sketch-draws

At its defaults it draws random numbers a sketch needs, against the number of indices, at n = 8. A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 1.51·10⁸ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 4,032 numbers at the same point — a factor of 3.74·10⁴ — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

sketch-draws is one function in lib/figures/tsketch.js — sketching a tensor — a random matrix larger than the object, and the structured one that is not. 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.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 1.51·10⁸ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 4,032 numbers at the same point — a factor of 3.74·10⁴ — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 1.51·10⁸ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 4,032 numbers at the same point — a factor of 3.74·10⁴ — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 1.93·10¹⁰ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 8,064 numbers at the same point — a factor of 2.4·10⁶ — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 1.18·10⁶ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 2,016 numbers at the same point — a factor of 585 — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 2.02·10⁷ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 3,024 numbers at the same point — a factor of 6665 — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

A dense Gaussian sketch of a mode-k unfolding multiplies an n × n^{d−1} matrix by a random one with n^{d−1} rows, so the random object is the size of the tensor divided by n — the line that crosses the tensor's own entry count at d = 2 and is 2.58·10⁹ by d = 8. A Khatri–Rao sketch replaces it with d − 1 small Gaussians per column, costing 6,048 numbers at the same point — a factor of 4.27·10⁵ — and is applied as mode products, so nothing the size of the tensor is ever formed. What it is not is Gaussian, which is what the previous figure has to measure rather than bound.

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.

9 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 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 146 of 287 generators — 131 print a residual and 15 are exempt with a published reason; 141 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.