low-rank-error
At its defaults it draws error of the best rank-k approximation to a 10×10 matrix. Approximation error against k on a logarithmic axis, with the measured error and the next singular value drawn as separate curves lying exactly on top of one another.
low-rank-error is one function in lib/figures/spectra.js —
spectra — sensitivity, the symmetric easy case, and rank as a decision. Everything below came out of it during this build, at
arguments taken from the essays rather than invented for this page. A figure here is the
figure a reader meets in an essay, and if the generator changes, this page changes with it.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
Approximation error against k on a logarithmic axis, with the measured error and the next singular value drawn as separate curves lying exactly on top of one another.
n: 10
The arguments are the ones A bound that holds with probability passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Approximation error against k on a logarithmic axis, with the measured error and the next singular value drawn as separate curves lying exactly on top of one another.
n: 12
The arguments are the ones Where the answer stops being in the data passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
Approximation error against k on a logarithmic axis, with the measured error and the next singular value drawn as separate curves lying exactly on top of one another.
What it checked while drawing
Every figure above asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions used to leave
no trace at all: a passing one returned true and the only evidence the figure had
checked anything was that nothing crashed. The list below is what they actually said, collected
by running this generator with an observer installed — not a description of
what it is believed to check.
35 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.
and in the Frobenius norm it is the tail from 2 on agree — asserted 11 times
the Frobenius error is never below the 2-norm error at k = 1 — asserted 11 times
the rank-1 error is σ2 agree — asserted 11 times
and no random rank-3 matrix beats the SVD's
matmul shapes agree
Against the rule
It draws a decomposition and prints its residual. It calls
svd, lowRank,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
Across the library: the rule bites on 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 factorise nothing.
Read from lib/residual-rule.js, which is the same body the gate enforces from,
and the gate's last check fails the build if this page and it disagree about any generator.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A bound that holds with probability
Every other guarantee on this site is deterministic. The randomised low-rank approximation offers one that holds with a probability, the seed changes the answer, and the honest figure is a band rather than a line.
Randomised, and the guarantee that changes kindRandomisation does not create structure
On a matrix whose singular values are all equal, a rank-ten randomised approximation has error 1.0 — and so does the optimal deterministic one. Neither achieved anything, and only one of them is usually sold with the implication that it might.
Eigenvalues, singular values, rankRank is a decision
A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.
Eigenvalues, singular values, rankSymmetry is worth more than precision
A symmetric matrix gives up its eigenvalues to full accuracy however ill-conditioned it is. An unsymmetric one can move them by the eighth root of a perturbation, so the rounding involved in merely storing the matrix shifts the spectrum by a hundredth.
Eigenvalues, singular values, rankThe best approximation there is
The error of the best rank-k approximation is not bounded by the next singular value. It is equal to it. That is an unusually sharp theorem, and it makes the theorem itself usable as an independent check on the computation.
Randomised, and the guarantee that changes kindThe dimension does not appear
A random projection preserves the lengths of a set of vectors to within a distortion that depends on how many vectors there are and not on how many coordinates each one has. That is the fact the whole field rests on, and it is genuinely surprising.
Regularisation, and the answer that is chosenWhere the answer stops being in the data
The Picard condition finds the index where a noisy right-hand side stops carrying signal, from the data alone, with no knowledge of the answer. It lands at 32 where the truncation that actually minimises the error is 28 — and at 45 where the best is 38. It overshoots at every noise level from 1% to 0.001%, and it overshoots for a reason.