In these proceedings, we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of k-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on k ≥ 2 complex eigenvalues. As a new result, we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of $M$ linear equations in $N$ unknowns: $(\bm a_k,\bm x)=b_k, \, k=1,\ldots,M$ on the $N-$sphere ${\bf x}^2=N$. We treat both the $N-$component vectors $\bm a_k$ and parameters $b_k$ as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case $M>N$ in the framework of the Kac-Rice approach combined with the Random Matrix Theory for Wishart Ensemble. Then we perform its asymptotic analysis as $N\to \infty$ at a fixed $\alpha=M/N>1$ in various regimes. In particular, this analysis allows to extract the Large Deviation Function for the density of the smallest Lagrange multiplier $\lambda_{min}$ associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value ${\cal E}_{min}$ of the loss function as $N\to \infty$. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the Large Deviation function for the density of ${\cal E}_{min}$ at $N\gg 1$ and {\it any} fixed ratio $\alpha=M/N>0$. As a by-product, we find the {\it compatibility threshold} $\alpha_c<1$ which is the value of $\alpha$ beyond which a large random linear system on the $N-$sphere becomes typically incompatible.

We use Kac-Rice method to analyze statistical features of an "optimization landscape" of the loss function in a random version of the Oblique Procrustes Problem, one of the simplest optimization problems of the least-square type on a sphere.