symsub_dist: Symmetric Subspace Distance

In Columbia-PRIME/PCPhelpers: Provides a bunch of functions to help with PCP experimentation.

Symmetric Subspace Distance

Description

see: http://www.paper.edu.cn/scholar/showpdf/NUT2cN2INTz0YxeQh (Wang et al., 2006) for more info

Usage

symsub_dist(U, V)

Details

Because all solutions may not successfully identify the correct number of patterns, symmetric subspace distance compares matrices of different dimensions (e.g., true score matrix 1,000 x 4 vs. solution scores 1,000 x 5). Symmetric subspace distance defines a distance measure between two linear subspaces, where the number of individual vectors is constant but the number of elements they contain may differ. The symmetric distance between m-dimensional subspace U and n-dimensional subspace V is defined as d(U, V) =sqrt{max (m, n)-sum_{i=1}^{m} sum_{j=1}^{n}left(u_{i}^{mathrm{T}} v_{j}right)^{2}} If two subspaces largely overlap, they will have a small distance; if they are almost orthogonal, they will have a large distance. To use this method to compare simulations with solutions, we took orthonormal bases of all included matrices and calculated the ratio between symmetric subspace distance and sqrt(max(m,n)). Two subspaces are similar if and only if this ratio is leq frac{1}{2}.

Columbia-PRIME/PCPhelpers documentation built on April 24, 2022, 7:57 p.m.