Inquiries about the "Fig. 2" from original paper "Greedy Sensor Placement With Cost Constraints"

[MakimaPhd]

Dear Prof. Travis Askham,

Thank you for open-sourcing your codes relating to the manuscript "Greedy Sensor Placement With Cost Constraints". Nevertheless, I fail to understand how the minimal reconstruction error is calculated in Fig. 2 from the original paper. I think theorem 1 derives an upper bound on the reconstruction error instead of a lower bound. Then how exactly the lower error bound is calculated confused me.

I will appreciate it if the code relating to "Fig. 2" is open-sourced. Looking forward to your early reply. Hoping you have a nice day!

[askhamwhat]

Hello, I cannot locate the file used but I could try to hack something together. Perhaps I can try to answer your question without producing the file though. The lower bound is indeed not related to Theorem 1. We are measuring error for the plot in terms of formula (4) in the paper: || Xtest- Xtest(:,J)(pinv(Xtrain(:,J))Xtrain) ||_F / || Xtest ||_F The lower bound is based on the fact that the matrix Xtest(:,J)(pinv(Xtrain(:,J))Xtrain) is rank |J|. The smallest that ||Xtest - B ||_F / ||Xtest||_F could be for any matrix B of rank |J| is sqrt(sigma(|J|+1)^2 + ... + sigma(end)^2), where sigma is the set of singular values of Xtest (cf the Eckhart Young theorem). This is a crude estimate of the best possible interpolation error because we are not looking for any B but for B of a specific form (specifically B = Xtest(:,J)*C where J is some set of good interpolation indices). Hope that clarifies, Travis