Estimating quality of projection

Suppose we are given a vector $v$ and vectors $\mu_i$:

$v = \mu_1+\mu_2+...+\mu_m$, where $\mu_i \in R^n$, all $\mu_i$ are of unit length.

Oracle will give me $k$ vectors $\mu_{j_1}, \mu_{j_2},...\mu_{j_k}$ from the original set such that when I project $v$ onto subspace spanned by these vectors the length of the projection is highest possible. In other words, from the set of all combinations of $k$ vectors from $[\mu_1,...\mu_n]$ the $[\mu_{j_1}, \mu_{j_2},...\mu_{j_k}]$ give highest length of projection. Lets denote by $v_{\text{proj}}$ projection of $v$ onto $[\mu_{j_1}, \mu_{j_2},...\mu_{j_k}]$

I want to estimate quality of projection before oracle gives me this $k$ vectors. I want to give upper bound on $||v - v_{\text{proj}}|| $

As far as I understood it is very difficult to obtain these $k$ vectors by myself. However, I know that for any two vectors $\mu_i, \mu_j$, $||\mu_i-\mu_j|| \leq \alpha$, where $\alpha$ is a given positive number.

Small values of $\alpha$ will tell me that all $\mu_i$ are close to each other and heading towards same direction. I would suspect then that projection will be good, and its length will be close to the length of original vector. How can I use this to give an upper bound $||v - v_{\text{proj}}|| $?

My attempts:

Without loss of generality lets assume that $k$ optimal vectors are first $k$ vectors in the list, i.e $\mu_1,\mu_2,...\mu_k$. Lets denote by $P$ projection operator on the space spanned by $\mu_1,\mu_2,...\mu_k$.

$\|v - v_{\text{proj}}\| = \|v - P(v)\| = \|v - P(\mu_1+\mu_2+...+\mu_m)\| = $

$\|v - P(\mu_1) - P(\mu_2) - ... - P(\mu_m)\| = $

$ \| v - \mu_1 - \mu_2 - ... - \mu_k - P(\mu_{k+1}) - P(\mu_{k+2}) - ... - P(\mu_m)\| = $

$\|\mu_{k+1} - P(\mu_{k+1}) + \mu_{k+2} - P(\mu_{k+2}) + ... + \mu_{m} - P(\mu_{m})\|$

$\|v - v_{\text{proj}}\| \leq \|\mu_{k+1} - P(\mu_{k+1})\| + \|\mu_{k+2} - P(\mu_{k+2}) + ... + \|\mu_{m} - P(\mu_{m})\|$

$\|v - v_{\text{proj}}\| \leq (m-k)\alpha$

So in order to make $\|v - v_{\text{proj}}\| \leq \epsilon$, we need $k \geq \frac{m\alpha - \epsilon}{\alpha}$

I am not satisfied with this result because $k$ grows linearly with $m$. I want it to grow much slower, something like $\log(m)$. My goal is to show that under some constraints on $\mu_i$, we need only approximately $\log(m)$ vectors to approximate $v$.

I think the bound can be improved substantially. First Cauchy inequality isn't very tight and second, I used $|\mu_{k+1} - P(\mu_{k+1})\| \leq \alpha$ which is also very loose.

I am open for additional constraints on $\mu_1,...\mu_m$ to achieve logarithmic growth

As Alex Ravsky has noted, we also need a constraint on $\alpha$ in order to achieve logarithmic growth. Assume that $k$ $\leq n$, $\mu_i$ is th $i$-th standard ort of the space $\mathbb{R}^n$, and $\alpha = \sqrt{2}$. Then $\|v - v_{\text{proj}}\| = \sqrt{m-k}$

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