2nd/10_Wiki/Topics/Domain_Programming/AI_and_ML/Symmetry-and-Invariance.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-symmetry-and-invariance	Symmetry and Invariance	10_Wiki/Topics	verified	self	Equivariance Group Equivariant Networks Invariant ML	none	A	0.9	applied	equivariance group-theory geometric-deep-learning neural-network		2026-05-10	pending	language framework python pytorch

Symmetry and Invariance

매 한 줄

"매 inductive bias 의 가장 powerful form". Physical / geometric world 의 symmetry (rotation, translation, permutation, scale) 을 architecture 에 baking → sample efficiency, generalization, interpretability 의 dramatic 개선. 2026 의 AlphaFold 3, EGNN, equivariant diffusion 의 mainstream.

매 핵심

매 invariance vs equivariance

• Invariance: f(g·x) = f(x) — 매 transformed input 의 output 의 동일.
• Equivariance: f(g·x) = g·f(x) — 매 input transform → output 의 same transform.
• 매 CNN 의 translation-equivariant (conv) + global pooling 의 translation-invariant.

매 group types

• Translation (T(d)): CNN 의 default.
• Rotation (SO(2), SO(3)): equivariant CNN (E(2)-CNN), spherical CNN.
• Permutation (S_n): GNN, DeepSets — 매 set / graph input.
• Scale: scale-equivariant network (less common).
• Lorentz / Galilean: physics-informed.

매 modern arch

• EGNN (Satorras 2021): E(n)-equivariant graph network — 매 molecule / particle.
• AlphaFold 3 (2024): SE(3)-equivariant — 매 protein structure.
• Equivariant diffusion (e.g., DiffDock 2024): 매 SE(3)-equivariant noise/denoise.
• Graph Transformers (GraphGPS): 매 permutation equivariant.

매 응용

1. Drug discovery — 매 molecule rotation 의 same energy.
2. Protein folding (AlphaFold).
3. Particle physics — 매 Lorentz symmetry.
4. 3D vision, point cloud (PointNet permutation, SE(3)-Transformer).
5. Robotics manipulation — 매 pose-equivariant policy.

💻 패턴

DeepSets (permutation-invariant)

import torch
import torch.nn as nn

class DeepSets(nn.Module):
    def __init__(self, in_dim, hidden, out_dim):
        super().__init__()
        self.phi = nn.Sequential(nn.Linear(in_dim, hidden), nn.ReLU(),
                                  nn.Linear(hidden, hidden))
        self.rho = nn.Sequential(nn.Linear(hidden, hidden), nn.ReLU(),
                                  nn.Linear(hidden, out_dim))
    def forward(self, x):  # x: (batch, set_size, in_dim)
        return self.rho(self.phi(x).sum(dim=1))

EGNN (E(n)-equivariant graph layer)

class EGNNLayer(nn.Module):
    def __init__(self, hidden):
        super().__init__()
        self.edge_mlp = nn.Sequential(nn.Linear(2*hidden + 1, hidden), nn.SiLU(),
                                       nn.Linear(hidden, hidden))
        self.coord_mlp = nn.Sequential(nn.Linear(hidden, hidden), nn.SiLU(),
                                        nn.Linear(hidden, 1))
        self.node_mlp = nn.Sequential(nn.Linear(2*hidden, hidden), nn.SiLU(),
                                       nn.Linear(hidden, hidden))
    def forward(self, h, x, edge_index):
        i, j = edge_index
        diff = x[i] - x[j]
        dist = (diff ** 2).sum(-1, keepdim=True)
        e = self.edge_mlp(torch.cat([h[i], h[j], dist], dim=-1))
        x_new = x + (diff * self.coord_mlp(e)).index_add_(0, i, ...)
        h_new = h + self.node_mlp(torch.cat([h, e.sum_aggr(j)], dim=-1))
        return h_new, x_new

Group convolution (rotation equivariance)

# e2cnn library
from e2cnn import gspaces, nn as enn

r2_act = gspaces.Rot2dOnR2(N=8)  # 8 discrete rotations
in_type = enn.FieldType(r2_act, 3 * [r2_act.trivial_repr])
out_type = enn.FieldType(r2_act, 16 * [r2_act.regular_repr])

conv = enn.R2Conv(in_type, out_type, kernel_size=5)
# input rotated by 45° → output 의 45°-rotated counterpart

Data augmentation as approximate equivariance

import torchvision.transforms as T

aug = T.Compose([
    T.RandomRotation(180),
    T.RandomHorizontalFlip(),
    T.RandomVerticalFlip(),
])
# 매 cheap baseline — works but no strict guarantee

SE(3)-Transformer attention (sketch)

# se3-transformer-pytorch
from se3_transformer_pytorch import SE3Transformer

model = SE3Transformer(
    dim=64, heads=8, depth=4,
    num_degrees=2, valid_radius=10
)
# input: features + 3D coords; output equivariant under rotation+translation

Invariance test (validation)

def test_invariance(model, x, group_action):
    y = model(x)
    y_g = model(group_action(x))
    assert torch.allclose(y, y_g, atol=1e-5), "not invariant"

def test_equivariance(model, x, group_action):
    y = model(x)
    y_g = model(group_action(x))
    assert torch.allclose(group_action(y), y_g, atol=1e-5), "not equivariant"

매 결정 기준

상황	Approach
매 image (translation)	CNN
매 set / graph	DeepSets / GNN
매 molecule / 3D points	EGNN, SE(3)-Transformer
매 protein structure	AlphaFold-style SE(3)
매 quick baseline	data augmentation
매 strict guarantee + small data	exact equivariant arch

기본값: 매 small data + clear symmetry → equivariant arch. 매 large data → augmentation often sufficient.

🔗 Graph

• 부모: Inductive-Bias
• 응용: AlphaFold · Point-Cloud
• Adjacent: Graph-Neural-Network · Transformer

🤖 LLM 활용

언제: 매 scientific ML task 의 architecture choice (molecule, physics, geometry), 매 small-data regime 의 inductive bias 강화. 언제 X: 매 plain text / language (no clear geometric symmetry — Transformer permutation-equivariance 만 충분), 매 huge data + flat structure (augmentation OK).

❌ 안티패턴

• 매 augmentation 만 의존 (data scarce): 매 strict equivariance 의 generalize 더 잘함 — 매 sample efficiency.
• Equivariant arch 의 over-engineering (data abundant): 매 large-data regime 에서 plain Transformer 가 따라잡음.
• Approximate equivariance 의 unstated: 매 floating-point + non-linearity 로 매 exact 가 깨질 수 있음 — 매 test_equivariance assertion 추가.

🧪 검증 / 중복

• Verified (Cohen & Welling 2016 Group Equivariant CNN; Satorras et al. 2021 EGNN; Bronstein et al. 2021 "Geometric Deep Learning"; Jumper et al. 2021/2024 AlphaFold 2/3).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — group theory + EGNN + SE(3) + AlphaFold context