vector_norm

→Tensor

vector_norm(x: Tensor, ord: int | float = 2, dim: int | list[int] | tuple[int, ...] | None = None, keepdim: bool = False, dtype: object = None)

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Compute a vector $p$ -norm along an axis.

Reduces along dim (or all elements if dim is None) using

\|x\|_p \,=\, \Big(\sum_i |x_i|^{\,p}\Big)^{1/p}

for any positive real $p$ . Special-cased values:

$p = 0$ — count of non-zero entries (not a true norm).
$p = 1$ — $\sum_i |x_i|$ .
$p = 2$ — Euclidean norm.
$p = +\infty$ — $\max_i |x_i|$ .
$p = -\infty$ — $\min_i |x_i|$ .

Parameters

xTensor

Input tensor.

ordint or float= 2

Order of the norm. Default 2.

dimint, list of int, tuple of int or None= None

Axis or axes to reduce. None reduces over every axis — the rank is retained, so keepdim=True yields one size-1 axis per input axis rather than a single one.

keepdimbool= False

If True, reduced dimensions are retained with size 1.

dtypedtype= None

Cast x to this dtype before reducing, so the whole computation — and the result — is in it; a complex dtype gives its real counterpart. It must be complex for a complex x and real otherwise, and must not narrow x. None (default) reduces in x's own dtype.

Returns

Tensor

Norm along the specified axes — real, also for a complex x.

Raises

TypeError

If x is an integer or bool tensor (a DtypeMismatch), or if dtype would narrow it or its kind (real / complex) differs.

ValueError

If dim names one axis twice.

Notes

All operations are routed through autograd-aware engine kernels, so gradients flow naturally even for non-integer $p$ (via $p$ -power and root). Where the norm is zero the gradient is zero — the subgradient the reference picks — rather than the nan the root's infinite slope there would give; so is a zero entry's for $p < 1$ .

A float16 or bfloat16 input is reduced in float32 and the result cast back, so a norm that fits the input dtype is not lost to an intermediate sum that does not: the 2-norm of 65 536 half-precision ones is 256, not inf.

A complex x is reduced through its magnitudes $|x_i|$ , as the definition says. Squaring the entries themselves gave a complex "norm" for ord=2 on Metal and an error on the CPU.

Examples

>>> import lucid
>>> from lucid.linalg import vector_norm
>>> vector_norm(lucid.tensor([3.0, 4.0]))
tensor(5.)
>>> vector_norm(lucid.tensor([1.0, -2.0, 3.0]), ord=1)
tensor(6.)
>>> vector_norm(lucid.tensor([3.0, 4.0]), dtype=lucid.float64)
tensor(5., dtype=lucid.float64)
>>> vector_norm(lucid.tensor([3 + 4j, 0j]))
tensor(5.)

Used by 1

lucid.nn.functional.normalization

>>> import lucid >>> from lucid.linalg import vector_norm >>> vector_norm(lucid.tensor([3.0, 4.0])) tensor(5.) >>> vector_norm(lucid.tensor([1.0, -2.0, 3.0]), ord=1) tensor(6.) >>> vector_norm(lucid.tensor([3.0, 4.0]), dtype=lucid.float64) tensor(5., dtype=lucid.float64) >>> vector_norm(lucid.tensor([3 + 4j, 0j])) tensor(5.)