Split \({\mathbb Q}^4\) into scalar packets and let
Dropping every off-diagonal block leaves \(I\). Each discarded entry is only \(1/100\), but the total error is not bounded by that single-entry size.
Reading the drawing
Block addresses identify output rows and input columns. The displayed example retains the contributions needed to distinguish individual block sizes from the assembled operator error.
Drawing labels and data 25 labels
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- \(100E\): twelve omitted entries
| Row / column | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| Row 1 | \(0\) | \(1\) | \(1\) | \(1\) |
| Row 2 | \(1\) | \(0\) | \(1\) | \(1\) |
| Row 3 | \(1\) | \(1\) | \(0\) | \(1\) |
| Row 4 | \(1\) | \(1\) | \(1\) | \(0\) |
- error norms, in units of \(1/100\)
- one block
- \(1\)
- spectral
- \(3\)
- Frobenius
- \(2 \sqrt{3}\)
\(\|E\|_F^2=12/10^4\), while \(E\mathbf1=(3/100)\mathbf1\).
Two norms answer different questions
There are twelve omitted entries, so \(\|E\|_F^2=12/10^4\). The all-one vector is an eigenvector with eigenvalue \(3/100\), while the three orthogonal directions have eigenvalue \(-1/100\). Hence \(\|E\|_2=3/100\).
For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient.
Input sensitivity is additional information
The packet perturbation estimate is
It can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly over unit vectors. A certificate for one input is not automatically a certificate for every right-hand side.
These are a posteriori checks for supplied projectors. They do not discover the best packet system or prove that a thresholding heuristic is globally optimal.
Read this as
A proposed compression needs a residual certificate in the requested norm and input scope. Small individual blocks can align into a larger collective error.