Condition Number
A metric that measures the sensitivity of a mathematical problem to input perturbations, defined as the ratio of the largest to the smallest eigenvalue of a matrix. In optimization it reflects the degree of curvature variation across different directions of the loss surface; a high condition number indicates an ill-conditioned problem.
Timeline (last 90 days)
In the context of algebraic geometry, the condition number measures the sensitivity of the solution set of a polynomial system under perturbations, and a bounded condition number rules out pathological cases such as severe bending, near self-intersection, or dimensional collapse of the solution set.
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.
The requirement of a bounded condition number may not be satisfied in many real scenarios, and whether the theoretical guarantees still hold after relaxing the condition is not discussed in the original paper.
The condition number is defined as the ratio of the matrix's largest eigenvalue to its smallest eigenvalue, κ = λmax/λmin.
All Facts (4)
In the context of algebraic geometry, the condition number measures the sensitivity of the solution set of a polynomial system under perturbations, and a bounded condition number rules out pathological cases such as severe bending, near self-intersection, or dimensional collapse of the solution set.
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.
50%UnverifiedThe requirement of a bounded condition number may not be satisfied in many real scenarios, and whether the theoretical guarantees still hold after relaxing the condition is not discussed in the original paper.
50%UnverifiedThe condition number is defined as the ratio of the matrix's largest eigenvalue to its smallest eigenvalue, κ = λmax/λmin.
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