Basins of attraction of minima that trap gradient descent

Characterize the basins of attraction of local minima in the empirical risk landscape R̂(θ) = (1/n) ∑_{i=1}^n ℓ_a(x_i·θ, x_i·θ*) for phase retrieval with Gaussian design x_i ∼ N(0, I_d), in the proportional regime n/d → α and at fixed overlap q = θ·θ*, to determine which minima are responsible for trapping gradient descent when a band of marginally stable minima exists.

Background

The paper analyzes the non-convex empirical risk landscape of phase retrieval and shows, via Kac–Rice and BBP analysis, that for certain (α, q) there exists a band of marginally stable local minima. Although algorithmic success correlates with the onset of a BBP instability at typical minima, the authors note that identifying which specific minima trap gradient descent requires a dynamical notion beyond landscape counts.

This motivates an explicit open problem: to characterize the basins of attraction of local minima in the high-dimensional phase retrieval landscape under Gaussian design, which would clarify which minima are dynamically relevant for trapping gradient descent in regimes where many marginally stable minima coexist.