Efficient numerical computations with tensor trains

[gh-honglizhaobob]

Following the progress with Ticket #30307, we propose to add computational functionalities with tensors via the tensor train decomposition, which is an idea originally named "Density Matrix Renormalization Group (DMRG)" and later popularized by Prof. Ivan Oseledets.

This functionality is especially useful for high dimensional tasks to prevent the curse of dimensionality (such as in Riemannian optimization), currently supported in MATLAB, Python, and C++. The MATLAB version is currently most versatile and robust.

We may consider internalizing this computational library for Sage, as an effort to formalize the currently available implementations. This addition can either be an additional child class under tensor, and overwrite the arithmetics; or a separate module. The backend is currently proposed to be TensorFlow?, however this may change / be flexible based on how the refactoring of Components progresses.

One can look at this github page for an example implementation (​https://github.com/oseledets/ttpy).

Ideally after the implementation of this ticket, users can specify a fixed rank tensor for a general manifold using tensor.modules, specify a ring and a frame, and perform useful and normally computationally intractable tasks using tensor train as a backend. Based on the progress of this ticket, one may also consider implementing other backends for storing a numerical tensor such as tensor rings and quantized tensor train.

For alternative tensor decomposition formats, please see the following survey papers:

• Tucker format: (​https://arxiv.org/abs/1810.01262)
• Tensor ring: (​https://arxiv.org/abs/1807.02513)
• Quantized tensor train (see section 3.5.1): (​https://refubium.fu-berlin.de/handle/fub188/3366)

For the theory behind why one can reduce a normally exponential (in the number of dimensions) task to a linear task, see (​https://epubs.siam.org/doi/abs/10.1137/090752286?journalCode=sjoce3).