Trace estimation — the series
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Counting 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.
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Counting what is inside a circle
A trace of a matrix nobody wants to form, integrated around a contour, gives an integer — how many eigenvalues are inside. It converges exponentially, it is estimated with random probes, and the probe block is a ceiling that the answer does not mention.
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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.
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The split nobody is in a position to choose
Hutch++ spends two thirds of its budget on a sketch and a third on probes, and the third is published as a constant. Swept across six rates of spectral decay at a fixed budget of 48 products, the best share is 0.45 on the fastest and 0.00 on the slowest — sketch nothing at all — and the published third costs between 1.09 and 4.11 times the best error. The decay that decides it is readable from the sketch's own singular values, for products the estimator was going to spend anyway.
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A rule that reads only its own probes
A trace estimator is a mean of independent samples, so its own standard error is estimable from the samples and a stopping rule needs nothing the estimator does not already have. Over forty draws it is calibrated in the middle and not at the edge: at a target relative standard error of 1% the median error reached is 5.3·10⁻³ and the worst of forty is 2.9·10⁻² — three times the target. And the cost of the target is the estimator's own square root: tightening it from 3% to 1% takes the median probe count from 75 to 696.
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The miss a normal table already priced
A trace estimator that stops when its own standard error reaches a target misses the target on about a third of draws, and the essay that measured it read the loose targets as the worst calibrated. Over 400 draws the loose target is the better covered — 76% at 10% against 64% at 3% — because the warm-up stops most of its runs with probes to spare. Where the criterion decides, the misses are a normal distribution's: a margin of c on the standard error buys what a normal table says, 94.8% at 1.96, and costs c² in probes, 3.86 times.
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A spread measured on probes it does not average
A trace estimator that stops on its own standard error misses its target a few points more often than a normal table says, because the runs that stop earliest are the ones that underestimated their noise. Spend a pilot of probes only on the spread, fix the number of probes to average in advance, and the selection is gone: with Student's margin the two-stage rule covers 64.8 to 73.3 per cent at one standard error where the table says 68.3. With the normal's margin and a pilot of four it covers 58.5. At one standard error it recovers one to five points for a fifth to two fifths more probes; at 1.96 there was nothing to recover, and the guarantee costs a tenth to double.
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A spread carried from the trace before
A two-stage trace estimator spends a pilot of probes learning its spread, and a computation that needs many traces of a slowly changing operator would rather pay for that once. Frozen at the first trace, the pilot is wrong by the sixteenth: on a spectrum that drifts from decay 0.9 to 0.86 the last estimate is inside a 3% target 52 times in a hundred against the table's 68, and at a faster drift 34. Carry instead the spread of the previous trace's own averaged probes — free, independent of this trace's, one step stale — and its sixteenth estimate covers between 64.5 and 69.5 per cent at every drift, for up to a quarter fewer products than a fresh pilot every time.