symmetry-discovery-questionnaire
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ChineseSymmetry Discovery Questionnaire
对称性发现问卷
What Is It?
这是什么？
This skill helps you discover hidden symmetries in your data through a structured collaborative process. Symmetries are transformations that leave important properties unchanged - and building them into neural networks dramatically improves performance (better sample efficiency, faster convergence, improved generalization).
You don't need to know group theory. This skill guides you through domain-specific questions to uncover what symmetries might be present.
该工具通过结构化的协作流程帮助你发现数据中隐藏的对称性。对称性是指不会改变重要属性的变换——将这些对称性融入神经网络能显著提升性能（更好的样本效率、更快的收敛速度、更强的泛化能力）。
你无需了解群论。该工具会引导你完成特定领域的问题，以揭示可能存在的对称性。
Workflow
工作流程
Copy this checklist and track your progress:
Symmetry Discovery Progress:
- [ ] Step 1: Classify your domain and data type
- [ ] Step 2: Analyze coordinate system choices
- [ ] Step 3: Test candidate transformations
- [ ] Step 4: Analyze physical constraints
- [ ] Step 5: Determine output behavior under transformations
- [ ] Step 6: Document symmetry candidatesStep 1: Classify your domain and data type
Ask user what their primary data type is. Use this table to identify likely symmetries and guide further questions. Images (2D grids) → likely translation, rotation, reflection. 3D data (point clouds, meshes) → likely SE(3), E(3). Molecules → E(3) + permutation + point groups. Graphs/Networks → permutation. Sets → permutation. Time series → time-translation, periodicity. Tabular → rarely symmetric. Physical systems → conservation laws imply symmetries. For detailed worked examples by domain, consult Domain Examples.
Step 2: Analyze coordinate system choices
Guide user through coordinate analysis questions: Is there a preferred origin? (NO → translation invariance). Is there a preferred orientation? (NO → rotation invariance). Is there a preferred handedness? (NO → reflection invariance). Is there a preferred scale? (NO → scale invariance). Is element ordering meaningful? (NO → permutation invariance). Document each answer with reasoning.
Step 3: Test candidate transformations
For each candidate transformation T, ask: "If I transform my input by T, should my output change?" If NO → invariance to T. If YES predictably → equivariance to T. If YES unpredictably → no symmetry. Use domain-specific checklists from Domain Transformation Tests. Test all relevant transformations systematically. For the detailed methodology behind this testing approach, see Methodology.
Step 4: Analyze physical constraints
Ask about conservation laws and physical symmetries. Noether's theorem: every conservation law implies a symmetry. Energy conserved → time-translation symmetry. Momentum conserved → space-translation symmetry. Angular momentum conserved → rotation symmetry. Ask: Are there physical conservation laws? Is system isolated from external reference frames? Are there gauge freedoms?
Step 5: Determine output behavior under transformations
Critical question: When input transforms, how should output transform? Classification labels → stay same (invariance). Bounding boxes → move with object (equivariance). Force vectors → rotate with system (equivariance). Scalar properties → stay same (invariance). Segmentation masks → transform with image (equivariance). This determines whether you need invariant or equivariant architecture.
Step 6: Document symmetry candidates
Create summary using Output Template. List identified symmetries with confidence levels. Note uncertain cases that need empirical validation. Identify non-symmetries (transformations that DO matter). Recommend next steps for validation and formalization. Quality criteria for this output are defined in Quality Rubric.
复制这份清单并跟踪你的进度：
Symmetry Discovery Progress:
- [ ] Step 1: Classify your domain and data type
- [ ] Step 2: Analyze coordinate system choices
- [ ] Step 3: Test candidate transformations
- [ ] Step 4: Analyze physical constraints
- [ ] Step 5: Determine output behavior under transformations
- [ ] Step 6: Document symmetry candidates步骤1：分类你的领域和数据类型
询问用户其主要数据类型是什么。使用下表识别可能存在的对称性并指导后续问题。图像（2D网格）→ 可能存在平移、旋转、反射对称性。3D数据（点云、网格）→ 可能存在SE(3)、E(3)对称性。分子→ E(3)+置换+点群对称性。图/网络→ 置换对称性。集合→ 置换对称性。时间序列→ 时间平移、周期性。表格数据→ 很少存在对称性。物理系统→ 守恒定律意味着存在对称性。如需按领域划分的详细实例，请查阅领域实例。
步骤2：分析坐标系选择
引导用户完成坐标系分析问题：是否存在首选原点？（否→平移不变性）。是否存在首选方向？（否→旋转不变性）。是否存在首选手性？（否→反射不变性）。是否存在首选尺度？（否→尺度不变性）。元素顺序是否有意义？（否→置换不变性）。记录每个答案及其推理依据。
步骤3：测试候选变换
对于每个候选变换T，询问：“如果我用T变换输入，输出应该改变吗？” 如果否→对T的不变性。如果是且可预测→对T的等变性。如果是且不可预测→无对称性。使用领域变换测试中的特定领域清单。系统地测试所有相关变换。有关该测试方法的详细方法论，请参阅方法论。
步骤4：分析物理约束
询问守恒定律和物理对称性相关问题。诺特定理：每一条守恒定律都对应一种对称性。能量守恒→时间平移对称性。动量守恒→空间平移对称性。角动量守恒→旋转对称性。询问：是否存在物理守恒定律？系统是否与外部参考系隔离？是否存在规范自由度？
步骤5：确定变换下的输出行为
关键问题：当输入发生变换时，输出应如何变化？分类标签→保持不变（不变性）。边界框→随物体移动（等变性）。力向量→随系统旋转（等变性）。标量属性→保持不变（不变性）。分割掩码→随图像变换（等变性）。这决定了你需要的是不变性还是等变性架构。
步骤6：记录对称性候选者
使用输出模板创建摘要。列出已识别的对称性及其置信度。标注需要实证验证的不确定案例。识别非对称性（即会产生影响的变换）。建议验证和形式化的后续步骤。该输出的质量标准定义在质量评估准则中。
Domain Transformation Tests
领域变换测试
Image Symmetries
图像对称性
| Transformation | Test Question | If NO → |
|---|---|---|
| Translation | Does object position matter for label? | Translation invariance |
| Rotation (90°) | Would rotated image have same label? | C4 symmetry |
| Rotation (any) | Would any rotation preserve label? | SO(2) symmetry |
| Horizontal flip | Would mirror image have same label? | Reflection |
| Scale | Would zoomed image have same label? | Scale invariance |
| 变换 | 测试问题 | 如果否→ |
|---|---|---|
| 平移 | 物体位置对标签有影响吗？ | 平移不变性 |
| 旋转（90°） | 旋转后的图像会有相同标签吗？ | C4对称性 |
| 旋转（任意角度） | 任意旋转都会保留标签吗？ | SO(2)对称性 |
| 水平翻转 | 镜像图像会有相同标签吗？ | 反射对称性 |
| 缩放 | 缩放后的图像会有相同标签吗？ | 尺度不变性 |
3D Data Symmetries
3D数据对称性
| Transformation | Test Question | If NO → |
|---|---|---|
| 3D Translation | Does absolute position matter? | Translation invariance |
| 3D Rotation | Does orientation matter? | SO(3) or SE(3) |
| Reflection | Does handedness matter? | O(3) or E(3) |
| Point permutation | Does point ordering matter? | Permutation invariance |
| 变换 | 测试问题 | 如果否→ |
|---|---|---|
| 3D平移 | 绝对位置有影响吗？ | 平移不变性 |
| 3D旋转 | 方向有影响吗？ | SO(3)或SE(3)对称性 |
| 反射 | 手性有影响吗？ | O(3)或E(3)对称性 |
| 点置换 | 点的顺序有影响吗？ | 置换不变性 |
Graph Symmetries
图对称性
| Transformation | Test Question | If NO → |
|---|---|---|
| Node relabeling | Does node ID matter, or just connectivity? | Permutation invariance |
| 变换 | 测试问题 | 如果否→ |
|---|---|---|
| 节点重标记 | 节点ID有影响，还是仅连接性有影响？ | 置换不变性 |
Molecular Symmetries
分子对称性
| Transformation | Test Question | If NO → |
|---|---|---|
| Rotation | Is property independent of orientation? | SO(3) |
| Translation | Is property independent of position? | Translation |
| Reflection | Are both enantiomers equivalent? | Include reflections |
| Atom permutation | Do identical atoms behave identically? | Permutation |
| 变换 | 测试问题 | 如果否→ |
|---|---|---|
| 旋转 | 属性是否与方向无关？ | SO(3)对称性 |
| 平移 | 属性是否与位置无关？ | 平移对称性 |
| 反射 | 两种对映异构体是否等效？ | 包含反射对称性 |
| 原子置换 | 相同原子的行为是否相同？ | 置换不变性 |
Temporal Symmetries
时间对称性
| Transformation | Test Question | If NO → |
|---|---|---|
| Time shift | Can pattern occur at any time? | Time-translation |
| Time reversal | Is forward same as backward? | Time-reversal |
| Periodicity | Do patterns repeat with period T? | Cyclic symmetry |
| 变换 | 测试问题 | 如果否→ |
|---|---|---|
| 时间偏移 | 模式可以在任意时间出现吗？ | 时间平移对称性 |
| 时间反转 | 正向与反向是否相同？ | 时间反转对称性 |
| 周期性 | 模式是否会以周期T重复？ | 循环对称性 |
Quick Reference
快速参考
The 5 Key Questions:
- Is there a preferred coordinate system? (origin, orientation, scale)
- Does element ordering matter?
- What transformations leave the label unchanged?
- What physical constraints apply?
- How should outputs transform when inputs transform?
Common Symmetry → Group Mapping:
- Rotation (2D, discrete) → Cyclic group Cₙ
- Rotation + reflection (2D) → Dihedral group Dₙ
- Rotation (2D, continuous) → SO(2)
- Rotation (3D) → SO(3)
- Rotation + translation (3D) → SE(3)
- Full Euclidean (3D) → E(3)
- Permutation → Symmetric group Sₙ
5个关键问题：
- 是否存在首选坐标系？（原点、方向、尺度）
- 元素顺序是否有意义？
- 哪些变换不会改变标签？
- 适用哪些物理约束？
- 当输入变换时，输出应如何变换？
常见对称性→群映射：
- 旋转（2D，离散）→ 循环群Cₙ
- 旋转+反射（2D）→ 二面体群Dₙ
- 旋转（2D，连续）→ SO(2)
- 旋转（3D）→ SO(3)
- 旋转+平移（3D）→ SE(3)
- 完整欧几里得变换（3D）→ E(3)
- 置换→ 对称群Sₙ
Output Template
输出模板
SYMMETRY CANDIDATE SUMMARY
==========================
Domain: [Data type]
Task: [Classification/Regression/Detection/etc.]
IDENTIFIED SYMMETRIES:
1. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
2. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
UNCERTAIN SYMMETRIES (need validation):
- [Transformation]: [Reason for uncertainty]
NON-SYMMETRIES (transformations that DO matter):
- [Transformation]: [Why it matters]
NEXT STEPS:
- Empirically validate uncertain symmetry candidates
- Map confirmed symmetries to mathematical groups
- Design architecture based on validated group structureSYMMETRY CANDIDATE SUMMARY
==========================
Domain: [Data type]
Task: [Classification/Regression/Detection/etc.]
IDENTIFIED SYMMETRIES:
1. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
2. [Transformation]: [Invariance/Equivariance]
- Evidence: [Why you believe this]
- Confidence: [High/Medium/Low]
UNCERTAIN SYMMETRIES (need validation):
- [Transformation]: [Reason for uncertainty]
NON-SYMMETRIES (transformations that DO matter):
- [Transformation]: [Why it matters]
NEXT STEPS:
- Empirically validate uncertain symmetry candidates
- Map confirmed symmetries to mathematical groups
- Design architecture based on validated group structure