randn

→Tensor

randn(size: _int | tuple[_int, ...] = (), dtype: DTypeLike = None, device: DeviceLike = None, requires_grad: _bool = False, generator: _C_engine.Generator | None = None)

source

Return a tensor of samples drawn from the standard normal distribution.

Each element is drawn independently from $\mathcal{N}(0, 1)$ , the standard Gaussian with zero mean and unit variance:

X_i \sim \mathcal{N}(0, 1), \quad f_X(x) = \frac{1}{\sqrt{2\pi}}\, e^{-x^2 / 2}

The distribution has mean $\mathbb{E}[X] = 0$ , variance $\operatorname{Var}[X] = 1$ , and all odd central moments zero.

Parameters

*sizeint or tuple[int, ...]

Shape of the output tensor.

dtypelucid.dtype

Floating-point data type. Defaults to the global default.

devicestr or lucid.device

Target device — "cpu" or "metal".

requires_gradbool

Enable autograd tracking. Default: False.

generatorlucid._C.engine.Generator

Explicit generator. Defaults to the global default generator.

Returns

Tensor

Tensor of shape size with $\mathcal{N}(0, 1)$ samples.

Notes

The engine uses the Box–Muller transform to convert pairs of uniform samples $(u_1, u_2) \sim U(0,1)^2$ into standard normals:

z_1 = \sqrt{-2\ln u_1}\,\cos(2\pi u_2), \quad z_2 = \sqrt{-2\ln u_1}\,\sin(2\pi u_2)

This is exact (not approximate) and is highly efficient on SIMD hardware because the two outputs $z_1, z_2$ can be produced simultaneously from one pair of uniform draws.

Xavier / Glorot initialisation (Glorot & Bengio, 2010) uses $\mathcal{N}(0, \sigma^2)$ with

\sigma = \sqrt{\frac{2}{n_\text{in} + n_\text{out}}}

which can be obtained as randn(fan_in, fan_out) * sigma.

He / Kaiming initialisation (He et al., 2015) for ReLU networks uses $\sigma = \sqrt{2 / n_\text{in}}$ .

Examples

>>> import lucid
>>> lucid.manual_seed(0)
>>> x = lucid.randn(1000)
>>> abs(float(x.mean())) < 0.1       # ≈ 0
True
>>> abs(float(x.std()) - 1.0) < 0.1  # ≈ 1
True
Kaiming initialisation for a linear layer:
>>> fan_in = 256
>>> w = lucid.randn(fan_in, 512) * (2.0 / fan_in) ** 0.5