linear

std.nn.linear · Level L0

Affine layer: a batch of rows times a weight matrix, plus a bias.

Y = X·W + b

Signature

linear(X: f64[m, k], W: f64[k, n], b: f64[n]) → f64[m, n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

Xf64[m, k]Wf64[k, n]bf64[n]MatMulXWAddYYf64[m, n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Equal to the reference X @ W + b in exact rational arithmetic, on all 40 test cases.
  • All 523 float64 results inside the running error bound; the closest uses 72% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
70%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
222

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls
—
Called by
mlp2mlp2
sha256:035bcc77537b3f00d8df34fc906bfe953607d33e7284fa6d326ad4a5a74aeff3

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol
Ω:std.nn.linear · Ω:linear
Pins
sha256:6a24fcda00e7db9e2f9f90ec0e027099376028e4d85cfb147956a07c28472828this graph alone
Evidence
sha256:bbe96fc870565212a6e1c02325867623b106456b0d6405980a7b35f2043d658fthe hash of its verification record
Needs
no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.