var_op

→TensorImplPtr

int var_op(const int & a, const int & axes, bool keepdims)

source edit

Python wrappers

pylucid.ops.varfunction

Compute the biased variance of a along axes.

Uses the population estimator $\sigma^2 = \frac{1}{N} \sum_i (x_i - \bar{x})^2$ — i.e. the denominator is $N$ , not $N - 1$ . This corresponds to the reference framework's unbiased=False (population) variant; the unbiased / sample variant is not exposed by this entry point. The standard deviation std_op is built directly on top by composing sqrt with this op so the autograd chain is automatic.

During the forward pass the mean is computed once with keepdims=true and broadcast to the full input shape so that VarBackward::apply can compute $x - \bar{x}$ without a second broadcast call. Both the input x and the broadcast mean are saved on the backward node.

Math

\sigma^2 = \frac{1}{N} \sum_{i \in \text{axes}} (x_i - \bar{x})^2, \qquad \frac{\partial \mathcal{L}}{\partial x_i} = \frac{2}{N}(x_i - \bar{x})\cdot \mathrm{broadcast}\!\left(\frac{\partial \mathcal{L}} {\partial \sigma^2}\right).

Shape

If keepdims is false, the reduced axes are removed; if true, they become size- $1$ .

See Also

std_op — square root of the variance.

Parameters

aTensorImplPtr

Input tensor of float dtype. Non-null.

axesstd::vector<int>

Axes to reduce. May be empty (reduce all axes). Negative indices wrap around ndim.

keepdimsbool

If true, retains reduced dims as size-1 entries; otherwise they are collapsed.

Returns

TensorImplPtr

Output tensor with the reduced shape. Autograd is wired when a->requires_grad() is true and GradMode is enabled.

Notes

Internal dispatch: IBackend::variance — Accelerate on CPU, MLX on GPU. The CPU implementation uses Welford-style accumulation for numerical stability on large $N$ . The empty-axis edge case (reduced == 0) clamps the count to $1$ to avoid a division by zero in the backward node; in that pathological case the gradient is not statistically meaningful but is well-defined.