In an SVD If 2 singular values ​of the matrix Σ are excluded, how many entries are freed from memory?

Question

My professor asked me the following question in order to solve a more theoretical question about dimensionality reduction. However, I cannot understand what steps I need to take to solve the following problem

"Given a matrix A (M x n), the SVD algorithm was applied and three matrices were obtained: UΣV in such a way that $$A = UΣV^t$$ Instead of storing matrix A in memory, only matrices U, Σ and V were kept. Since matrix Σ is an r x r diagonal matrix, to save space it was stored as a vector.

Let M = 100, n =40 and r = 8.

If 2 singular values ​​of the matrix Σ are excluded, how many entries are freed from memory?"

The answer options are 280, 282 or 296

I know the question may sound a little vague, but I'm really stuck on solving this problem.

Answer 1

Since $A$ is $m \times n$, the dimensions of $U$, $\Sigma$, and $V^t$ would be:

• $U$: $m \times r$
• $\Sigma$: $r \times r$ (only the diagonal stored)
• $V^t$: $r \times n$

So the original number of entries stored would have been $mr + r + rn = 800 + 8 + 320 = 1128$.

If you remove two singular values, then $r$ is reduced from $8$ to $6$, and $m$ and $n$ remain the same, so the new count is $mr + r + rn = 600 + 6 + 240 = 846$, which means you have saved $1128 - 846 = 282$ entries.