refactor(topics): 멀티 에이전트용 지식 재편 — _Common(공통 기본기) + Domain_* 구조
에이전트 8종(대화형/프로그래머 C·S/디자이너/설계자/기획자/QA/PD/PM)에게 [공통 기본 능력 + 롤별 Specialty] 2층으로 지식을 주입하기 위한 재분류. 문서 내용·포맷은 무수정, 폴더 이동만 (6,372개 문서 수 보존 확인). - Topic_Programming → Domain_Programming (내부 구조 보존) - Topic_Graphic → Domain_Design - Topic_Business → Domain_Product - Topic_General → Domain_General - _Common 신설: Math(구 Topic_Math_Specialty), Reasoning(구 General/From_Thinking & Reasoning), Reasoning_Creativity(구 General/From_창의성), Communication(Poetic_Blog_Writing + From_writing) - 타 도메인의 From_* 폴더는 유지 (출처 표기일 뿐, 이미 도메인에 맞게 분류된 문서) - 빈 폴더 정리 (memory/procedures) - 에이전트→폴더 매핑은 workspace의 .astra/agent-knowledge-map.json (9개 에이전트) Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Antigravity Agent
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---
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id: wiki-2026-0508-symmetry-and-invariance
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title: Symmetry and Invariance
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Equivariance, Group Equivariant Networks, Invariant ML]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.9
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verification_status: applied
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tags: [equivariance, group-theory, geometric-deep-learning, neural-network]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: python
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framework: pytorch
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---
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# Symmetry and Invariance
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 invariance vs equivariance
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- **Invariance**: f(g·x) = f(x) — 매 transformed input 의 output 의 동일.
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- **Equivariance**: f(g·x) = g·f(x) — 매 input transform → output 의 same transform.
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- 매 CNN 의 translation-equivariant (conv) + global pooling 의 translation-invariant.
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### 매 group types
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- **Translation** (T(d)): CNN 의 default.
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- **Rotation** (SO(2), SO(3)): equivariant CNN (E(2)-CNN), spherical CNN.
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- **Permutation** (S_n): GNN, DeepSets — 매 set / graph input.
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- **Scale**: scale-equivariant network (less common).
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- **Lorentz / Galilean**: physics-informed.
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### 매 modern arch
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- **EGNN** (Satorras 2021): E(n)-equivariant graph network — 매 molecule / particle.
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- **AlphaFold 3** (2024): SE(3)-equivariant — 매 protein structure.
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- **Equivariant diffusion** (e.g., DiffDock 2024): 매 SE(3)-equivariant noise/denoise.
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- **Graph Transformers** (GraphGPS): 매 permutation equivariant.
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### 매 응용
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1. Drug discovery — 매 molecule rotation 의 same energy.
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2. Protein folding (AlphaFold).
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3. Particle physics — 매 Lorentz symmetry.
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4. 3D vision, point cloud (PointNet permutation, SE(3)-Transformer).
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5. Robotics manipulation — 매 pose-equivariant policy.
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## 💻 패턴
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### DeepSets (permutation-invariant)
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```python
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import torch
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import torch.nn as nn
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class DeepSets(nn.Module):
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def __init__(self, in_dim, hidden, out_dim):
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super().__init__()
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self.phi = nn.Sequential(nn.Linear(in_dim, hidden), nn.ReLU(),
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nn.Linear(hidden, hidden))
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self.rho = nn.Sequential(nn.Linear(hidden, hidden), nn.ReLU(),
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nn.Linear(hidden, out_dim))
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def forward(self, x): # x: (batch, set_size, in_dim)
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return self.rho(self.phi(x).sum(dim=1))
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```
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### EGNN (E(n)-equivariant graph layer)
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```python
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class EGNNLayer(nn.Module):
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def __init__(self, hidden):
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super().__init__()
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self.edge_mlp = nn.Sequential(nn.Linear(2*hidden + 1, hidden), nn.SiLU(),
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nn.Linear(hidden, hidden))
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self.coord_mlp = nn.Sequential(nn.Linear(hidden, hidden), nn.SiLU(),
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nn.Linear(hidden, 1))
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self.node_mlp = nn.Sequential(nn.Linear(2*hidden, hidden), nn.SiLU(),
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nn.Linear(hidden, hidden))
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def forward(self, h, x, edge_index):
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i, j = edge_index
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diff = x[i] - x[j]
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dist = (diff ** 2).sum(-1, keepdim=True)
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e = self.edge_mlp(torch.cat([h[i], h[j], dist], dim=-1))
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x_new = x + (diff * self.coord_mlp(e)).index_add_(0, i, ...)
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h_new = h + self.node_mlp(torch.cat([h, e.sum_aggr(j)], dim=-1))
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return h_new, x_new
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```
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### Group convolution (rotation equivariance)
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```python
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# e2cnn library
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from e2cnn import gspaces, nn as enn
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r2_act = gspaces.Rot2dOnR2(N=8) # 8 discrete rotations
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in_type = enn.FieldType(r2_act, 3 * [r2_act.trivial_repr])
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out_type = enn.FieldType(r2_act, 16 * [r2_act.regular_repr])
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conv = enn.R2Conv(in_type, out_type, kernel_size=5)
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# input rotated by 45° → output 의 45°-rotated counterpart
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```
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### Data augmentation as approximate equivariance
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```python
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import torchvision.transforms as T
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aug = T.Compose([
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T.RandomRotation(180),
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T.RandomHorizontalFlip(),
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T.RandomVerticalFlip(),
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])
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# 매 cheap baseline — works but no strict guarantee
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```
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### SE(3)-Transformer attention (sketch)
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```python
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# se3-transformer-pytorch
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from se3_transformer_pytorch import SE3Transformer
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model = SE3Transformer(
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dim=64, heads=8, depth=4,
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num_degrees=2, valid_radius=10
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)
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# input: features + 3D coords; output equivariant under rotation+translation
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```
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### Invariance test (validation)
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```python
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def test_invariance(model, x, group_action):
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y = model(x)
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y_g = model(group_action(x))
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assert torch.allclose(y, y_g, atol=1e-5), "not invariant"
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def test_equivariance(model, x, group_action):
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y = model(x)
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y_g = model(group_action(x))
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assert torch.allclose(group_action(y), y_g, atol=1e-5), "not equivariant"
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| 매 image (translation) | CNN |
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| 매 set / graph | DeepSets / GNN |
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| 매 molecule / 3D points | EGNN, SE(3)-Transformer |
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| 매 protein structure | AlphaFold-style SE(3) |
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| 매 quick baseline | data augmentation |
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| 매 strict guarantee + small data | exact equivariant arch |
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**기본값**: 매 small data + clear symmetry → equivariant arch. 매 large data → augmentation often sufficient.
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## 🔗 Graph
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- 부모: [[Inductive-Bias]]
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- 응용: [[AlphaFold]] · [[Point-Cloud]]
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- Adjacent: [[Graph-Neural-Network]] · [[Transformer]]
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## 🤖 LLM 활용
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**언제**: 매 scientific ML task 의 architecture choice (molecule, physics, geometry), 매 small-data regime 의 inductive bias 강화.
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**언제 X**: 매 plain text / language (no clear geometric symmetry — Transformer permutation-equivariance 만 충분), 매 huge data + flat structure (augmentation OK).
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## ❌ 안티패턴
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- **매 augmentation 만 의존 (data scarce)**: 매 strict equivariance 의 generalize 더 잘함 — 매 sample efficiency.
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- **Equivariant arch 의 over-engineering (data abundant)**: 매 large-data regime 에서 plain Transformer 가 따라잡음.
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- **Approximate equivariance 의 unstated**: 매 floating-point + non-linearity 로 매 exact 가 깨질 수 있음 — 매 test_equivariance assertion 추가.
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## 🧪 검증 / 중복
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- 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).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — group theory + EGNN + SE(3) + AlphaFold context |
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