Introduction to Non-Negative Matrix Factorization Answers 1. What is one of the main advantages of using Non-negative Matrix Factorization (NMF) over Principal Component Analysis (PCA)? A. NMF avoids the occurrence of negative values in its resulting components. B. NMF guarantees unique solutions for all datasets. C. NMF is faster to compute than PCA for any dataset size. D. NMF can handle extraordinarily large datasets without issues. Answer: NMF avoids the occurrence of negative values in its resulting components. (A) NMF provides components that have an intuitive meaning by ensuring non-negativity, which can be a limitation in PCA. 2. Which of the following is a significant disadvantage of Non-negative Matrix Factorization (NMF)? A. NMF requires no parameter adjustments for effective application. B. NMF is inherently scalable to any size of dataset. C. NMF solutions may not be unique, making interpretation more complex. D. NMF always converges to the global optimum. Answer: NMF solutions may not be unique, making interpretation more complex. (C) One of the disadvantages of NMF is that the resulting solutions may not be unique, leading to potential difficulties in interpretation. 3. What factor is crucial when choosing the rank (number of components) in NMF? A. The rank can be any arbitrary number without impact on results. B. Higher ranks always yield better results regardless of data. C. The rank should always be minimized to reduce computation time. D. Choosing an appropriate rank can significantly affect the modeling outcome. Answer: Choosing an appropriate rank can significantly affect the modeling outcome. (D) Selecting the right rank is vital for optimal results in NMF, as it directly affects the representation of the original data. 4. How does the initialization of W and H matrices impact NMF? A. Good initialization can prevent convergence issues. B. The initialization has no impact on the final solution of NMF. C. Poor initialization guarantees that NMF won't converge. D. Initialization only affects computation time, not outcomes. Answer: Good initialization can prevent convergence issues. (A) Suitable initialization approaches can influence the convergence and the quality of the final solution in NMF. 5. What common issue might arise during the application of NMF? A. NMF always converges quickly to the final solution. B. NMF applications do not require any adjustment of its parameters. C. The choice of algorithms does not affect computational speed. D. Algorithms may not converge to an optimal solution depending on data characteristics. Answer: Algorithms may not converge to an optimal solution depending on data characteristics. (D) The convergence of NMF algorithms can be affected by various factors, including data characteristics and initial settings. 6. What is the main goal of Non-Negative Matrix Factorization (NMF)? A. To remove negative values from datasets B. To maximize the size of the data matrix C. To increase the dimensionality of data D. To find a decomposition such that V ≈ WH Answer: To find a decomposition such that V ≈ WH (D) NMF primarily aims to decompose the data matrix into two lower-rank matrices that approximate the original data. 7. What does the coefficient matrix (H) represent in NMF? A. The basis elements along with their weights B. The operational complexity of NMF C. The data points in non-negative form D. The errors in reconstruction Answer: The basis elements along with their weights (A) The coefficient matrix (H) contains the weights used to describe how each basis element contributes to the data points. 8. Which of the following is NOT a typical application of NMF? A. Clustering and categorizing documents B. Identifying patterns in gene expression data C. Extracting features from images D. Increasing the computational complexity of data analysis Answer: Increasing the computational complexity of data analysis (D) NMF is not used to increase computational complexity; rather, it aims for efficient data representation. 9. How does the Alternating Least Squares (ALS) algorithm operate in the context of NMF? A. By updating one matrix while holding the other fixed B. By applying random transformations to the matrices C. By simultaneously updating both W and H matrices D. By using linear regression techniques Answer: By updating one matrix while holding the other fixed (A) The ALS algorithm iteratively updates one matrix while keeping the other fixed, focusing on reducing reconstruction error. 10. What is the basis matrix (W) in NMF commonly used for? A. To hold negative values for non-negative data B. To define the components on which data points are represented C. To increase the dimensions of the data matrix D. To represent the error metrics of the decomposition Answer: To define the components on which data points are represented (B) The basis matrix (W) contains the elements from which the data points are composed. 11. Which of the following statements about Non-Negative Matrix Factorization is true? A. NMF focuses only on linear relationships in data B. NMF ensures extracted components are interpretable C. NMF requires a negative data matrix for processing D. NMF can result in matrices with negative values Answer: NMF ensures extracted components are interpretable (B) One of the key benefits of NMF is that the resulting components often have intuitive and interpretable meanings. 12. In the context of NMF, what is the Frobenius norm used for? A. To calculate the error rates of clustering B. To compare data matrices of different sizes C. To illustrate the dimensions of the data D. To optimize the approximation of V from W and H Answer: To optimize the approximation of V from W and H (D) The Frobenius norm is a common measure used in NMF to optimize the approximation of the original data matrix. 13. Which of the following is true regarding multiplicative updates in NMF? A. They generate new data points outside the original data range B. They are an efficient method for updating W and H C. They involve addition rather than multiplication D. They are slower than the ALS algorithm Answer: They are an efficient method for updating W and H (B) Multiplicative updates are known for being simple and fast when applied to update the W and H matrices.