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Let $V\in\mathbb{R}^{n\times m}$, where $n>m$, be a random matrix of standard normal gaussians. Given an arbitrary vector $y\in\mathbb{R}^n$, I need to understand the distance $||y-p_V(y)||^2$, where $p_V$ is the projection onto the range of $V$, meaning $p_V = V(V^TV)^{-1}V^T$ as a matrix. Through a slightly different lens, I would like to understand the distance of $y$ from $R(V)$.

Given that $V$ is random, this makes the above distance a random variable whose distribution (or at least expected value) is important to my research. This feels like a not-so-novel problem but I couldn't find a solution on my own.

How can I approach this? Thanks in advance!

J. W. Tanner
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Nadav Schweiger
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1 Answers1

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Up to formalizing why $R(V)$ is a uniformly sampled subspace (which should follow from the fact that standard Gaussians are spherically uniform), I think I got it:

By rotation we can assume that $R(V)$ is $\mathbb{R}^m$, and change the question to be "sampling a vector $y$ uniformly from the sphere $S^{n-1}$, what is $\sum_{i=1}^m y_i^2$?". By this answer, we get that, if $B\sim\text{Beta}\left(\frac{n}{2}, \frac{n}{2}\right)$ then: $$ y_i^2 = 4B^2-4B+1 \implies \mathbb{E}y_i^2 = 4\cdot \frac{n+1}{4n} - 4\cdot\frac{1}{2}+1 = \frac{1}{n} $$ Where the computation of the expected value of $B^2$ relies on this answer.

Therefore we get an error of $1-\frac{m}{n}$.

I would appreciate help with first part if anyone has a direction.

Nadav Schweiger
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