, classification of handwritten digits, ST-HOSVD achieved a speedup factor of $50$ over T-HOSVD during the training phase, and reduced the classification time and storage costs, while not significantly affecting the classification error. The third application demonstrates the effectiveness of ST-HOSVD in compressing results from a numerical simulation of a partial differential equation. In such problems, ST-HOSVD inevitably can greatly improve the running time. We present an example wherein the $2$ hour $45$ minute calculation of T-HOSVD was reduced to just over one minute by ST-HOSVD, representing a speedup factor of $133$, while even improving the memory consumption.