{
  "slug": "cayley-orthogonal-transform",
  "title": "斜对称矩阵 Cayley 变换构造正交矩阵",
  "difficulty": "Medium",
  "difficulty_breakdown": {
    "concept": 3,
    "implementation": 3,
    "math_derivation": 3,
    "boundary_complexity": 3
  },
  "category": "Math",
  "tags": [
    "LinearAlgebra",
    "OrthogonalMatrix",
    "CayleyTransform",
    "TorchLeet"
  ],
  "description_md": "### 题目描述\n实现正交权重参数化（Orthogonal RNN / ViT）中常用的 Cayley 变换：给定任意方阵参数 `w`（形状 `[N, N]`），先构造斜对称矩阵（Skew-Symmetric Matrix）$A = W - W^\\top$（满足 $A^\\top = -A$），再计算正交矩阵 $Q = (I - A)(I + A)^{-1}$（满足 $Q^\\top Q = I$ 且 $\\det Q = +1$）。\n\n### 函数签名\n```python\ndef cayley_orthogonal_matrix(w: torch.Tensor) -> torch.Tensor:\n```\n\n### 示例\n```python\n# 输入\nw = [[0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]]\n\n# 输出 (Tensor, shape=[3, 3], dtype=float32)\n[[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]\n```\n\n### 参数规格与约束\n- **`w`**：`torch.Tensor`，测试规格 `shape=[3, 3], dtype=float32`\n- **返回值**：`torch.Tensor`，输出规格 `shape=[3, 3], dtype=float32`\n- **精度容差**：`atol=0.0001, rtol=0.0001`\n> 题目来源：改编自 [TorchLeet](https://github.com/Exorust/TorchLeet)（MIT License），已改写为函数式评测接口与中文题面。",
  "entry_point": "cayley_orthogonal_matrix",
  "starter_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import List, Tuple, Dict, Optional\n\ndef cayley_orthogonal_matrix(w: torch.Tensor) -> torch.Tensor:\n    # 请在此处编写你的实现代码\n    pass\n",
  "reference_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import List, Tuple, Dict, Optional\n\ndef cayley_orthogonal_matrix(w: torch.Tensor) -> torch.Tensor:\n    n = w.shape[0]\n    eye = torch.eye(n, dtype=w.dtype)\n    a = w - w.t()\n    return torch.linalg.solve((eye + a).t(), (eye - a).t()).t()\n",
  "test_code": "import torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import List, Tuple, Dict, Optional\n\ncases = [\n    (\"零矩阵对应单位正交阵 I\", torch.zeros(3, 3)),\n    (\"对称矩阵斜对称部分为 0 对应 I\", torch.ones(3, 3)),\n    (\"2x2 旋转生成手算\", torch.tensor([[0.0, 0.5], [-0.5, 0.0]])),\n    (\"随机 5x5 参数生成正交阵\", torch.randn(5, 5)),\n]\nfor idx, (name, *args) in enumerate(cases, 1):\n    t0 = time.time()\n    try:\n        exp = ref_obj(*args)\n        act = user_obj(*args)\n        match, err, max_diff = compare_outputs(act, exp, atol=atol, rtol=rtol)\n        passed, msg = match, err\n    except Exception as e:\n        passed, msg = False, f\"Runtime Exception: {str(e)}\"\n        act, exp, max_diff = None, None, None\n    cases_results.append({\n        \"case_id\": idx, \"case_name\": name, \"passed\": passed,\n        \"time_ms\": round((time.time() - t0) * 1000, 2), \"message\": msg,\n        \"user_output_summary\": summarize_obj(act) if act is not None else None,\n        \"expected_summary\": summarize_obj(exp) if exp is not None else None,\n        \"max_diff\": max_diff,\n    })\n",
  "test_cases": [
    {
      "name": "零矩阵对应单位正交阵 I",
      "input_desc": "真实参数契约用例 (w)",
      "params": [
        {
          "name": "w",
          "type": "torch.Tensor",
          "value": "torch.tensor([[0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]], dtype=torch.float32)"
        }
      ],
      "raw_inputs": {
        "w": "torch.tensor([[0.0, 0.0, 0.0], [0.0, 0.0, 0.0], [0.0, 0.0, 0.0]], dtype=torch.float32)"
      }
    },
    {
      "name": "对称矩阵斜对称部分为 0 对应 I",
      "input_desc": "真实参数契约用例 (w)",
      "params": [
        {
          "name": "w",
          "type": "torch.Tensor",
          "value": "torch.tensor([[1.0, 1.0, 1.0], [1.0, 1.0, 1.0], [1.0, 1.0, 1.0]], dtype=torch.float32)"
        }
      ],
      "raw_inputs": {
        "w": "torch.tensor([[1.0, 1.0, 1.0], [1.0, 1.0, 1.0], [1.0, 1.0, 1.0]], dtype=torch.float32)"
      }
    },
    {
      "name": "2x2 旋转生成手算",
      "input_desc": "真实参数契约用例 (w)",
      "params": [
        {
          "name": "w",
          "type": "torch.Tensor",
          "value": "torch.tensor([[0.0, 0.5], [-0.5, 0.0]], dtype=torch.float32)"
        }
      ],
      "raw_inputs": {
        "w": "torch.tensor([[0.0, 0.5], [-0.5, 0.0]], dtype=torch.float32)"
      }
    }
  ],
  "time_limit": 5.0,
  "memory_limit_mb": 512,
  "atol": 0.0001,
  "rtol": 0.0001,
  "forbidden_modules": [
    "os",
    "sys",
    "subprocess",
    "socket",
    "shutil",
    "pathlib",
    "ctypes",
    "multiprocessing"
  ],
  "forbidden_calls": [
    "eval",
    "exec",
    "compile",
    "open",
    "__import__",
    "input",
    "breakpoint"
  ],
  "is_featured": false,
  "editorial_md": "#### 💡 核心算法原理与数学推导\n**斜对称矩阵 Cayley 变换构造正交矩阵**（`cayley_orthogonal_matrix`）聚焦于 **Math** 中的关键计算逻辑：\n\n实现正交权重参数化（Orthogonal RNN / ViT）中常用的 Cayley 变换：给定任意方阵参数 `w`（形状 `[N, N]`），先构造斜对称矩阵（Skew-Symmetric Matrix）$A = W - W^\\top$（满足 $A^\\top = -A$），再计算正交矩阵 $Q = (I - A)(I + A)^{-1}$（满足 $Q^\\top Q = I$ 且 $\\det Q = +1$）。\n\n在实现中，函数接收参数 `w` (`torch.Tensor`)，目标是计算出类型与形状严格符合 `torch.Tensor` 契约的结果。\n\n#### 📐 输入输出张量变换与步骤拆解\n- **输入基准规格**：`w` 规格为 `torch.Tensor(shape=[3, 3], dtype=float32)`。\n- **期望输出规格**：`torch.Tensor(shape=[3, 3], dtype=float32)`。\n\n**具体推导与计算步骤如下：**\n1. 计算中间量 `n`：通过表达式 `w.shape[0]` 完成张量或数值变换。\n2. 计算中间量 `eye`：通过表达式 `torch.eye(n, dtype=w.dtype)` 完成张量或数值变换。\n3. 计算中间量 `a`：通过表达式 `w - w.t()` 完成张量或数值变换。\n4. 返回最终结果：输出 `torch.linalg.solve((eye + a).t(), (eye - a).t()).t()`，严格对齐函数签名要求的返回类型与维度。\n\n#### 📊 复杂度分析\n- **时间复杂度**：$O(N \\cdot D^2 + D^3)$，主要由矩阵乘积与线性代数求逆/奇异值分解开销主导。\n- **空间复杂度**：$O(N \\cdot D + D^2)$，用于存储核矩阵/协方差矩阵及分解结果张量。\n\n#### ⚠️ 关键实现细节与易错点\n- **输入输出类型与形状契约**：请严格按照函数签名返回对应类型（`torch.Tensor`、`Tuple` 或标量数值），避免改变张量维度。\n- **边界输入鲁棒性**：注意单样本（`Batch=1`）、零值输入或极端超参数下的计算行为，禁止硬编码特定测试用例常量。\n- **向量化算子优先**：优先使用 PyTorch 原生向量化张量算子，避免低效的逐元素循环。\n\n#### 💻 官方标准参考实现\n```python\nimport torch\nimport torch.nn as nn\nimport torch.nn.functional as F\nimport math\nfrom typing import List, Tuple, Dict, Optional\n\ndef cayley_orthogonal_matrix(w: torch.Tensor) -> torch.Tensor:\n    n = w.shape[0]\n    eye = torch.eye(n, dtype=w.dtype)\n    a = w - w.t()\n    return torch.linalg.solve((eye + a).t(), (eye - a).t()).t()\n```"
}