1. Nonlinear optimization theory: The basic idea of manifold optimization is to transform complex nonlinear optimization problems into a series of simple local linear optimization problems. This transformation process needs the help of nonlinear optimization theory, such as KKT condition, subgradient and other concepts and methods.
2. Differential geometry: The core of manifold optimization is to find a low-dimensional manifold structure to approximate a high-dimensional nonlinear function. This requires the help of some basic concepts in differential geometry, such as tangent space, cotangent space, Riemannian metric and so on.
3. Machine learning theory: Manifold optimization is widely used in machine learning, especially kernel method. Therefore, some basic theories of machine learning, such as support vector machine and kernel method, are also important theoretical foundations of manifold optimization.
4. Numerical analysis: The solution of manifold optimization usually needs some numerical optimization algorithms, such as gradient descent method and Newton method. Therefore, some basic theories of numerical analysis, such as convergence and stability, are also important theoretical foundations of manifold optimization.
5. Signal processing: Manifold optimization is also widely used in signal processing, especially in blind source separation and image denoising. Therefore, some basic theories of signal processing, such as Fourier transform and wavelet transform, are also important theoretical foundations of manifold optimization.