## 2 Materials and Methods
### Theoretical Backgrou
#### 2.1.3 Non-Negative Matrix Factorization (NMF)  
NMF is based on the idea that humans recognize objects by combining parts of the object's information, representing object information as features and semantic variables [7]. A piece of object information is factorized into two-part information composed of non-negative values. The purpose of NMF is to reduce object information by capturing only common features. It efficiently represents a large amount of information by representing the entire object through a combination of such part information. The NMF process involves finding the non-negative matrix factor W and H for the non-negative matrix V [7]. Non-negative Matrix Factorization (NMF) factorizes the original matrix into the product of two lower-dimensional matrices:  
\[\text{V}\approx\text{WH}\]  
Here, V is the original data matrix, and W and H are the two matrices resulting from the factorization. All elements of these matrices are non-negative. 1. Initialize two matrices, W and H, with the same size as the original matrix V and all elements being non-negative. The size of W is set to [m, r] and H to [r, n]. Here, r is a given hyperparameter and is generally equal to or smaller than the number of columns in the original matrix. 2. Update W and H iteratively. For each update step, first fix W and update H, then fix H and update W. These updates are carried out in a direction that minimizes the difference between the original matrix V and the product of matrices W and H.
3. Continue this process until a certain number of iterations are reached, or the product of W and H becomes sufficiently close to V. The method of updating H is as follows:  
\[\text{H}<\text{-H}\,.^{*}\,((\text{W}^{\prime}\,\text{*}\,\text{V})\,./(\text{ W}^{\prime}\,\text{*}\,\text{W}\,\text{*}\,\text{H}+\text{epsilon}))\]  
Here, '. *' represents element-wise multiplication, and './' represents element-wise division. Epsilon is a very small constant to prevent division by zero. The method of updating W is as follows:  
\[\text{W}<\text{-W}\,.^{*}\,((\text{V}\,\text{*}\,\text{H})\,./(\text{W}\, \text{*}\,\text{H}\,\text{*}\,\text{H}^{\prime}+\text{epsilon}))\]  
Through this process, matrices W and H that are very close to the original matrix V can be obtained. Here, each column of W forms a basis representing features of the original data, and each row of H represents coefficients for the basis. NMF is used in this way to extract the structure of the original data and is utilized in various fields such as image analysis, text mining, and audio signal processing. The \"\(<\)-\" symbol is used to signify the assignment of updated values to the matrices H and W, indicating the replacement of their current values with newly computed ones. The apostrophe in \"H\" denotes the matrix H prior to its update. 4. Once the updates are completed, the product of W and H approximates the original matrix V. Here, W represents the 'basis vectors', and H represents the 'weights' of these vectors. The two constructed matrices can well represent the patterns of the original matrix. 5. In this way, NMF factorizes the structure of the original matrix into two smaller matrices, understanding the pattern of the data, and performs tasks such as prediction and classification based on this. Thus, NMF is useful in extracting meaningful structures from original data. Under the condition that all elements of the original matrix are non-negative, NMF has the advantage of reflecting 'part' information well. This is especially useful in cases where each pixel or word in data like images or text independently has meaning. Also, unlike other dimension reduction techniques in machine learning and data analysis, NMF has the advantage of ease of interpretation, given that the resulting matrices are non-negative. Due to these features, NMF is used in various fields such as image recognition, text mining, gene expression pattern analysis. The goal of NMF is to minimize the error function.