Local Property Restoring

In this paper we present the new notion of local restoring of properties. A restorer gets implicit access, in the form of oracle queries, to a function (which may correspond to an object such as a graph, a code or a matrix) that is close to having a property, such as function monotonicity or graph bipartiteness. The restorer then, using a small number of queries to the function, can locally compute the value of a “corrected” function in any desired point. The corrected function is required to be both close to the original function and strictly have the property. In the case of error correcting codes, this corresponds to local self-correcting. Here we extend the definition and study for general functions and properties. We define the new notion and go on to give restorers for properties in the dense graph model: Bipartiteness and ρ-Clique, and for function monotonicity. We also show some general relations between property restoring and other notions such as property testing and tolerant property testing.

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