Section 9.6 ends with two warnings. Dropping small singular values gives a lower-rank matrix that stays close to the original, and the singular vectors of a data matrix depend on how the columns are scaled and on whether the columns have been centred. Change either and the important directions change. The chapter does not say how many singular values to keep, and the field has no procedure that settles the number for every use. Compression, denoising and interpretation want different cuts, and a cut defended by one of those purposes can remove the structure another one needs.
Take a position on this question: what procedure should choose the number of singular values to keep, and what should the resulting low-rank object be allowed to claim?
In your answer: say what the singular values are computed from, and why that choice of scaling and centring is the one your purpose needs; state the rule that picks the cut and what quantity it optimises; say what the truncated matrix can and cannot promise about the matrix it came from and about the rows behind it; and name one way your rule misleads you even when every step is followed.
Rubric. Tick each criterion your answer meets, then save the self-assessment. The platform does not grade it.
Worked solution
There is no single correct answer, so this is a model of the shape a complete answer takes, not a key. A level-3 answer starts from the purpose and lets the purpose fix the preprocessing. For compression of a fixed table, it keeps the columns as they are, because the reconstruction is compared with the original entries, and a centring step would change the object being reconstructed. For finding directions in a data table, it centres each column first, because a large mean would otherwise appear as a leading direction that describes where the table sits rather than how it varies; if the columns carry different units, it also scales them, and it says that the scaling is part of the definition of the directions. Then it states a rule with a quantity in it. One defensible rule keeps the smallest number of singular values whose squares reach per cent of the sum of all squared singular values, and reports the remaining fraction as the discarded part. Another cuts after the last singular value that is at least ten times the next one, and reports the gap it used. Either rule is a decision, written before the cut is inspected. The answer then separates the two claims. The truncated matrix is close to the original in the squared-entry sense, and the size of the discarded singular values measures that distance; this claim is about a matrix already in hand. The leading directions describe new rows only if those rows are drawn like the rows used to fit them, after the same centring and scaling, and only if the leading singular values are separated enough that small changes in the table do not rotate the directions. The answer names a failure. A table whose entries are mostly noise has singular values of similar size, so a rule that keeps a fixed fraction of the total keeps noise and reports a small discarded part while describing nothing. A single duplicated row can create a gap that looks like structure. Centring can remove a level that a reader needed, and scaling can promote a column that was measured in small units but carries little information. The answer also says what the honest measurement of the reconstruction error costs: rows held back from the fit, or a split of the table, and a report of the error on those rows rather than on the rows that chose the cut. A level-2 answer has the rule and the purpose but blurs the matrix claim with the data claim. A level-1 answer names a cut without a quantity. A level-0 answer repeats the warnings of section 9.6 and stops.