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Optimization over bounded-rank matrices through a desingularization enables joint global and local guarantees
Quentin Rebjock, Nicolas Boumal
Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a nonsmooth and nonconvex algebraic variety. Existing techniqu…
Smooth, globally Polyak-Åojasiewicz functions are nonlinear least-squares
Nicolas Boumal, Christopher Criscitiello, Quentin Rebjock
The Polyak-Åojasiewicz (PÅ) condition is often invoked in nonconvex optimization because it allows fast convergence of algorithms beyond strong convexity. A function $f \colon \m…
Sensor network localization has a benign landscape after low-dimensional relaxation
Christopher Criscitiello, Andrew D. McRae, Quentin Rebjock +1
We consider the sensor network localization problem, which is closely related to multidimensional scaling and Euclidean distance matrix completion. Given a ground truth configurati…
Synchronization on circles and spheres with nonlinear interactions
Christopher Criscitiello, Quentin Rebjock, Andrew D. McRae +1
We consider the dynamics of points on a sphere in () which attract each other according to a function of their inner products. When is linear (…
Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices
Quentin Rebjock, Nicolas Boumal
Trust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite. However, if the minima are not isolated, then the Hessian there cannot be pos…
Fast convergence to non-isolated minima: four equivalent conditions for functions
Quentin Rebjock, Nicolas Boumal
Optimization algorithms can see their local convergence rates deteriorate when the Hessian at the optimum is singular. These singularities are inescapable when the optima are non-i…