Overview of Mode-based Tensor Decomposition, Algorithm and Implementation Examples

Overview of Mode-based Tensor Decomposition

Mode-based tensor decomposition is a method of decomposing a multidimensional data tensor into a product of lower-rank tensors, which can be used specifically to decompose the tensor and extract latent structures and patterns in the data set. Tensor decomposition can also be viewed as a multidimensional extension of matrix decomposition (e.g., SVD).

The following is a basic overview of the Modi type tensor decomposition.

1. Tensor:

A tensor is a multidimensional array, which may include scalars (zero-order tensors), vectors (first-order tensors), and matrices (second-order tensors). Modi type tensor decomposition is usually concerned with tensors of third-order or higher.

2. CP decomposition (canonical decomposition):

One of the common methods of model-type tensor decomposition is the CP decomposition, which approximates a tensor as a linear combination of multiple rank-1 tensors (outer products), specifically of the form.

\[ X \approx \sum_{r=1}^{R} a^{(1)}_r \circ a^{(2)}_r \circ a^{(3)}_r \]

where \(X\) is the original tensor, \(R\) is the rank, \(a^{(i)}_r\) is the vector, and \(\circ\) is the outer product.

3. ALS (Alternating Least Squares):

Modes type tensor decomposition such as CP decomposition is usually formulated as an optimization problem using the least squares method, etc. ALS is one of the methods to solve this optimization problem, which is an approach to optimize each mode one by one, and since ALS iteratively repeats optimization for each mode, convergence is relatively ALS is considered to have relatively high convergence because it iteratively repeats optimization for each mode.

4. Tensor Train Decomposition:

Another method of model-type tensor decomposition is Tensor Train Decomposition. This method decomposes a tensor into a product of multiple tensor trains and is effective for high-dimensional tensors.

Modi type tensor decomposition is widely used to capture the features and structure of data, and in specific applications, the best tensor decomposition method is selected according to the nature and distribution of the data.

Algorithms related to Mode-based tensor decomposition

Several algorithms exist for modal tensor decomposition. The following is a list of typical modal tensor decomposition algorithms.

1. the CANDECOMP/PARAFAC (CP) decomposition:

CP decomposition approximates a tensor as the sum of multiple rank-1 tensors, such as Alternating Least Squares (ALS), which is often used to optimize for each mode. It is also known as non-negative tensor factorization (NMF). For details, please refer to “CP (CANDECOMP/PARAFAC) Decomposition Overview, Algorithm and Implementation Examples“.

2. the Tucker decomposition:

The Tucker decomposition decomposes a tensor with different ranks for each mode. The Tucker decomposition is more flexible than the CP decomposition, but tends to be more computationally expensive. See “Tucker Decomposition Overview, Algorithm, and Implementation Examples” for details.

3 Higher Order Singular Value Decomposition (HOSVD):

HOSVD can be regarded as a special case of the Tucker decomposition: In HOSVD, singular value decomposition (SVD) is performed for each mode, and the tensor is decomposed by the combination of the basis matrix and the corresponding singular values in each mode. For details, see “Overview of Higher Order Singular Value Decomposition (HOSVD), Algorithm and Example Implementation.

4 Tensor Train Decomposition:

Tensor Train Decomposition decomposes a tensor into a product of multiple tensor trains. This method is effective for tensors whose models have high dimensionality, and a so-called Tensor Train rank is maintained for each column. See “Tensor Train Decomposition Overview, Algorithm and Example Implementation” for more details.

The algorithm to be selected will generally be based on considerations such as computational cost, flexibility of decomposition, and control of ranks. The optimal algorithm will depend on the specific problem one is trying to solve.

Application of Mode-based tensor decomposition

Modi type tensor decomposition has been widely applied in many different areas. The following are examples of applications.

1. image processing:

Modi type tensor decomposition is used to decompose multi-dimensional image tensors to extract latent patterns and features. For example, tensor decomposition helps to understand the features of image data in tasks such as face recognition, object detection, and image completion.

2. sensor networks:

In a sensor network, multiple sensors generate multidimensional data. Modal tensor decomposition is used to extract potential features and trends from the tensor data obtained from these sensors, e.g., environmental monitoring, sensor fusion, and anomaly detection.

3. neuroscience:

Neuroscience research yields complex data with temporal and spatial information, and Modi type tensor decomposition is used to analyze functional brain networks and patterns of brain activity.

4. social network analysis:

Social media and online communication data are multidimensional in terms of users, time, and content. Modi type tensor decomposition is used to extract potential topics, trends, and influences from these data.

5. cloud computing:

In a cloud computing environment, multiple resources and metrics vary with time and location. Modetype tensor decomposition is used to understand cloud resource usage patterns and cloud service performance.

These are only a few examples, and modetype tensor decomposition can be a highly applicable algorithm applied to analyze data and extract features in a variety of fields.

Example implementation of Mode-based tensor decomposition

As an example of implementing a modetype tensor decomposition, we show a CP decomposition (canonical decomposition) using TensorLy, a Python tensor decomposition library TensorLy is integrated with scientific computing libraries such as NumPy and SciPy, which makes it easy to implement tensor decomposition.

First, install TensorLy.

pip install tensorly

Next, the following is a simple implementation of CP 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))

# CP decomposition
rank = 2  # Rank
factors = tl.decomposition.parafac(tensor, rank=rank)

# Construction of Recovery Tensor
reconstructed_tensor = tl.kruskal_to_tensor(factors)

# Display Results
print("Original Tensor:")
print(tensor)
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 of rank 2 is generated and then CP decomposed. tl.decomposition.parafac function is a CP decomposition function provided by TensorLy that decomposes the tensor for a given rank. Running the above code will display the original tensor, the decomposed matrix for each mode, and the tensor reconstructed from the decomposition.

Challenges of Mode-based Tensor Decomposition and their Countermeasures

Moda-type tensor decomposition, like most data analysis methods, has some challenges. Below we discuss some of the challenges of the model-type tensor decomposition and how to overcome them.

1. rank selection:

Challenge: Rank is an important hyperparameter in tensor decomposition, and it is difficult to select an appropriate rank.
Solution: Use cross-validation and information criterion (AIC, BIC, etc.) to evaluate the appropriate complexity of the model and select appropriate ranks. Automated methods for rank selection have also been proposed.

2. computational cost:

Challenge: Tensor decomposition, especially for high ranks, can be computationally expensive, which is especially problematic for large, high-dimensional data.
Solution: Computational cost can be reduced by using approximation and rank reduction methods. Parallel computing and the use of GPUs can also be considered.

3. model overfitting:

Challenge: High-ranked models may fit data including noise, and there is a risk of overlearning.
Solution: Overfitting can be prevented by introducing a regularization term or by selecting an appropriate rank. It is also important to evaluate model performance using cross-validation.

4. initial value dependence:

Challenge: Results may vary depending on the initial values of the model, leading to convergence to a locally optimal solution.
Solution: Either start with several different initial values and select the best solution, or devise an algorithm initialization method to reduce the effect of initial values.

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“