Łojasiewicz inequality

In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rⁿ, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K

$\operatorname {dist} (x,Z)^{\alpha }\leq C|f(x)|.$

Here α can be large.

The following form of this inequality is often seen in more analytic contexts: with the same assumptions on ƒ, for every p ∈ U there is a possibly smaller open neighborhood W of p and constants θ ∈ (0,1) and c > 0 such that

$|f(x)-f(p)|^{\theta }\leq c|\nabla f(x)|.$

A special case of the Łojasiewicz inequality, due to Polyak, is commonly used to proof linear convergence of gradient descent algorithms.

References

• Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Publications Mathématiques de l'IHÉS (67): 5–42, ISSN 1618-1913, MR 0972342
• Ji, Shanyu; Kollár, János; Shiffman, Bernard (1992), "A global Łojasiewicz inequality for algebraic varieties", Transactions of the American Mathematical Society, 329 (2): 813–818, doi:10.2307/2153965, ISSN 0002-9947, JSTOR 2153965, MR 1046016