160 single-probe estimates of one 40×40 trace, from the two standard probe distributions
At its defaults it draws 160 single-probe estimates of one 40×40 trace, from the two standard probe distributions. Two clouds of points, each one estimate of the trace from one random vector. Both are centred on the true value of 171.14 — the estimator has no bias at all — and the dashed bands are the standard deviation each distribution is known in closed form to have: 59.14 for the ±1 probe and 72.01 for the normal one. Measured over these draws they come out 56.68 and 68.57.
probe-spread is one function in lib/figures/trace.js —
counting the diagonal without looking — two probes, and one of them is free. 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.
Two clouds of points, each one estimate of the trace from one random vector. Both are centred on the true value of 171.14 — the estimator has no bias at all — and the dashed bands are the standard deviation each distribution is known in closed form to have: 59.14 for the ±1 probe and 72.01 for the normal one. Measured over these draws they come out 56.68 and 68.57.
n: 120
The arguments are the ones A rate that belongs to the matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two clouds of points, each one estimate of the trace from one random vector. Both are centred on the true value of 644.26 — the estimator has no bias at all — and the dashed bands are the standard deviation each distribution is known in closed form to have: 198.6 for the ±1 probe and 217.1 for the normal one. Measured over these draws they come out 227.1 and 205.3.
n: 20
The arguments are the ones A rate that belongs to the matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two clouds of points, each one estimate of the trace from one random vector. Both are centred on the true value of 71.955 — the estimator has no bias at all — and the dashed bands are the standard deviation each distribution is known in closed form to have: 26.87 for the ±1 probe and 35.73 for the normal one. Measured over these draws they come out 25.01 and 35.
n: 10
The arguments are the ones Counting what cannot be looked at passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Two clouds of points, each one estimate of the trace from one random vector. Both are centred on the true value of 29.29 — the estimator has no bias at all — and the dashed bands are the standard deviation each distribution is known in closed form to have: 11.27 for the ±1 probe and 17.61 for the normal one. Measured over these draws they come out 10.64 and 20.47.
What it checked while drawing
Every figure above checked its own claims on the way to being drawn, and a claim that failed
would have stopped the picture rather than shipped a wrong one. Those checks 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.
9 distinct claims across 4 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.
a size the repeated products can afford
and so is the Gaussian one
and the ratio of the two spreads is a property of the matrix
enough draws for a spread to be visible and countable
matmul shapes agree
the Gaussian spread measured and the Gaussian spread in closed form agree
the Rademacher probe is unbiased
the Rademacher spread measured and the Rademacher spread in closed form agree
with the Rademacher probe the tighter of the two
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 217
of 397 generators —
199 print a residual and
18 are exempt with a published reason;
180 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 rate that belongs to the matrix
Hutchinson's fitted exponent sits near a half on every spectrum measured. Hutch++'s runs from −7.15 to −0.67 across the same four budgets, decided entirely by how fast the singular values fall — so one of the two methods has a convergence rate and the other has a rate per matrix. The ±1 probe's advantage moves the same way, from 1.56× at n = 10 to 1.09× at n = 120.
Randomised, and the guarantee that changes kindCounting what cannot be looked at
The trace is n additions and one of the most expensive quantities in the subject to estimate, because the matrices whose trace is wanted are never stored. Hutchinson's estimator is unbiased with one line of algebra — and its variance depends on which random vector is used, by a factor that is a property of the matrix, and on a diagonal matrix one choice is exact from the first probe and the other is not.