I'm trying to create a kernelized latent factor model for some given data $R_{i, j} \in \mathbb{R} \; \forall i, j \in S$. This lead me to performing MLE, which lead to minimizing an objective function. However, before I can minimize the objective function, I need to know if it is convex. Here's how I arrived at the objective function:
\begin{align*} \arg\max_{U, V} P(\mathcal{E}, U, V) & = \arg\max_{U, V} \prod_{i, j \in S} P(\varepsilon_{i, j}|u_i, v_j) P(u_i)P(v_j) \\ & = \arg\min_{U, V} \underbrace{\sum_{i, j \in S} \frac{1}{\sigma_{\varepsilon}^2}(K_{i,j} - R_{i,j})^2 + \frac{1}{\sigma_{u}^2}||u_i||^2 + \frac{1}{\sigma_{v}^2}||v_j||^2}_{L(U, V)} && \text{taking log} \\ \end{align*} Where \begin{align*} K_{i,j} & = \Phi(u_i)^{\top}\Phi(v_j) = \exp(-\gamma ||u_i - v_j||^2) && \text{The RBF kernel} \\ R_{i, j} & = K_{i, j} + \varepsilon_{i, j} && \text{Latent factor model for given data } R_{i, j} \\ \varepsilon_{i, j} & \sim \mathcal{N}(0, \sigma_{\varepsilon}^2) && \text{Noise comes from a normal distribution centered at 0} \\ u_i & \sim \mathcal{N}(0, \sigma_u^2) && \text{Prior on } u_i \\ v_j & \sim \mathcal{N}(0, \sigma_v^2) && \text{Prior on } v_j \\ u_i & = \text{The }i\text{-th row of U} \\ v_j & = \text{The }j\text{-th row of V} \\ \end{align*} Now, I want to see if this $L$ is biconvex (i.e., if just $u_i$ or $v_j$ is a variable, and everything else is constant, is the function convex?) However, I'm not so great at matrix calculus, so I keep messing up the derivatives. How do I derive whether or not the hessian matrix, $\nabla_{u_i}^2 L$ is PSD and prove that the objective $L$ is/isn't convex?