normalization.rst

Normalization

import torch

We define two kind of normalizations: component and norm.

Definition

component

component normalization refers to tensors with each component of value around 1. More precisely, the second moment of each component is 1.

⟨x_i²⟩ = 1

Examples:

• [1.0, -1.0, -1.0, 1.0]
• [1.0, 1.0, 1.0, 1.0] the mean don't need to be zero
• [0.0, 2.0, 0.0, 0.0] this is still fine because ∥x∥² = n

torch.randn(10)

norm

norm normalization refers to tensors of norm close to 1.

∥x∥ ≈ 1

Examples:

• [0.5, -0.5, -0.5, 0.5]
• [0.5, 0.5, 0.5, 0.5] the mean don't need to be zero
• [0.0, 1.0, 0.0, 0.0]

torch.randn(10) / 10**0.5

There is just a factor $\sqrt{n}$ between the two normalizations.

Motivation

Assuming that the weights distribution obey

⟨w_i⟩ = 0

⟨w_iw_j⟩ = σ²δ_ij

It imply that the two first moments of x ⋅ w (and therefore mean and variance) are only function of the second moment of x

$$\langle x \cdot w \rangle &= \sum_i \langle x_i w_i \rangle = \sum_i \langle x_i \rangle \langle w_i \rangle = 0$$$$\langle (x \cdot w)^2 \rangle &= \sum_{i} \sum_{j} \langle x_i w_i x_j w_j \rangle$$$$&= \sum_{i} \sum_{j} \langle x_i x_j \rangle \langle w_i w_j \rangle$$$$&= \sigma^2 \sum_{i} \langle x_i^2 \rangle$$

Testing

You can use e3nn.util.test.assert_normalized to check whether a function or module is normalized at initialization:

from e3nn.util.test import assert_normalized
from e3nn import o3
assert_normalized(o3.Linear("10x0e", "10x0e"))