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Is there an intuitive meaning for the spectral norm of a matrix? Why would an algorithm calculate the relative recovery in spectral norm between two images (i.e. one before the algorithm and the other after)? Thanks

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2 Answers 2

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The spectral norm (also know as Induced 2-norm) is the maximum singular value of a matrix. Intuitively, you can think of it as the maximum 'scale', by which the matrix can 'stretch' a vector.

The maximum singular value is the square root of the maximum eigenvalue or the maximum eigenvalue if the matrix is symmetric/hermitian

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    $\begingroup$ thanks - this helps me conceptually. $\endgroup$
    – val
    Aug 29, 2012 at 6:56
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    $\begingroup$ This is specifically important since noise will be amplified by this value. For a very nice and intuitive understanding of this I recommend slides 21 and 22 of ee263.stanford.edu/lectures.html . $\endgroup$
    – divB
    May 4, 2016 at 7:35
  • $\begingroup$ @divB, of which lecture? $\endgroup$
    – jds
    Oct 10, 2018 at 0:29
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    $\begingroup$ 27, "SVD and applications" $\endgroup$
    – divB
    Oct 10, 2018 at 22:03
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    $\begingroup$ This is only true for symmetric matrices. math.stackexchange.com/questions/1437569/… $\endgroup$ Mar 28, 2020 at 2:09
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Let us consider the singular value decomposition (SVD) of a matrix $X = U S V^T$, where $U$ and $V$ are matrices containing the left and right singular vectors of $X$ in their columns. $S$ is a diagonal matrix containing the singular values. A intuitive way to think of the norm of $X$ is in terms of the norm of the singular value vector in the diagonal of $S$. This is because the singular values measure the energy of the matrix in various principal directions.

One can now extend the $p$-norm for a finite-dimensional vector to a $m\times n$ matrix by working on this singular value vector:

\begin{align} \|X\|_p &= \left( \sum_{i=1}^{\text{min}(m,n)} \sigma_i^p \right)^{1/p} \end{align}

This is called the Schatten norm of $X$. Specific choices of $p$ yield commonly used matrix norms:

  1. $p=0$: Gives the rank of the matrix (number of non-zero singular values).
  2. $p=1$: Gives the nuclear norm (sum of absolute singular values). This is the tightest convex relaxation of the rank.
  3. $p=2$: Gives the Frobenius norm (square root of the sum of squares of singular values).
  4. $p=\infty$: Gives the spectral norm (max. singular value).
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    $\begingroup$ thanks - can we really talk about Schatten p=0 for matrices? since we would be looking at 1/0...? $\endgroup$
    – val
    Aug 29, 2012 at 7:02
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    $\begingroup$ Yes we can. And it is equal to the rank of the matrix. Can you clarify your question a bit more? $\endgroup$ Aug 29, 2012 at 13:41
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    $\begingroup$ Hi - i was referring to the Schatten norm equation above: the exponent is 1/p. If p=0 we have 1/0. thanks.. p must > 1. $\endgroup$
    – val
    Aug 29, 2012 at 17:31
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    $\begingroup$ Think of $p \rightarrow 0$. Then any nonzero singular values will lead to $1$, while $0$ singular values will give $0$ anyway. Hence, you will end up with the number of non-zero singular values. However, for $p \in [0,1]$, the Schatten norm is not a "norm" since it does not satisfy all the properties. However, it is still common practice to call it a "norm". $\endgroup$ Aug 29, 2012 at 18:10
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    $\begingroup$ I just want to point out the confusion in your notation, the same notation: $||A||_2$ is also being used as spectral norm of a Matrix, which is the $p=\infty$ in your answer. I don't know what is Schatten Norm but one thing is universally agreed is that, matrix is an operator, and its norm should be defined in an operator fashion. $\endgroup$ Nov 15, 2018 at 19:26

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