Overview of Non-Negative Tensor Factorization (NTF), its algorithm and examples of implementation

Non-Negative Tensor Factorization (NTF) Overview

Non-Negative Tensor Factorization (NTF) is a method for obtaining a representation of multidimensional data, which decomposes a tensor (multidimensional array) into non-negative elements. and signal analysis, feature extraction, and dimensionality reduction.

The objective of NTF is to decompose a given tensor \(X\) into a product of \(r\) nonnegative matrices (or tensors), specifically expressed as follows.

\[ X \approx A_1 \otimes A_2 \otimes \ldots \otimes A_N \]

where \(X\) is a \(N\)-dimensional tensor, \(A_1, A_2, \ldots, A_N\) is a nonnegative matrix or tensor, and \(\otimes\) denotes the tensor product. \(r\) is the rank of the decomposition or the number of components.

Algorithms related to Non-Negative Tensor Factorization (NTF)

Several algorithms exist for Non-Negative Tensor Factorization (NTF). The following is a list of typical algorithms related to NTF.

1. ALS-NMF (Alternating Least Squares – Non-Negative Matrix Factorization):

ALS-NMF is an extension of the Non-Negative Matrix Factorization (NMF) algorithm to NTF, which performs nonnegative decomposition of a tensor by alternately updating the nonnegative matrix. 1. the ALS-NMF is a relatively simple and easy-to-understand method. For details, please refer to “Alternating Least Squares for Non-Negative Matrix Factorization (ALS-NMF): Overview, Algorithm, and Example Implementation“.

2. HOSVD (Higher Order Singular Value Decomposition):

HOSVD is a method for decomposing tensors using higher-order singular value decomposition as described in “Overview of Singular Value Decomposition (SVD) and examples of algorithms and implementations“, in which the tensor is divided into multiple modes (dimensions) and singular value decomposition is performed for each mode. 3. Beta-Divergence based decomposition

3. Beta-Divergence based Methods:

In tensor decomposition, Beta-Divergence is used to preserve element-by-element non-negativity. This includes KL-divergence and I-divergence, which use gradient and multiplicative update methods to optimize under non-negative constraints.

4. Tensor Train (TT) Decomposition:

Tensor Train decomposition is a method for decomposing a tensor into a product of several smaller tensors (core tensors); this method is particularly effective for high-dimensional tensors and provides a low-rank approximation. For more information on TTD, see “Tensor Train Decomposition Overview, Algorithm and Example Implementation“.

It is important to select a suitable algorithm depending on the nature of the tensor and the purpose. ALS-NMF and Beta-Divergence based Methods are widely used as alternatives, but the most effective method for a particular problem depends on the problem and should be carefully considered. However, the most effective method for a particular problem depends on the problem and should be carefully considered.

Application of Non-Negative Tensor Factorization (NTF)

Non-Negative Tensor Factorization (NTF) has been widely applied in various fields. The following is a list of typical applications of NTF.

1 Image Processing and Feature Extraction:

NTF is used for image data decomposition and feature extraction. For example, it is sometimes used to decompose multispectral imaging (MSI) data to extract features of different materials.

2. audio signal processing:

NTF is useful in the analysis of speech signals and the decomposition of musical data, and has been applied to source separation of different instruments and feature extraction of musical data.

3. Text Mining and Topic Modeling:

In the analysis of text data, NTF is used for topic modeling and extracting latent structures of text data, and methods based on non-negative matrix factorization (NMF) and NTF of documents are employed.

4. analysis of brain imaging data:

In the analysis of brain imaging data (e.g. fMRI data), NTF is used to extract different brain regions and potential patterns of brain activity.

5. recommendation systems:

NTF is employed to decompose user behavior and evaluation data to extract potential user preferences and item characteristics.

6. chemical data analysis:

In chemical data analysis, NTFs are used to extract the characteristics of different substances from multiple spectral data.

These applications demonstrate the usefulness of NTF to effectively decompose data and signals with non-negativity and extract potential features. Other applications of NTFs include the medical field, environmental monitoring, economics, social network analysis, and many other fields.

For an example implementation of Non-Negative Tensor Factorization (NTF)

Below is a simple example of NTF implementation using Python and NumPy. This example uses the ALS-NMF (Alternating Least Squares – Non-Negative Matrix Factorization) approach.

import numpy as np

def als_nmf(X, rank, max_iter=100, tol=1e-4):
    # X: input tensor
    # rank: Rank of decomposition (number of components)
    # max_iter: number of repetitions
    # tol: convergence threshold
    
    # Tensor Dimensions
    N = len(X.shape)
    
    # Initialize random non-negative matrix
    factors = [np.random.rand(X.shape[i], rank) for i in range(N)]
    
    for _ in range(max_iter):
        # Update matrices with non-negative constraints for each mode
        for mode in range(N):
            # Loop all modes
            indices = [i for i in range(N) if i != mode]
            
            # Compute tensor convolution
            t_prod = np.ones_like(X)
            for i in indices:
                t_prod = np.multiply(t_prod, factors[i])
            
            # Non-negative matrix factor update
            factors[mode] = np.multiply(factors[mode], np.tensordot(X, t_prod, axes=(indices, indices)) / np.tensordot(t_prod, t_prod, axes=(indices, indices)))
            
        # convergence judgment (judgement)
        reconstruction = np.tensordot(factors[0], np.tensordot(factors[1], factors[2], axes=(1, 0)), axes=(1, 0))
        error = np.linalg.norm(X - reconstruction)
        if error < tol:
            break
    
    return factors

# Test Data Generation
np.random.seed(42)
tensor_shape = (5, 5, 5)
X = np.random.rand(*tensor_shape)

# NTF Execution
rank = 3
factors = als_nmf(X, rank)

# Displays the decomposed matrix
for i, factor in enumerate(factors):
    print(f"Factor {i + 1}:n{factor}n")

In this example, the tensor is decomposed based on the ALS-NMF approach. In real-world applications, parameters and initialization methods will need to be tailored to specific data, and the use of more advanced algorithms and libraries (e.g., TensorLy, HOOI described in “HOOI (High-Order Orthogonal Iteration) Overview, Algorithm, and Implementation Examples“, etc.) may also be considered.

Challenges of Non-Negative Tensor Factorization (NTF) and how to deal with them

Several challenges exist in the NTF. The challenges and their solutions are described below.

1. nonconvexity and local solutions:

Challenge: The optimization problem of NTF is nonconvex and there may be multiple local solutions.
Solution: By running the problem multiple times with different initial values, we can search for different local solutions and increase the likelihood of convergence to the optimal solution. Random initialization or methods that vary the initial value constraints may also be employed.

2. computational cost and scalability:

Challenge: NTF for high-dimensional tensor data is computationally expensive and difficult to converge in practical time.
Solution: Apply fast and efficient algorithms and approximation methods. Distributed processing or the use of GPUs may also improve the computational speed.

3. selecting appropriate ranks:

Challenge: It is difficult to select an appropriate rank (rank of decomposition) for a tensor.
Solution: There are methods to select ranks using cross-validation or information criterion. This prevents the model from becoming overly complex and allows the appropriate rank to be estimated.

4. data sparsity:

Challenge: When tensor data is sparse (many entries are zero), effective decomposition is difficult.
Solution: To cope with sparse data, algorithms that consider sparse constraints and methods that introduce sparse factor matrices have 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“