In
real algebraic geometry In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomia ...
, 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
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
''U'' in R
''n'', and let ''Z'' be the
zero locus of ƒ. Assume that ''Z'' is not empty. Then for any
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
''K'' in ''U'', there exist positive constants α and ''C'' such that, for all ''x'' in ''K''
:
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
:
A special case of the Łojasiewicz inequality, due to , is commonly used to prove linear
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
of
gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
algorithms.
References
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{{DEFAULTSORT:Lojasiewicz inequality
Inequalities
Mathematical analysis
Real algebraic geometry