Overview of CP (CANDECOMP/PARAFAC) Decomposition, Algorithm and Example Implementation

CP (CANDECOMP/PARAFAC) Decomposition Overview

CP decomposition (CANDECOMP/PARAFAC) is a type of tensor decomposition and is one of the decomposition methods for multidimensional data. CP decomposition approximates a tensor as the sum of multiple rank-1 tensors. It is usually applied to tensors of three or more dimensions, and we will use a three-dimensional tensor as an example here.

A 3-dimensional tensor \(\mathbf{X}\) is divided into three modes (dimensions), and the CP decomposition of the tensor \(\mathbf{X}\) is expressed 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 rank 1 matrices or vectors. They are associated with the first, second, and third modes of the tensor, respectively, where (R) denotes the rank.

The goal of the CP decomposition is to find a sum of rank 1 tensors that reproduces the given tensor \(\mathbf{X}\) as accurately as possible; each rank 1 tensor corresponds to a basis and coefficients for different modes, which when combined reproduce the original tensor.

Algorithms related to CP (CANDECOMP/PARAFAC) decomposition

Several algorithms have been proposed for CP decomposition (CANDECOMP/PARAFAC). Typical algorithms include Alternating Least Squares (ALS) and Gradient Descent. Each algorithm is described below.

1. Alternating Least Squares (ALS):

ALS is one of the most commonly used algorithms in CP decomposition, and its basic idea is to optimize a matrix of rank 1 tensors for each mode in turn. The basic idea is to optimize the rank 1 tensor matrix for each mode in turn.

• • For each mode, find the optimal matrix by fixing the matrices of the other modes.
  • Repeat optimization for all modes alternately.
  • Repeat until the convergence condition is satisfied.

For more information, see “Alternating Least Squares (ALS) Overview, Related Algorithms, and Example Implementations.

2. Gradient Descent:

The gradient method is an algorithm also used in CP decomposition, which searches for optimal parameters using gradient descent when updating the matrix of rank 1 tensor for each mode. The procedure is as follows

• • Calculate the gradient of the loss function for the matrix of the rank 1 tensor.
  • Update the parameters in the opposite direction of the gradient.
  • Repeat until the convergence condition is satisfied.

For more information on the gradient method, see also “Overview of the Gradient Method and Examples of Algorithms and Implementations.

These algorithms should be chosen appropriately depending on the rank of the tensor and the nature of the data; ALS is usually considered to be relatively easy to optimize for each mode and less dependent on initial values, while Gradient Descent is widely used as part of numerical optimization methods and has the characteristic of moving in the direction of minimizing the objective function using the gradient. Both rely on appropriate initial values and hyperparameters. These algorithms may yield different results in performance comparisons and adaptations for specific problems or data in tensor decomposition.

Application Examples of CP (CANDECOMP/PARAFAC) Decomposition

CP decomposition (CANDECOMP/PARAFAC) is widely used for tensor data analysis in various fields. Examples of their applications are described below.

1. signal processing:

CP decomposition is used to extract signal components and features in signal processing such as audio and image data. For example, it can be used to decompose waveforms of audio data or feature maps of image data to understand potential patterns and structures.

2. chemistry:

In the analysis of chemical data, CP decomposition is used to analyze spectral and chromatogram data for multiple chemical components. This allows spectra and chromatograms of different components to be separated and quantitative information to be obtained.

3. machine learning:

In dimensionality reduction and feature extraction of tensor data, CP decomposition has been applied to machine learning tasks. For example, high-dimensional data such as customer behavior data or sensor data may be decomposed to extract latent features and used as input for models.

4. brain science:

In the analysis of brain activity, CP decomposition of brain data such as EEG (electroencephalography) and fMRI (functional magnetic resonance imaging) enables us to understand different spatial patterns and temporal dynamics.

5. interpolation of tensor data:

CP decomposition is also used to interpolate and complement incomplete tensor data, and to complement missing data based on known latent structures.

These are only a few examples, and CP decomposition can be a method with a wide range of applications in data analysis, feature extraction, dimensionality reduction, and modeling.

Example Implementation of CP (CANDECOMP/PARAFAC) Decomposition

A simple example of implementing CP decomposition is shown 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 Disassembly
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, and when the above code is executed, the original tensor The above code will display the original tensor, the decomposed matrix for each mode, and the tensor reconstructed from the decomposition.

Challenges of CP decomposition and how to address them

Several challenges exist in CP decomposition, and there are several responses to address these.

1. rank selection:

Challenge: CP decomposition requires choosing the rank of the tensor (the number of dimensions in the decomposition); if the rank is too low, the model will not capture enough data, and if it is too high, there is a risk of over-fitting.
Solution: Cross-validation and information criterion can be used to select the best rank.

2. computational cost:

Challenge: Decomposition of higher-order tensors is computationally expensive, and for large data sets and high-dimensional tensors, the computation is very costly.
Solution: Similarity algorithms and parallel processing can be used to speed up the computation, and optimization methods exist for specific problems.

3. effect of noise:

Challenge: If the data contains noise, CP decomposition is sensitive to noise, making accurate decomposition difficult.
Solution: It is important to minimize the effect of noise through data preprocessing and outlier detection, and regularization and anomaly detection methods can be used to reduce the effect of noise.

4. dependence on initial values:

Challenge: The CP decomposition algorithm depends on initial values, and different initial values may yield different results.
Solution: There are ways to start with several different initial values and select the most appropriate result, or random initial values can be used to reduce initial value dependence.

5. dealing with missing data:

Challenge: If the input tensor contains missing data, the performance of CP decomposition may be degraded.
Solution: Missing data can be addressed using missing value completion algorithms or tensor completion methods, or by applying CP decomposition after tensor completion.

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“