Algebraic Complete Intersection
A concept in algebraic geometry referring to a real algebraic variety formed as the common zero set of D-d polynomials whose gradients are linearly independent on a dense open set, so that the dimension can be read directly from the number of equations and the total degree is given by Bézout's theorem.
Timeline (last 90 days)
The paper "Testing Algebraic Complete Intersections" studies the statistical problem of testing, given independent and identically distributed samples from a probability distribution on a high-dimensional real space, whether that distribution is concentrated near an algebraic complete intersection.
An algebraic complete intersection is defined as the common zero set of D-d polynomials in R[x_1,...,x_D], where these polynomials have linearly independent gradients on a dense open set, in which case the zero set has dimension exactly d.
For an algebraic complete intersection whose polynomials have degrees d_i, its total degree is given by Bezout's theorem as the product of the d_i.
The family of general smooth manifolds is too vast and lacks a finite parameterization, which makes testing the general manifold hypothesis computationally infeasible; this is the motivation for the paper tightening the manifold family to algebraic complete intersections.
The paper requires the algebraic complete intersection to have a specified dimension, bounded degree, and bounded condition number; this restriction sets a threshold for computability and statistical stability, at the cost that the hypothesis class is narrowed and may fail to cover some real complex data distributions.
All Facts (5)
The paper "Testing Algebraic Complete Intersections" studies the statistical problem of testing, given independent and identically distributed samples from a probability distribution on a high-dimensional real space, whether that distribution is concentrated near an algebraic complete intersection.
50%UnverifiedAn algebraic complete intersection is defined as the common zero set of D-d polynomials in R[x_1,...,x_D], where these polynomials have linearly independent gradients on a dense open set, in which case the zero set has dimension exactly d.
50%UnverifiedFor an algebraic complete intersection whose polynomials have degrees d_i, its total degree is given by Bezout's theorem as the product of the d_i.
50%UnverifiedThe family of general smooth manifolds is too vast and lacks a finite parameterization, which makes testing the general manifold hypothesis computationally infeasible; this is the motivation for the paper tightening the manifold family to algebraic complete intersections.
50%UnverifiedThe paper requires the algebraic complete intersection to have a specified dimension, bounded degree, and bounded condition number; this restriction sets a threshold for computability and statistical stability, at the cost that the hypothesis class is narrowed and may fail to cover some real complex data distributions.
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