Every entry in a matrix counts for the L2-spectral radius similarity. Suppose that A1,…,Ar,B1,…,Br are real n×n-matrices. Set A⊗2=A⊗A. Define the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) to be the number
ρ(A1⊗B1+⋯+Ar⊗Br)ρ(A⊗21+⋯+A⊗2r)1/2ρ(B⊗21+⋯+B⊗2r)1/2. Then the L2-spectral radius similarity is always a real number in the interval [0,1], so one can think of the L2-spectral radius similarity as a generalization of the value |⟨u,v⟩|∥u∥⋅∥v∥ where u,v are real or complex vectors. It turns out experimentally that if A1,…,Ar are random real matrices, and each Bj is obtained from Aj by replacing each entry in Bj with 0 with probability 1−α, then the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) will be about √α. If u=(A1,…,Ar),v=(B1,…,Br), then observe that |⟨u,v⟩|∥u∥⋅∥v∥≈√α as well.
Suppose now that A1,…,Ar are random real n×n matrices and C1,…,Cr are the m×m submatrices of A1,…,Ar respectively obtained by only looking at the first m rows and columns of A1,…,Ar. Then the L2-spectral radius similarity between A1,…,Ar and C1,…,Cr will be about √m/n. We can therefore conclude that in some sense C1,…,Cr is a simplified version of A1,…,Ar that more efficiently captures the behavior of A1,…,Ar than B1,…,Br does.
If A1,…,Ar,B1,…,Br are independent random matrices with standard Gaussian entries, then the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) will be about 1/√r with small variance. If u,v are random Gaussian vectors of length r, then |⟨u,v⟩|∥u∥⋅∥v∥ will on average be about c/√r for some constant c, but |⟨u,v⟩|∥u∥⋅∥v∥ will have a high variance.
These are some simple observations that I have made about the spectral radius during my research for evaluating cryptographic functions for cryptocurrency technologies.
Your notation is confusing me. If r is the size of the list of matrices, then how can you have a probability of 1-r for r>=2? Maybe you mean 1-1/r and sqrt{1/r} instead of 1-r and sqrt{r} respectively?
Thanks for pointing that out. I have corrected the typo. I simply used the symbol r for two different quantities, but now the probability is denoted by the symbol α.
Every entry in a matrix counts for the L2-spectral radius similarity. Suppose that A1,…,Ar,B1,…,Br are real n×n-matrices. Set A⊗2=A⊗A. Define the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) to be the number
ρ(A1⊗B1+⋯+Ar⊗Br)ρ(A⊗21+⋯+A⊗2r)1/2ρ(B⊗21+⋯+B⊗2r)1/2. Then the L2-spectral radius similarity is always a real number in the interval [0,1], so one can think of the L2-spectral radius similarity as a generalization of the value |⟨u,v⟩|∥u∥⋅∥v∥ where u,v are real or complex vectors. It turns out experimentally that if A1,…,Ar are random real matrices, and each Bj is obtained from Aj by replacing each entry in Bj with 0 with probability 1−α, then the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) will be about √α. If u=(A1,…,Ar),v=(B1,…,Br), then observe that |⟨u,v⟩|∥u∥⋅∥v∥≈√α as well.
Suppose now that A1,…,Ar are random real n×n matrices and C1,…,Cr are the m×m submatrices of A1,…,Ar respectively obtained by only looking at the first m rows and columns of A1,…,Ar. Then the L2-spectral radius similarity between A1,…,Ar and C1,…,Cr will be about √m/n. We can therefore conclude that in some sense C1,…,Cr is a simplified version of A1,…,Ar that more efficiently captures the behavior of A1,…,Ar than B1,…,Br does.
If A1,…,Ar,B1,…,Br are independent random matrices with standard Gaussian entries, then the L2-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) will be about 1/√r with small variance. If u,v are random Gaussian vectors of length r, then |⟨u,v⟩|∥u∥⋅∥v∥ will on average be about c/√r for some constant c, but |⟨u,v⟩|∥u∥⋅∥v∥ will have a high variance.
These are some simple observations that I have made about the spectral radius during my research for evaluating cryptographic functions for cryptocurrency technologies.
Your notation is confusing me. If r is the size of the list of matrices, then how can you have a probability of 1-r for r>=2? Maybe you mean 1-1/r and sqrt{1/r} instead of 1-r and sqrt{r} respectively?
Thanks for pointing that out. I have corrected the typo. I simply used the symbol r for two different quantities, but now the probability is denoted by the symbol α.