Local Reconstruction

Years and Authors of Summarized Original Work

• 2010; Saks, Seshadhri

Problem Definition

Consider some massive dataset represented as a function f : D ↦ R, where D is discrete and R is an arbitrary range. This dataset could be as varied as an array of numbers, a graph, a matrix, or a high-dimensional function. Datasets are often useful because they possess some property of interest. An array might be sorted, a graph might be connected, a matrix might be orthogonal, or a function might be convex. These properties are critical to the use of the dataset. Yet, due to unavoidable errors (say, in storing the dataset), these properties might not hold any longer. For example, a sorted array could become unsorted because of roundoff errors.

Can we find a function g : D ↦ R that satisfies the property and is “sufficiently close” to f? Let us formalize this question. Let $\mathcal{P}$ denote a property, which we define as a subset of functions. We define a distance between functions, \(\mathsf{dis...