This article mainly simulates four NMF algorithms: KL-NMF, incremental KL-NMF, EUC-NMF, and incremental EUC-NMF.
The continuous development of non-negative matrix factorization theory provides new ways to solve the problem of blind source separation, and it has quickly become a hot issue in the field of blind source separation. This chapter mainly introduces the mathematical models and related theories of blind source separation and nonnegative matrix factorization.
Nonnegative matrix factorization is an important method for processing data. Its essence is a method that uses nonnegative constraints to obtain data representation. The NMF theory can be described as follows: decompose a non-negative matrix into the product of two non-negative matrices, that is, for an arbitrary non-negative matrix V, the NMF algorithm can decompose it into the product of a non-negative matrix W and a non-negative matrix H. The mathematical model of NMF can be expressed as follows
The NMF algorithm is a new multi-variable statistical analysis method with a simple principle, a concise algorithm, and is easy to understand and execute. In order to obtain the final desired decomposition matrix, the objective function must be optimized and the update rules of the factor matrix during the iteration process must be established. Only in this way can the purpose of matrix decomposition be finally achieved. Many important problems in signal processing can be attributed to certain parameters or direct signal estimation problems under given conditions, and blind source separation is no exception. These parameter or signal estimates minimize or maximize a given objective function, also called the cost function. This process is called optimization. Optimization given certain conditional constraints is called constrained optimization. In other words, solving parameters or estimating signals translates into establishing an appropriate objective function.
Simulation results
nmf_kl: final_err=0.014312
nmf_alg: final err=0.014312
nmf_euc: final_err=0.8626
nmf_alg: final err=0.8626
nmf_kl: final_err=1.9262e-06
nmf_alg: final err=0.00012495
nmf_euc: final_err=0.14091
nmf_alg: final err=0
