Overview of Tucker decomposition and examples of algorithms and implementations

Overview of Tucker Decomposition

Tucker decomposition is a decomposition method for multidimensional data and is a type of tensor decomposition.Tucker decomposition approximates a tensor as a product of several low-rank tensors. Typically, the Tucker decomposition of a tensor \( \mathbf{X} \) is expressed as follows.

\[
\mathbf{X} \approx \mathbf{G} \times_1 \mathbf{U}_1 \times_2 \mathbf{U}_2 \times_3 \mathbf{U}_3
\]

where \(\mathbf{G}\) is the core tensor and (\(\mathbf{U}_1, \mathbf{U}_2, \mathbf{U}_3\) are the matrices (or tensors) corresponding to mode 1, mode 2 and mode 3, respectively. This decomposition can have different ranks for each mode.

Because the Tucker decomposition can have different ranks for each mode, it is useful for capturing the complex structure of high-dimensional data; on the other hand, it is important to choose the rank of the Tucker decomposition appropriately, and choosing the right rank can be difficult. If the rank is too low, the structure of the data cannot be adequately captured, while too high a rank creates the risk of overlearning.

Algorithms related to Tucker decomposition

Different approaches and algorithms exist for Tucker decomposition. Typical algorithms are described below.

1. HOSVD (High-Order Singular Value Decomposition):

HOSVD is the most basic method for obtaining the Tucker decomposition, which performs a singular value decomposition for each mode of the tensor, thereby obtaining the core tensor and a matrix for each mode. It is computationally efficient, but can be difficult to select ranks and account for certain constraints. For details, see “HOSVD (High-Order Singular Value Decomposition): Overview, Algorithm, and Example Implementation.

2. HOOI (High-Order Orthogonal Iteration):

HOOI is a sequential approximation of the Tucker decomposition using an iterative method and an Alternating Least Squares (ALS) approach, which iteratively updates rank by rank, typically using HOSVD results as initial values. For details, see “HOOI (High-Order Orthogonal Iteration) Overview, Algorithm, and Implementation Examples.

3 TTM (Tensor-Train Matrix):

TTM is a type of iterative method in Tucker decomposition that compresses a tensor by applying a rank to each mode of the tensor. It is said to be particularly efficient for large tensors. For details, see “TTM (Tensor-Train Matrix) Overview, Algorithm and Example Implementation.

4. Randomized Algorithms:

Recently, randomized algorithms have been applied to tensor decomposition as well, using random projections to transform a tensor into a low-rank form, which is then used as the basis for Tucker decomposition. See “Overview of Random Algorithms for Tensor Decomposition and Examples of Implementations” for more details.

5. Tensor Power Method:

The Tensor Power Method is a method for estimating the rank of a tensor. This method iteratively updates the eigenvectors of the tensor and is used to estimate the rank. For details, please refer to “Tensor Power Method Overview, Algorithm and Example Implementation.

These algorithms are used in the Tucker decomposition, and the appropriate method should be selected according to the nature of the tensor and the particular aspect of the problem. In particular, careful algorithm selection and parameter tuning are needed to address issues related to rank selection, computational cost, and convergence.

Application of Tucker decomposition

Tucker decomposition has been applied to data analysis and feature extraction in various fields. The following are examples of such applications.

1. image processing:

By applying Tucker decomposition to image data of three dimensions or more, complex image structures and features can be extracted, for example, in the analysis of medical images and earth observation data.

2. linguistics:

Text data can be modeled as a multidimensional tensor and the Tucker decomposition can be used to extract latent structures and topics in the text. This has applications in topic modeling and text mining.

3. sensor networks:

Tucker decomposition may be applied to multidimensional data from sensor networks to understand their structure and impact in different modes, e.g., for sensor data analysis and anomaly detection.

4. brain science:

Multiple brain activity data (e.g., EEG, fMRI) may be modeled as tensors and Tucker decomposition may be used to extract different brain regions and temporal patterns, which contribute to understanding brain function and studying brain diseases.

5. chemistry:

In the analysis of chemical data, Tucker decomposition is used to analyze spectral and chromatogram data for different chemical components, whereby the features of different components are separated and chemical processes are understood.

6. machine learning:

In dimensionality reduction and feature extraction of tensor data, Tucker decomposition is applied to machine learning tasks, for example, extracting latent patterns and features from high-dimensional data for model training and inference.

Example implementation of Tucker decomposition

A simple example of implementing Tucker decomposition is shown using TensorLy, a Python tensor decomposition library; TensorLy is integrated with scientific computing libraries such as NumPy and SciPy, which makes the implementation of tensor decomposition easier.

First, install TensorLy.

pip install tensorly

Next, the following is a simple implementation of Tucker decomposition using TensorLy.

import numpy as np
import tensorly as tl

# tensor generation
shape = (3, 3, 3)  # Tensor Shape
tensor = tl.tensor(np.arange(np.prod(shape)).reshape(shape))

# Tucker decomposition
rank = (2, 2, 2)  # Rank of each mode
core, factors = tl.decomposition.tucker(tensor, rank=rank)

# Construction of Recovery Tensor
reconstructed_tensor = tl.tucker_to_tensor((core, factors))

# Display Results
print("Original Tensor:")
print(tensor)
print("nCore Tensor:")
print(core)
print("nFactors:")
for mode, factor in enumerate(factors):
    print(f"Factor-{mode + 1}:n{factor}")
print("nReconstructed Tensor:")
print(reconstructed_tensor)

In this example, a 3x3x3 tensor is generated, Tucker decomposition is performed, and the tl.decomposition.tucker function is a Tucker decomposition function provided by TensorLy that decomposes the tensor for a given rank.

Running the above code will display the original tensor, the core tensor, the decomposed matrix for each mode, and the tensor reconstructed from the decomposition.

Challenges of Tucker decomposition and how to address them

There are several challenges to the Tucker decomposition, and countermeasures have been studied to address them. The main challenges of the Tucker decomposition and their countermeasures are described below.

1. rank selection:

Challenge: Selecting an accurate rank is difficult; too low a rank does not accurately capture the structure of the data, and too high a rank creates the risk of over-learning.
Solution: Use cross-validation and information criteria (AIC, BIC, etc.) to select appropriate ranks. Automated methods for rank selection have also been proposed.

2. computational costs:

Challenge: The higher the rank, the higher the computational cost of the Tucker decomposition. This is especially inefficient for large tensors and high-dimensional data.
Solution: The computational cost can be reduced by using approximation and rank reduction techniques, and parallel computing and the use of GPUs can also be considered.

3. initial value dependence:

Challenge: Solutions may converge depending on initial values, and starting from different initial values may lead to different final solutions.
Solution: Either start with several different initial values and select the best result, or devise an initialization method to reduce the influence of initial values.

4. modes with different ranks:

Challenge: Tucker decompositions with different ranks for each mode are common, but their handling is complex.
Solution: A method to adjust ranks for each mode or a method to select ranks for each mode has been proposed.

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“