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Document Associé À Des Manifestations Scientifiques Année : 2023

Orthogonalization schemes in tensor train format

Résumé

In the framework of tensor spaces, we consider orthogonalization kernels to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. In particular, we investigate experimentally the loss of orthogonality of six orthogonalization methods, namely Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Gram approach, and the Householder transformation. To tackle the curse of dimensionality, we represent tensor with low-rank approximation using the Tensor Train (TT) formalism, and we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed precision. After describing the algorithm structure and properties, we illustrate experimentally that the theoretical bounds for the loss of orthogonality in the classical matrix computation round-off analysis results are maintained, with the unit round-off replaced by the TT-rounding precision. The computational analysis for each orthogonalization kernel in terms of the memory requirement and the computational complexity measured as a function of the number of TT-rounding, which happens to be the computational most expensive operation, completes the study.
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Dates et versions

hal-04394115 , version 1 (15-01-2024)

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  • HAL Id : hal-04394115 , version 1

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Martina Iannacito, Olivier Coulaud, Luc Giraud. Orthogonalization schemes in tensor train format. The 29th Biennial Conference on Numerical Analysis 2023, Jun 2023, Glasgow, United Kingdom. ⟨hal-04394115⟩
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