Overview of PARAFAC2 (Parallel Factor 2) Decomposition, Algorithm and Implementation Example

PARAFAC2 (Parallel Factor 2) Decomposition Overview

PARAFAC2 (Parallel Factor 2) decomposition is a method of tensor decomposition and is a type of mode-based tensor decomposition described in “Overview of Mode-based Tensor Decomposition, Algorithm and Implementation Examples“. The usual PARAFAC (canonical decomposition) approximates tensors of three or more dimensions as a sum of lower-rank tensors, but PARAFAC2 can be applied to tensors of more general geometry.

PARAFAC2 decomposes a tensor as two low-rank tensors. Specifically, it decomposes the third-order tensor \(\mathbf{X}\) as follows.

\[
\mathbf{X} \approx \sum_{r=1}^{R} \mathbf{a}_r \circ \mathbf{b}_r \circ \mathbf{c}_r
\]

where \(\circ\) represents the outer product, and \(\mathbf{a}_r\)、\(\mathbf{b}_r\)、\(\mathbf{c}_r\) are vectors, respectively. In this decomposition, the three vectors \(\mathbf{a}_r\)、\(\mathbf{b}_r\)、\(\mathbf{c}_r\) correspond to different modes of the tensor.

The PARAFAC2 approach is to approximate the tensor based on the appropriate ranks for each of the different modes, which makes PARAFAC2 useful for tensors with complex geometry data, which can have different ranks for different modes.

Different algorithms have been proposed for actual implementation, including the Alternating Least Squares (ALS) approach described in “Alternating Least Squares (ALS) Overview and Related Algorithms and Example Implementations and “Overview of Gradient Methods and Related PARAFAC2 is used in conjunction with, for example, singular value decomposition of tensor data, which is a useful approach for extracting the structure of the data.

Algorithms related to PARAFAC2 (Parallel Factor 2) decomposition

The algorithms associated with the PARAFAC2 (Parallel Factor 2) decomposition will typically use optimization methods or iterative methods to decompose the tensor. An overview of these methods is given below.

1. Alternating Least Squares (ALS):

ALS is a commonly used method for PARAFAC2 decomposition, where the basic idea is to alternately optimize the factor matrix for each mode. The specific procedure is as follows

• • Mode 1 optimization: Fix the factor matrix associated with mode 1 and perform a low-rank decomposition of the portion of the tensor related to the remaining modes.
  • Mode 2 optimization: Fix the factor matrix associated with mode 2 and perform a low-rank decomposition of the portion of the tensor related to the remaining modes.
  • Repeat the optimization for each mode in turn.

See Alternating Least Squares (ALS) Overview and Related Algorithms and Example Implementations for details.

2. Gradient Descent:

The gradient method calculates the gradient of each element of the tensor and updates the factor matrix based on the gradient. This method allows for the decomposition of the tensor through an iterative optimization process. For more details on the gradient method, please refer to “Overview of the Gradient Method, Algorithms, and Examples of Implementations.

3. Non-Negative Tensor Factorization (NTF):

Another approach to adding non-negative constraints in PARAFAC2 is to incorporate the Non-Negative Tensor Factorization (NTF) method, or to introduce constraints that preserve a non-negative factor matrix. See “Non-Negative Tensor Factorization (NTF) Overview, Algorithms, and Example Implementations” for more details.

It is important to select the appropriate algorithm depending on the rank of the tensor and the nature of the data, and to tailor the algorithm to the specific problem, such as initialization, convergence conditions, and introduction of regularization. Cross-validation and hyperparameter tuning will also be considered in algorithm selection and parameterization.

Application of PARAFAC2 (Parallel Factor 2) Decomposition

The PARAFAC2 (Parallel Factor 2) decomposition has been used to analyze tensor data in many different fields. The following are examples of applications of the PARAFAC2 decomposition.

1. neuroscience:

PARAFAC2 decomposition is used to analyze tensor data of brain activity. In the analysis of EEG data with multiple time-series channels and functional MRI data, decomposition for different spatial and frequency modes is performed to extract the potential structure of brain activity.

2. sensor networks:

PARAFAC2 decomposition is useful when analyzing multidimensional data (e.g., time series data, spatial data) derived from sensor networks to identify structures and influences associated with different modes.

3. clinkout:

PARAFAC2 has also been applied in the analysis of chemical data. For example, in the analysis of chromatographic fingerprints data, PARAFAC2 is used to identify different samples and chemical components.

4. communication networks:

In the analysis of communication networks, PARAFAC2 is applied to decompose multi-factor tensor data to understand structures and trends. For example, PARAFAC2 is used in social network analysis and communication trend analysis.

5. image processing:

When analyzing image tensor data, PARAFAC2 decomposition is used to capture features and spatial structure. For example, it is applied to detect facial features and objects from multi-dimensional image tensors.

PARAFAC2 decomposition has been used for data analysis and feature extraction in various fields, and in particular, its ability to simultaneously extract latent structures associated with different modes has contributed to its diverse applications.

Example Implementation of PARAFAC2 (Parallel Factor 2) Decomposition

A simple example of PARAFAC2 implementation is shown using TensorLy, a Python tensor decomposition library; TensorLy is integrated with scientific computing libraries such as NumPy and SciPy, making it an easy tool for implementing tensor decomposition.

First, install TensorLy.

pip install tensorly

Next, the following will be a simple implementation of the PARAFAC2 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))

# PARAFAC2 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 PARAFAC2 decomposed. tl.decomposition.parafac function is a PARAFAC decomposition function provided by TensorLy that decomposes a 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.

PARAFAC2 (Parallel Factor 2) Decomposition Challenges and their Remedies

Several challenges exist in the PARAFAC2 (Parallel Factor 2) decomposition, and countermeasures are being considered to address them.

1. rank selection:

Challenge: Rank selection is critical in tensor decomposition, and choosing the right rank is difficult. If the rank is too small, the model cannot capture the complexity, and if the rank is too large, there is a risk of overlearning.
Solution: Use cross-validation or information criterion (AIC, BIC, etc.) to select appropriate ranks. Automated methods for rank selection have also been proposed.

2. initial value dependence:

Challenge: Tensor decomposition may depend on initial values and may converge to different final solutions when starting from different initial values.
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.

3. computational cost:

Challenge: Tensor decomposition, especially for high ranks, is computationally expensive and inefficient for large data sets and high-dimensional tensors.
Solution: Use of approximation and rank reduction techniques can reduce computational cost, and parallel computing and GPU utilization can also be considered.

4. handling of non-negative constraints:

Challenge: When data are non-negative, it is difficult to obtain a factor matrix that satisfies the non-negative constraints.
Solution: Combine with non-negative value tensor decomposition (NTF) or introduce methods that take non-negative value constraints into account.

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“