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I know the proof for two random vectors$\def\a{\mathbf{a}} \def\b{\mathbf{b}} \def\c{\mathbf{c}}$ (say $\a$ and $\b$) in high dimension becomes orthogonal, i.e. $\langle \a,\b\rangle = 0$. I am keen to know what happens when $\a$ and $\b$ are correlated. Can they be quasi-orthogonal? In that case, how would be the inner product be defined?

Clarification: I am following the proof shown here. The statement clearly says that both the vectors are randomly drawn (as a result I assume there is no correlation involved). But in my case, I have $\|\a−\c\|_2=\|\b\|_2.$ Now, I want to know whether in high dimension, $\a$ and $\b$ can be treated as orthonormal or not. Can this be proven that they are orthogonal (if they are)?

amWhy
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God_Help
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  • How are $a,,b,,c$ sampled so that $a-c$ has the same length as $b$? I'll update my answer once I understand the assumed joint distribution of $a,,b$. – J.G. Mar 06 '20 at 07:50
  • You can consider a as an original signal and c as a reconstructed signal (consider reconstructed using OMP-based algorithm). b can be treated as reconstruction error. So, all three vectors would have same length. – God_Help Mar 06 '20 at 08:01

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Since $a=c+b$ and $c\cdot b=0$, $a\cdot b=b^2\ge0$ and $a\cdot c=c^2\ge0$.

J.G.
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  • Thanks for the answer. I am trying to follow the answer given here. The statement clearly says that both the vectors are randomly drawn (as a result I assume there is no correlation involved). But in my case, I have $||\mathbf{a} - \mathbf{c}||_2 = ||\mathbf{b}||_2$. Now, I want to know whether in high dimension, a and b can be treated as orthonormal or not. Can this be proven that they are orthogonal (if they are)? – God_Help Mar 06 '20 at 03:05
  • @God_Help See edit. – J.G. Mar 06 '20 at 08:12
  • Thanks a lot. I have some difficulty in understanding why did you write $c • b =0$? Are you considering orthogonality between them? – God_Help Mar 06 '20 at 08:25
  • @God_Help See the third bullet point here. – J.G. Mar 06 '20 at 08:26
  • That helps. Thanks a lot for the answer. – God_Help Mar 06 '20 at 08:28