Optimization landscape in the simplest constrained random least-square problem

Abstract We analyze statistical features of the ‘optimization landscape’ in a random version one simplest constrained optimization problems least-square type: finding best approximation for solution system M linear equations N unknowns: ( k , x ) = b 1, …, on -sphere 2 . treat both -component vectors and parameters as independent mean zero real Gaussian variables. First, we derive exact expressions number stationary points loss function overcomplete case > framework Kac-Rice approach combined with matrix theory Wishart ensemble. Then perform its asymptotic analysis → ∞ at fixed α / 1 various regimes. In particular, this allows to extract large deviation density smallest Lagrange multiplier λ min associated problem, way find most probable value. This can be further used predict minimal value E min ∞. Finally, develop an alternative based replica trick conjecture form ≫ any ratio 0. As by-product, compatibility threshold c < which is beyond becomes typically incompatible.