rank-accuracy
At its defaults it draws how many columns a decade of accuracy costs, measured and predicted, at q = 0.500. The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.
rank-accuracy is one function in lib/figures/kernel.js —
one block — where a matrix with no zero entries turns out to have five useful columns. 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.
gap: 1
The arguments are the ones A block nobody can call sparse 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.
gap: 0.5
The arguments are the ones A block nobody can call sparse 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 0.5. The measured curve is a straight line at 0.70 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 5.66 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 8.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.
gap: 0.25
The arguments are the ones A rank that is a number of digits 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 0.25. The measured curve is a straight line at 0.84 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 10.34 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 12.7× about the constant, which is the safe direction for a quantity you have to allocate storage from.
gap: 8
The arguments are the ones A rank that is a number of digits 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 8. The measured curve is a straight line at 0.32 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 1.00 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 3.0× about the constant, which is the safe direction for a quantity you have to allocate storage from.
gap: 2
The arguments are the ones An accuracy that is a backward error 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.
The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 2. The measured curve is a straight line at 0.50 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 2.11 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 4.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.
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.
13 distinct claims across 6 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.
the geometry's bound is a bound at ε = 10^-2 — asserted 7 times
a block size the dense SVD below is affordable at
a convergent expansion, which is what admissible means
a decade of accuracy costs a fixed number of columns
a gap at which the expansion converges
a kernel this file defines
and it is loose by a factor rather than by an order of magnitude
Against the rule
It draws a decomposition and prints its residual. It calls
rankAgainstAccuracy,
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 141
of 264 generators —
126 print a residual and
15 are exempt with a published reason;
123 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 block nobody can call sparse
A 96 × 96 block of a kernel matrix has ninety-six nonzero singular values and five that matter. It has no zero entries, it is not described by fewer numbers than it contains, and neither of the two ways this collection already knows to make a large matrix affordable applies to it.
Neither sparse nor denseA rank that is a number of digits
Ask a kernel block for two digits and it costs two columns; ask for fourteen and it costs nine. The curve is a straight line at 0.55 columns a decade, and the bound the geometry gives is a straight line too — at 3.32, which is the same shape and six times the price.
Two errors, and whose fault they areAn accuracy that is a backward error
Every backward error on this site is something an algorithm produced and somebody then measured. This one is a line in the program. Solving with a compressed matrix gives a residual that is the compression's own error, at a slope of 1.000 over ten decades, so the knob that sets the storage sets the backward error directly.
Methods that were designed apartFour knobs and one floor
A truncation, a Tikhonov parameter, a step count and a randomised rank, on one problem with an answer that is known. Their best errors are 0.1445, 0.1406, 0.1426 and 0.1449 — a spread of 3% across four methods that share no arithmetic.
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, 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.
Neither sparse nor denseThe kernel with nothing to compress
Hold the geometry fixed at q = ½, fix the wavelength, and scale the picture up by sixteen. A smooth kernel needs six columns at every scale. An oscillatory one needs twelve, sixteen, twenty-two, thirty-three, fifty-three, and there is no scale at which it stops.
Least squares, and the road not to takeThe projection and the right angle
The least-squares solution is the one whose residual is perpendicular to everything the columns can reach. That is not a mnemonic — it is an equation, Aᵀr = 0, and the computed answer satisfies it to 10⁻¹⁶.
Neither sparse nor denseThe size the rank does not notice
Sample a kernel block at 32, 64, 128 and 256 points a side and it needs five columns, five, five and five. Sample the touching block next to it at the same four sizes and it needs nine, eleven, twelve and thirteen. Same kernel, same accuracy, one number and a logarithm.
Regularisation, and the answer that is chosenWhen the answer is a choice
A backward-stable least-squares solve of this problem returns an answer whose relative error is 5.5·10⁸. Nothing went wrong. The singular values decay exponentially with no gap anywhere in them, the data does not determine the answer, and something outside the data has to choose — which is the computation rather than a preliminary to it.
Neither sparse nor denseWhich pairs are allowed to be small
A hierarchical representation is a partition of the matrix into blocks, and the rule that produces it reads four numbers per pair of index clusters and not one entry of the matrix. On a 256-square it yields 112 blocks, 66 of them stored as two thin factors, none of rank above five.