Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

oversampling-band

One function in the hbuild library, called 9 times across 5 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 11 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws what extra columns buy a construction that never sees the matrix, over five seeds. The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 8 there is, which a decomposition of every block would find. With no oversampling the construction is 11.6 times behind it and the spread across seeds is 44 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 12 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

oversampling-band is one function in lib/figures/hbuild.js — built from products — a few hundred applications of an operator that is never formed. 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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 8 there is, which a decomposition of every block would find. With no oversampling the construction is 11.6 times behind it and the spread across seeds is 44 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 12 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻⁸10⁻⁷10⁻⁶10⁻⁵extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-8 representationfive draws, mean and rangea band, not a lineexcess at p = 012excess at p = 83spread at p = 00.44spread at p = 80.12the optimum of this rank7.8·10⁻⁸one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 8 there is, which a decomposition of every block would find. With no oversampling the construction is 11.6 times behind it and the spread across seeds is 44 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 12 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

k: 8

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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 8 there is, which a decomposition of every block would find. With no oversampling the construction is 11.6 times behind it and the spread across seeds is 44 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 12 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻⁸10⁻⁷10⁻⁶10⁻⁵extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-8 representationfive draws, mean and rangea band, not a lineexcess at p = 012excess at p = 83spread at p = 00.44spread at p = 80.12the optimum of this rank7.8·10⁻⁸one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 8 there is, which a decomposition of every block would find. With no oversampling the construction is 11.6 times behind it and the spread across seeds is 44 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 12 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

k: 12

The arguments are the ones Built from products alone 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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 12 there is, which a decomposition of every block would find. With no oversampling the construction is 50.0 times behind it and the spread across seeds is 146 per cent of the mean. Eight extra columns bring it to 3.4× and the spread to 16 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-12 representationfive draws, mean and rangea band, not a lineexcess at p = 050excess at p = 83.4spread at p = 01.5spread at p = 80.16the optimum of this rank9.5·10⁻¹²one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 12 there is, which a decomposition of every block would find. With no oversampling the construction is 50.0 times behind it and the spread across seeds is 146 per cent of the mean. Eight extra columns bring it to 3.4× and the spread to 16 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

k: 4

The arguments are the ones Built from products alone 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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 4 there is, which a decomposition of every block would find. With no oversampling the construction is 15.0 times behind it and the spread across seeds is 50 per cent of the mean. Eight extra columns bring it to 2.9× and the spread to 24 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻⁴10⁻³10⁻²10⁻¹extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-4 representationfive draws, mean and rangea band, not a lineexcess at p = 015excess at p = 82.9spread at p = 00.5spread at p = 80.24the optimum of this rank2.6·10⁻⁴one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 4 there is, which a decomposition of every block would find. With no oversampling the construction is 15.0 times behind it and the spread across seeds is 50 per cent of the mean. Eight extra columns bring it to 2.9× and the spread to 24 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

k: 6

The arguments are the ones Built from products alone 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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 6 there is, which a decomposition of every block would find. With no oversampling the construction is 13.8 times behind it and the spread across seeds is 16 per cent of the mean. Eight extra columns bring it to 2.7× and the spread to 23 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻⁶10⁻⁵10⁻⁴10⁻³extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-6 representationfive draws, mean and rangea band, not a lineexcess at p = 014excess at p = 82.7spread at p = 00.16spread at p = 80.23the optimum of this rank5·10⁻⁶one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 6 there is, which a decomposition of every block would find. With no oversampling the construction is 13.8 times behind it and the spread across seeds is 16 per cent of the mean. Eight extra columns bring it to 2.7× and the spread to 23 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

k: 10

The arguments are the ones Built from products alone 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.

What extra columns buy a construction that never sees the matrix, over five seedsThe band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 10 there is, which a decomposition of every block would find. With no oversampling the construction is 15.7 times behind it and the spread across seeds is 85 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 5 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.048121610⁻¹⁰10⁻⁹10⁻⁸10⁻⁷extra columns in the sample, p‖A − A_H‖ ⁄ ‖A‖the best rank-10 representationfive draws, mean and rangea band, not a lineexcess at p = 016excess at p = 83spread at p = 00.85spread at p = 80.049the optimum of this rank9.6·10⁻¹⁰one seed shows the meanand five show the risk

The band is the range across five seeds and the line is their mean; the flat lower line is the best representation of rank 10 there is, which a decomposition of every block would find. With no oversampling the construction is 15.7 times behind it and the spread across seeds is 85 per cent of the mean. Eight extra columns bring it to 3.0× and the spread to 5 per cent. Both halves matter and only one of them is visible in a single run: what oversampling buys is a better answer and a more predictable one, and a figure drawn from one seed would have shown the first and hidden the second. No draw is ever better than the flat line, which is the check that the comparison is honest — a sample cannot beat the decomposition it is approximating.

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.

11 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.

and no draw beats the best representation of the same rank, at p = 0 — asserted 6 times

a rank the leaf blocks can hold

an oversampling inside the range this is worth drawing over

by a factor rather than a few per cent, at every rank the sweep runs

matmul shapes agree

oversampling closes the gap to the best representation of the same rank

Against the rule

It draws a decomposition and prints its residual. It calls oversamplingSweep, 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.

Randomised, and the guarantee that changes kind

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 kind

Built from products alone

A 512-square hierarchical representation, at a relative error of 4·10⁻⁷, from 256 applications of an operator that is never assembled. The compression route reads 262,144 entries; this one reads none, and pays for it with a factor of seven against the representation the entries would have given.

 Randomised, and the guarantee that changes kind

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.

 Orthogonality, measured

Orthogonal is a number

"Q is orthogonal" is a claim about a measurable quantity, ‖QᵀQ − I‖, and on the eight-by-eight Hilbert matrix two standard algorithms return 10⁻¹⁵ and 1 for it. The one that returns 1 still reconstructs the matrix perfectly, which is why nothing warns you.

 Randomised, and the guarantee that changes kind

The sketch that is not the answer

Sketch-and-solve throws away the original problem and keeps the small one's answer, which is why its answer moves with the seed. Use the same sketch as a preconditioner instead and the condition number the iteration sees is the same number at every κ from a hundred to ten billion — identically the same, to nine digits, because the spectrum cancels out of it.

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