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Mike Izbicki
Mike Izbicki
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Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from the mean to the subspace spanned by the samples. I suspect that this distance grows as $O(1-\sqrt{k/d})$ with high probability, but I'm having trouble proving it.

This seems like the sort of problem that should already be solved somewhere, but I don't know where to look. Everything I've found on random projections assumes a centered distribution, but what makes this problem interesting is the non-zero mean.

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from the mean to the subspace spanned by the samples. I suspect that this distance grows as $O(1-\sqrt{k/d})$ with high probability, but I'm having trouble proving it.

This seems like the sort of problem that should already be solved somewhere, but I don't know where to look. Everything I've found on random projections assumes a centered distribution, but what makes this problem interesting is the non-zero mean.

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from the mean to the subspace spanned by the samples.

This seems like the sort of problem that should already be solved somewhere, but I don't know where to look. Everything I've found on random projections assumes a centered distribution, but what makes this problem interesting is the non-zero mean.

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Mike Izbicki
Mike Izbicki
  • 395
  • 1
  • 2
  • 15

distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from the mean to the subspace spanned by the samples. I suspect that this distance grows as $O(1-\sqrt{k/d})$ with high probability, but I'm having trouble proving it.

This seems like the sort of problem that should already be solved somewhere, but I don't know where to look. Everything I've found on random projections assumes a centered distribution, but what makes this problem interesting is the non-zero mean.