Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then what is the upper bound on $$\frac{\max \{|u_i^\top y|: i = 1....n\}}{\min \{|u_i^\top y|: i = 1....n\}} \; \forall i=1,...,n$$ in terms of maximum and minimum singular value? I have asked similar question at https://math.stackexchange.com/questions/3854456/ratio-of-maximum-to-minimum-value
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$\begingroup$ I have difficulties to make sense of your intention. The expression $y = X\beta + \epsilon$ seems to indicate a lineat model in statistics. Can you give a solution to your problem for the most simple special case $y_i = \beta_0 + \beta_1 x_i + \epsilon_i$? Do there exist bounds you have found? $\endgroup$– Dieter KadelkaCommented Oct 8, 2020 at 10:15
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$\begingroup$ Yes, it is a linear model. I am not able to find any sort of bounds so far. $\endgroup$– newbieCommented Oct 8, 2020 at 16:38
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