Overview of Tensor Train Decomposition and examples of algorithms and implementations.

Overview of Tensor Train Decomposition

Tensor Train Decomposition (TT decomposition) is one of the methods for dimensionality reduction and data compression of multidimensional tensors and is an approach that provides an efficient data representation by approximating a tensor as a product of multiple low-rank tensors.

TT decomposition is achieved by transforming a tensor into a multi-dimensional column vector and reconstructing the column vector into a specific product (tensor column), where each element of the tensor can be represented as an inner product of the tensor columns.

An overview of the TT decomposition is given below.

1. column vectorisation of the tensor: the original tensor is transformed into a multi-dimensional column vector. In this process, each dimension of the tensor is expanded into a single column vector.

2. reconstruction by product of tensor columns: the column-vectorised tensor is reconstructed into a specific product (tensor column). In this process, each tensor column is represented as a low-rank tensor.

3. low-rank approximation of the tensor sequence: a low-rank approximation of each tensor sequence is performed in order to represent each element of the tensor sequence as an inner product of low-rank tensors. In this case, the rank of the tensor sequence is either pre-specified or determined automatically.

4. product of approximated tensor sequences: the low-rank approximated tensor sequences are multiplied to approximate the original tensor. The product enables an efficient and accurate representation of the original tensor.

TT decomposition is used in various domains such as machine learning and signal processing, as it provides an efficient data representation while exploiting the high dimensionality of the tensor.

Algorithms related to Tensor Train Decomposition

Algorithms for computing the TT decomposition are mainly based on iterative optimisation methods. This section describes the algorithm for computing the TT decomposition.

1. column vectorisation: transform the original tensor into a multi-dimensional column vector.

2. iterative optimisation: an iterative optimisation method is used to compute the TT decomposition. The method performs a series of steps that improve the approximation of the tensor.

3. low-rank approximation of each tensor sequence: in each step of the iteration, a low-rank approximation of each tensor sequence is computed. Typically, techniques such as singular value decomposition (SVD) as described in “Overview of Singular Value Decomposition (SVD) and examples of algorithms and implementations” are used to approximate the tensor sequence as a product of low-rank tensors.

4. iterative iteration: the above steps are repeated until a given convergence criterion is reached. The number of iterations is adjusted according to the desired accuracy and convergence performance.

5. selection of TT ranks: in TT decomposition, the selection of the rank (TT rank) in each dimension of the tensor sequence is important. This can be determined by an iterative optimisation method or specified a priori.

6. reconstruction of the approximated tensor: once the TT decomposition has converged, the approximated tensor is reconstructed. This yields a TT decomposition that efficiently represents the original tensor.

The TT decomposition algorithm is efficient even when the dimension of the tensor is high and is used in many application areas.

Examples of the application of Tensor Train Decomposition.

The following are examples of applications of TT decomposition.

1. image processing:

Image data compression: TT decomposition is used to efficiently compress image data, e.g. high-resolution images. This reduces the costs of image storage and transmission.

2. signal processing:

Analysis of speech data: TT decomposition is used to efficiently represent high-dimensional data, such as spectrograms of speech signals, for processing such as feature extraction and noise reduction.

3. machine learning:

Analysis of tensor data: TT decomposition is used to analyse data in tensor form, such as sensor data or biomedical engineering data, for dimensionality reduction and feature extraction. This facilitates data visualisation and pattern recognition.

4. quantum chemistry:

Calculation of molecular orbitals: TT decomposition is used to efficiently represent higher dimensional data such as molecular orbitals and the electronic structure of molecules. This speeds up the calculation of the electronic structure of molecules.

5. tensor networks:

Learning tensor networks: the TT decomposition is used to efficiently represent data in tensor form and is applied to learning and reasoning about tensor networks.

Example implementation of Tensor Train Decomposition

The implementation of Tensor Train Decomposition (TT decomposition) is relatively complex and is done using numerical and tensor computation libraries. A simple example of a TT decomposition implementation in Python is given below. This example uses the NumPy library.

import numpy as np

def tt_decomposition(tensor, rank):
    """
    Functions to perform TT decomposition
    :param tensor: Tensors applying the TT decomposition
    :param rank: List of TT ranks.
    :return: TT decomposed tensor sequence
    """
    # Number of dimensions of the tensor
    num_dims = len(rank) - 1
    
    # Initialisation of tensor sequence
    tensor_list = []
    
    # Column vectorisation of tensors.
    tensor_flat = np.reshape(tensor, (-1,))
    
    # TT Disassembly
    start_index = 0
    for i in range(num_dims):
        current_dim = tensor.shape[i]
        next_dim = tensor.shape[i + 1]
        
        # Number of elements in tensor sequence
        num_elements = rank[i] * current_dim * rank[i + 1]
        
        # Get the elements of the tensor column.
        tensor_list.append(np.reshape(tensor_flat[start_index:start_index + num_elements], (rank[i], current_dim, rank[i + 1])))
        
        start_index += num_elements
    
    return tensor_list

# Tensor example.
tensor = np.random.rand(2, 3, 4)

# Designation of TT ranks
rank = [1, 2, 3, 4, 1]

# Perform TT decomposition
tt_tensors = tt_decomposition(tensor, rank)

# Display of TT-resolved tensor columns.
for i, tt_tensor in enumerate(tt_tensors):
    print(f"Tensor {i + 1}:")
    print(tt_tensor)
    print()

The code performs a TT decomposition on the given tensor and displays the TT-decomposed tensor sequence; the TT rank is pre-specified and the TT decomposition approximates the tensor as a product of TT-ranked low-rank tensors.

Challenges and measures for Tensor Train Decomposition.

Tensor Train Decomposition (TT decomposition) has several challenges, and methods have been proposed to address them.

1. extension to higher dimensional tensors: the TT decomposition is computationally expensive as the dimensionality of the regular tensors increases. There is a need to develop effective TT decomposition methods for high dimensional tensors.

Solution: efficient algorithms and data structures for TT decomposition of high-dimensional tensors are being developed, e.g. TT-HOSVD.

2. choice of appropriate TT rank: in TT decomposition, the choice of appropriate TT rank is important, as too low a rank reduces approximation accuracy, while too high a rank increases computational cost.

Solution: techniques such as cross-validation and information criterion can be used to select an appropriate TT rank. Algorithms for automatic rank determination have also been proposed.

3. dealing with nonlinearity of tensors: as the TT decomposition is a linear model, it can be difficult to approximate adequately for non-linear data and relationships.

Solution: non-parametric approaches and non-linear TT decomposition methods are being developed, as well as data pre-processing and extension methods.

4. increased computational cost: TT decomposition can be computationally expensive. Computation time increases, especially when the dimension of the tensor is high or the size of the tensor is large.

Solution: distributed or parallel processing can be used to reduce the computational cost, or approximate TT decomposition methods can be used.

Reference Information and Reference Books

For more information on optimization in machine learning, see also “Optimization for the First Time Reading Notes” “Sequential Optimization for Machine Learning” “Statistical Learning Theory” “Stochastic Optimization” etc.

Reference books include Optimization for Machine Learning

“Machine Learning, Optimization, and Data Science“

“Linear Algebra and Optimization for Machine Learning: A Textbook“