Probabilistic bounds — where it appears
Named by 22 essays across 5 fields — each of them below, with the objects they name alongside it.
A bound that holds with probability
Every other guarantee in this collection 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.
The 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.
An answer that changes with the seed
A randomised rank-k solve is a truncation computed in a random subspace, and it reaches the same floor as the deterministic ones. What it does not do is return the same answer twice — a factor of 1.84 across four seeds at rank 8, and 1.02 at the rank where the method is best.
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.
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.
What a single draw cannot report
With no oversampling the construction's error is 11.6 to 50 times the best representation of the same rank, and the spread across five seeds runs from 16 to 146 per cent of the mean. Sixteen extra columns bring the excess to between 2.0 and 2.9 at every rank measured and the spread to between 4 and 20 per cent — and the second number is the one a single run cannot report and the one that decides whether the first is a measurement.
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.
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.
The rank a certificate charges
A randomised range finder can choose its own rank: grow the basis a column at a time and stop when ten fresh probes all come back short. With the published safety factor it never stopped early in any draw measured, and on a matrix whose singular values fall by 0.8 a step it stopped at rank 30 for a tolerance the best rank-11 approximation already meets. The nineteen extra columns are three separate prices — four for building the basis from random vectors, five because a probe reads more than the spectral norm, and ten for the constant — and the spectrum decides which of them dominates.
A sketch that finds the columns it can see
A sparse sketch with one nonzero in each row is twenty times cheaper to apply than a Gaussian one, and on a matrix whose important directions are spread across its columns it finds the same range: a median error of 0.45 against 0.43. Put the same ten directions into ten particular columns and it is eight times worse — 3.33 against 0.41, with a worst draw of 7.1 — because two important columns hashed to one bucket are one direction. Three nonzeros a row repair it at a sixth of the Gaussian's cost, and a randomised Hadamard transform never had the problem.
The variation that comes with a seed
A randomised low-rank approximation's error moves by 31% between draws and by 10⁻¹⁵ between partitionings of one draw. In the one field whose answer already comes as a band, the machine is inside the width of the line — and it is still there.
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.
The tail a sample never reaches
Hager's estimator is exactly right on four random matrices in five, and that share is stable — between 80.5 and 87.5 per cent across nine sizes. The worst underestimate is not stable at all: it falls every time more matrices are drawn, from 0.746 at sixty to 0.377 at four hundred, and the matrix built to defeat the estimator sits five times below anything four hundred draws found.
The leverage that did not move
A one-nonzero sketch fails on a matrix whose leading directions sit on ten particular columns, and coherence — the largest column leverage — is the statistic that names the failure. Turn the directions away from their columns by a hundredth of a radian and the sketch's median error falls from 3.47 to 1.23 times σ₁₁ while the coherence stays at 6.40 to three figures. Giving the heaviest columns buckets of their own repairs the rest, but only when it reserves more buckets than the rank: ten reserved leave 1.16, sixteen reach 0.36, below the Gaussian's 0.41.
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.
The rank that stops being typical
A random 2 × 2 × 2 tensor has rank two with probability π/4 and rank three otherwise, and the sentence has no analogue for matrices. It is the first of a family. An n × n × 2 tensor is a pencil of two slices, and it has rank n exactly when the pencil's eigenvalues are all real, n + 1 otherwise. Over draws, the share of rank n is 0.786 at n = 2, 0.500 at 3, 0.264 at 4, 0.039 at 6, 0.004 at 8 and none of 4,000 at 10 — falling like e^(−0.087n²) — because the mean number of real eigenvalues grows only like the square root of n, to Edelman, Kostlan and Shub's closed form within two per cent. Both ranks stay typical in theory; in practice the lower one disappears.
Columns that steer a random start
A condition estimator whose first vector is random cannot be built against, and concentrating the hidden column did not hide it: its own entries steer the first sign vector onto it. Give the steering to other columns — each equal on both rows of every pair the hidden column alternates over — and the random start is fooled after all. The prediction was a factor of one over the square root of the number of steering columns, on a share of seeds that falls as they lose to the hidden column, and the block of two fooled as often. The factor is whatever the builder chooses; the share, at a fixed factor of one half, rises with every column added, from 9 per cent with four to 77 with thirty-four. And the block of two is not fooled as often: it is fooled on the square of the single start's share, and the block of four on its fourth power — independent guesses, which is the one thing that still works.
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.
A drift put back at the table's price
A trace estimator that carries its spread from the trace before loses one step of drift, and the drift is visible in the spreads themselves, so the repair proposed was to multiply the carried spread by the ratio of the last two. Over 400 drifting sequences it works: averaged over the third to sixteenth traces it covers 67.8 to 68.9 per cent at one standard error at every drift, where the plain carry falls to 65.4 and the table says 68.3. It costs what the table says closing that gap costs — 10.5 per cent more products at the fastest drift against a price of 11.9 — so at that drift it spends more than a fresh pilot every trace. Its factor is noisy, scattering the spread it uses two and a half times as widely as the plain carry does, and on a sequence that does not drift that noise costs nothing at all. And the single-trace comparison the proposal named turned out to be smaller than its own draw noise.
Promotion buys rank, not safety
A range finder that grows its basis from cheap sparse probes and certifies it with dense ones was made safe on a coherent matrix by promoting a refusing dense probe into the basis — ten promotions for ten coherent directions, none for none. On matrices mixed between the two, the promotions fade as predicted: ten at the coherent end, nine at a mixture angle of 0.1, none from 0.8. But the danger they guard against does not fade with them. It is gone at an angle of 0.05, where the rule that never promotes already certifies every draw correctly. What promotion buys there is rank — a basis of 30 columns instead of 38 — for four fifths more probe work, and the threshold that decides it is a price, not a safety margin.
A turn is seen one trace late
A carried spread multiplied by the ratio of the last two spreads restored a drifting sequence's coverage, and the prediction for a sequence whose drift turns was a dip of four or five points on the trace after the turn, gone a trace later, and a guard that removes it for nothing. Run on the turn the prediction described — the spread rising and then falling — there is no dip: the extrapolation promises more rise, uses a spread eleven per cent too large, and covers 70.5 per cent against a table of 68.3. Run on the mirror turn, the spread falling and then rising, the dip is there and larger than predicted: a spread a quarter too small and 59.5 per cent coverage. Either way it is gone a trace later and the sequence's average stays within a point. The guard costs nothing and does not help, because no rule that reads past spreads can see a turn before the trace that shows it.
The fall sets the undershoot
A trend-carried spread meets a turn sharper than the drift before it — the decay rising 0.02 a trace and then falling 0.06 — and the prediction was a spread about 0.6 of the truth on the trace after the turn, coverage under 55 per cent, a fresh pilot there restoring it, and the sequence's average within two points. All four hold: 0.617, 48.5 per cent, 71.0 per cent, and 67.7 against the table's 68.3. What fails is the rule the earlier essay gave for the error, the square of the last ratio. The undershoot is the last ratio over the next one, and the coverage it costs is the normal table's at that fraction.
Named alongside it
The objects these essays reach for when they reach for this one.
Spectral decayHutchinson's estimatorRandom probeTrace estimationMatrix-freeRandom projectionRandomised SVDStopping criterionFlop countSketchingLow-rank approximationRange finder