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Jaehong Oh

∑Notes

T14 — Gradient Flow Convergence (Łojasiewicz)

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  1. #1

    The projected gradient flow on the constraint manifold ,

    u˙=−ΠΣm∇E(u),\dot u = -\Pi_{\Sigma_m} \nabla \mathcal{E}(u),
  2. #2

    Exponential rate via Łojasiewicz. For analytic functions on compact semi-algebraic sets (which includes ), the Łojasiewicz–Simon inequality holds: for some and ,

    ∣E(u)−E(u∗)∣1−θ≤C∥ΠΣm∇E(u)∥|\mathcal{E}(u) - \mathcal{E}(u^*)|^{1-\theta} \leq C \|\Pi_{\Sigma_m}\nabla \mathcal{E}(u)\|