تقرير
On some orthogonalization schemes in Tensor Train format
| العنوان: | On some orthogonalization schemes in Tensor Train format |
|---|---|
| المؤلفون: | Coulaud, Olivier, Giraud, Luc, Iannacito, Martina |
| سنة النشر: | 2022 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Computer Science - Distributed, Parallel, and Cluster Computing, Mathematics - Numerical Analysis |
| الوصف: | 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 experimentally study 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 overcome the curse of dimensionality, we represent tensors with a low-rank approximation using the Tensor Train (TT) formalism. In addition, we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the structure and properties of the algorithms, we illustrate their loss of orthogonality with numerical experiments. The theoretical bounds from the classical matrix computation round-off analysis, obtained over several decades, seem to be maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirements and the computational complexity measured as a function of the number of TT-rounding, which happens to be the most computationally expensive operation, completes the study. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2211.08770 |
| رقم الأكسشن: | edsarx.2211.08770 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |