A Representation–Optimization Trichotomy for Objectives Parameterized via Matrix Exponential Maps

The matrix exponential parameterization J (θ) = Je(exp(θ)), with θ in a matrixLie algebra g ⊂ Rn×n , arises in geometric optimization, structured reinforce-ment learning, and rotation-valued signal processing. A central computationalquestion is how the gradient Lipschitz constant L(R)—which governs step-sizeselection and convergence rates of first-order methods—scales with the searchradius R. We prove a representation–optimization trichotomy: the growth rateof L(R) is determined entirely by the algebraic type of g. For compact algebras(g ⊆ u(n), e.g. so(n), su(n)), L(R) = O(1) uniformly in R, enabling radius-independent step sizes. For algebras containing a hyperbolic element (e.g. gl(n),sl(n)), L(R) = O(e2R ); this rate is sharp on gl(n) via an explicit witnessconstruction. Non-compact algebras without hyperbolic elements (e.g. se(3))occupy an intermediate polynomial regime L(R) = O(R2 ), established by adirect block-matrix analysis.As a consequence, Lie-Projected Gradient Ascent√(LPG) achieves O(1/ T ) convergence for compact algebras with a closed-forme 2 /ε2 ) toO(n2 K) projection step, and reduces sample complexity from O(n2e g /ε ) in Lie Group Markov Decision Processes. Numerical experiments onO(dSO(3)K and SE(3) confirm the predicted scaling behaviours.