Antigravity Agent
161 lines
4.8 KiB
Markdown
161 lines
4.8 KiB
Markdown
---
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id: wiki-2026-0508-linear-algebra
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title: Linear Algebra
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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: [Linear Algebra for ML, Vectors and Matrices, SVD, Eigendecomposition]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [math, linear-algebra, numpy, svd, eigen, pca, ml-foundations]
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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: { language: Python, framework: numpy/torch }
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---
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# Linear Algebra
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## 매 한 줄
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> **"매 ML은 행렬 곱셈이다"**. Vector·Matrix·Tensor 위에서 projection, rotation, decomposition으로 모든 게 표현된다.
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## 매 핵심
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### 매 객체
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- Scalar, Vector ∈ ℝⁿ, Matrix ∈ ℝᵐˣⁿ, Tensor (high-rank).
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- Norms: ‖x‖₁ (sparsity), ‖x‖₂ (energy), ‖x‖∞.
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- Inner product, outer product, cosine similarity.
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### 매 연산 핵심
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- Matmul C=AB. shape (m,k)·(k,n)=(m,n).
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- Transpose, inverse, pseudoinverse (Moore-Penrose).
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- Determinant (scaling), trace (diagonal sum, eigen sum).
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- Rank: 독립 column 수. low-rank → 압축 가능.
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### 매 분해
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- **Eigendecomposition** A = QΛQ⁻¹ (square, diagonalizable). PCA covariance.
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- **SVD** A = UΣVᵀ (any matrix). 가장 일반적.
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- **QR** Gram-Schmidt. least squares 안정.
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- **Cholesky** A = LLᵀ (symm. PD). 빠른 solve, GP, Kalman.
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- **LU** general solve.
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### 매 ML 응용
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1. **PCA**: covariance eigen / data SVD → top-k.
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2. **Linear regression**: x̂ = (XᵀX)⁻¹Xᵀy 또는 SVD pseudoinverse.
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3. **Recommendation MF**: A ≈ UVᵀ.
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4. **Word embeddings**: LSA SVD, word2vec implicit MF.
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5. **Attention**: softmax(QKᵀ/√d)V — 전부 matmul.
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## 💻 패턴
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### NumPy 핵심
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```python
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import numpy as np
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A = np.random.randn(4, 3); x = np.random.randn(3)
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y = A @ x # matmul
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G = A.T @ A # 3x3 Gram
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inv = np.linalg.inv(G)
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sol = np.linalg.solve(G, A.T @ y) # 안정적인 normal eq
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```
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### SVD & truncated rank-k
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```python
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U, S, Vt = np.linalg.svd(A, full_matrices=False)
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k = 2
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A_k = U[:, :k] @ np.diag(S[:k]) @ Vt[:k]
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```
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### Eigen / PCA
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```python
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X = np.random.randn(1000, 10)
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Xc = X - X.mean(0)
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cov = Xc.T @ Xc / (len(X) - 1)
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vals, vecs = np.linalg.eigh(cov) # symmetric → eigh
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order = np.argsort(-vals)
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PCs = vecs[:, order[:3]]
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Z = Xc @ PCs # (N, 3)
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```
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### Least squares 4가지
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```python
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# 1) normal eq (불안정)
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b1 = np.linalg.inv(A.T @ A) @ A.T @ y
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# 2) solve (better)
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b2 = np.linalg.solve(A.T @ A, A.T @ y)
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# 3) lstsq (SVD-based, 가장 안정)
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b3, *_ = np.linalg.lstsq(A, y, rcond=None)
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# 4) pseudoinverse
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b4 = np.linalg.pinv(A) @ y
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```
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### einsum (general tensor)
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```python
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# batch matmul (B,M,K)·(B,K,N) → (B,M,N)
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C = np.einsum("bmk,bkn->bmn", X, Y)
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# attention scores
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scores = np.einsum("bqd,bkd->bqk", Q, K) / np.sqrt(Q.shape[-1])
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```
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### Norms / cosine
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```python
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def cosine(a, b):
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return (a @ b) / (np.linalg.norm(a) * np.linalg.norm(b) + 1e-12)
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```
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### PyTorch (autograd)
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```python
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import torch
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A = torch.randn(4, 3, requires_grad=True)
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loss = (A @ x - y).pow(2).sum()
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loss.backward() # dL/dA
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U, S, Vt = torch.linalg.svd(A, full_matrices=False)
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```
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### Cholesky (GP / Kalman)
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```python
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L = np.linalg.cholesky(K + 1e-6 * np.eye(n)) # K SPD + jitter
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alpha = np.linalg.solve(L.T, np.linalg.solve(L, y))
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```
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## 매 결정 기준
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| 작업 | 함수 |
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|---|---|
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| 일반 solve | `np.linalg.solve` |
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| Least squares | `np.linalg.lstsq` (SVD) |
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| Symm. eigen | `eigh` |
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| 일반 eigen | `eig` |
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| 일반 분해 | `svd` |
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| SPD solve 빠르게 | Cholesky |
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| Sparse 큰 행렬 | `scipy.sparse.linalg` (eigsh, svds) |
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| GPU | torch.linalg / cupy |
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**기본값**: 정확도/안정성은 SVD/Cholesky, 속도는 solve, 빠른 prototype은 lstsq.
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## 🔗 Graph
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- 부모: [[Mathematics]], [[Numerical-Methods]]
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- 변형: [[SVD]], [[Eigendecomposition]], [[QR-Decomposition]], [[Cholesky]]
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- 응용: [[PCA]], [[Linear-Regression]], [[Latent-Semantic-Analysis-LSA]], [[Attention]]
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- Adjacent: [[Tensor]], [[Numpy]], [[Optimization]], [[Calculus]]
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## 🤖 LLM 활용
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**언제**: 식 유도, einsum 변환, 함수 선택, shape debug.
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**언제 X**: numerical conditioning / iterative solver tuning은 전문가.
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## ❌ 안티패턴
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- `inv(A) @ b` 대신 `solve(A, b)` 안 씀
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- Symm 행렬에 일반 `eig` (느림+정확도)
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- Large dense에 raw SVD (메모리) → randomized/truncated
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- Loop matmul (vectorize)
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- 차원/축 mismatch — `einsum`으로 명시
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- Float32 누적 오차 (PCA covariance) → float64 또는 standardize
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## 🧪 검증 / 중복
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- Verified (Strang, Trefethen NLA, NumPy/PyTorch docs). 신뢰도 A.
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- 중복: 없음.
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## 🕓 Changelog
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| 날짜 | 변경 |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — einsum, lstsq, Cholesky 패턴 추가 |
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