Dynamical Tensor Approximation
نویسندگان
چکیده
For the approximation of time-dependent data tensors and of solutions to tensor differential equations by tensors of low Tucker rank, we study a computational approach that can be viewed as a continuous-time updating procedure. This approach works with the increments rather than the full tensor and avoids the computation of decompositions of large matrices. In this method, the derivative is projected onto the tangent space of the manifold of tensors of Tucker rank (r1, . . . , rN ) at the current approximation. This yields nonlinear differential equations for the factors in a Tucker decomposition, suitable for numerical integration. Approximation properties of this approach are analyzed.
منابع مشابه
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors
We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch & Lubich, 2007 and 2010) to the recently proposed Hierarchical Tucker tensor format (HT, Hackbusch & Kühn, 2009) and the Tensor Train format (TT, Oseledets, 2011), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamic...
متن کاملDynamical approximation of hierarchical Tucker and tensor-train tensors
We extend results on the dynamical low-rank approximation for the treatment of time-dependent matrices and tensors (Koch & Lubich, 2007 and 2010) to the recently proposed Hierarchical Tucker tensor format (HT, Hackbusch & Kühn, 2009) and the Tensor Train format (TT, Oseledets, 2011), which are closely related to tensor decomposition methods used in quantum physics and chemistry. In this dynamic...
متن کاملBEST APPROXIMATION IN QUASI TENSOR PRODUCT SPACE AND DIRECT SUM OF LATTICE NORMED SPACES
We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...
متن کاملNumerical approximation of multi-dimensional PDEs in quantized tensor spaces
Modern methods of tensor-product approximation by separation of variables allow an efficient lowparametric calculus of functions and operators in higher dimensions. Most common separable representations combine the Tucker, canonical, tensor train (TT) and the more general MPS-type decompositions. The idea of data quantization makes it possible to represent (approximate) the multivariate functio...
متن کاملThe Optimization Landscape for Fitting a Rank-2 Tensor with a Rank-2 Tensor∗
The ability to approximate a multivariate function/tensor as a sum of separable functions/tensors is quite useful, but algorithms to do so exhibit unpleasantly interesting behavior known as swamps. Such swamps have been the bane of users for decades and need to be understood so they can be alleviated. In previous work, we developed and applied dynamical systems concepts to analyze the simplest ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Matrix Analysis Applications
دوره 31 شماره
صفحات -
تاریخ انتشار 2010