2nd/10_Wiki/Topics/AI_and_ML/Matrix-Factorization.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-matrix-factorization	Matrix Factorization	10_Wiki/Topics	verified	self	MF Matrix Decomposition Latent Factor Models Collaborative Filtering MF	none	A	0.93	applied	ml recommender svd nmf als collaborative-filtering dimensionality-reduction		2026-05-10	pending	language framework python scipy|surprise|implicit|sklearn

Matrix Factorization

한 줄: 큰 행렬 R ≈ U·Vᵀ로 분해해 잠재 요인(latent factor)을 학습 — 추천 시스템·차원 축소·이미지 압축의 핵심.

핵심

• 목표: R (m×n, 희소) → U (m×k) · Vᵀ (k×n), k ≪ min(m,n).
• 변형: SVD (full/truncated), NMF (≥0 제약, 해석성), ALS (희소 explicit/implicit 피드백), FunkSVD (RMSE optimize), BPR (ranking loss).
• 손실: explicit → MSE on observed; implicit → weighted MSE w/ confidence (Hu-Koren-Volinsky).
• 정규화: L2 (λ‖U‖² + λ‖V‖²) — 과적합 방지. λ 튜닝 필수.
• Cold-start: side feature 결합 (FM, hybrid), content-based fallback.

결정 기준

데이터	알고리즘	라이브러리
작은 dense matrix, 차원 축소	Truncated SVD	sklearn.decomposition.TruncatedSVD
토픽 모델 / 해석성	NMF	sklearn.decomposition.NMF
Explicit ratings (1-5)	SVD/SVD++	surprise
Implicit feedback (click/play)	ALS / BPR	implicit
사이드 피처 포함	Factorization Machines	xlearn, pyFM
대규모 분산	Spark ALS	pyspark.ml.recommendation
딥러닝 추천	NCF / Two-Tower	TF Recommenders, PyTorch

💻 패턴

Truncated SVD

from sklearn.decomposition import TruncatedSVD
svd = TruncatedSVD(n_components=50, random_state=0)
U = svd.fit_transform(X)        # (n_samples, 50)
V = svd.components_             # (50, n_features)
explained = svd.explained_variance_ratio_.sum()

NMF (해석 가능 토픽)

from sklearn.decomposition import NMF
nmf = NMF(n_components=10, init="nndsvda", l1_ratio=0.5, max_iter=500)
W = nmf.fit_transform(X_tfidf)  # 문서×토픽
H = nmf.components_             # 토픽×단어

ALS implicit (BM25 weighting)

import implicit, scipy.sparse as sp
from implicit.nearest_neighbours import bm25_weight
user_item = sp.csr_matrix((vals, (uids, iids)))
weighted = bm25_weight(user_item, K1=100, B=0.8).tocsr()
model = implicit.als.AlternatingLeastSquares(factors=64, regularization=0.05, iterations=20)
model.fit(weighted)
ids, scores = model.recommend(user_id=42, user_items=user_item[42], N=10)

Surprise (explicit ratings)

from surprise import SVD, Dataset, Reader, accuracy
from surprise.model_selection import train_test_split, GridSearchCV
data = Dataset.load_from_df(df[["user","item","rating"]], Reader(rating_scale=(1,5)))
gs = GridSearchCV(SVD, {"n_factors":[50,100], "lr_all":[0.005], "reg_all":[0.02,0.1]},
                  measures=["rmse"], cv=3)
gs.fit(data); print(gs.best_score["rmse"], gs.best_params["rmse"])

From-scratch SGD (FunkSVD)

def funk_svd(R_obs, k=20, lr=0.005, reg=0.02, epochs=20):
    m, n = R_obs.shape
    U = np.random.normal(0, 0.1, (m, k)); V = np.random.normal(0, 0.1, (n, k))
    for _ in range(epochs):
        for i, j, r in R_obs:  # list of (user, item, rating)
            err = r - U[i] @ V[j]
            U[i] += lr * (err * V[j] - reg * U[i])
            V[j] += lr * (err * U[i] - reg * V[j])
    return U, V

Cold-start: Hybrid with content

# 신규 아이템: content embedding으로 V_new 초기화
from sentence_transformers import SentenceTransformer
emb = SentenceTransformer("all-MiniLM-L6-v2").encode(item_descriptions)
V_new = projection_layer(emb)   # learned mapping → factor space

🔗 Graph

• 상위: Recommender-Systems · Linear-Algebra-Foundations · Dimensionality-Reduction
• 관련: SVD · ALS · Collaborative-Filtering · Two-Tower

🤖 LLM 활용

• LLM 임베딩 → MF의 V를 부분 초기화 → cold-start 완화.
• 잠재 factor 해석: top-N 아이템을 LLM에 던져 "이 그룹 공통 테마는?" — 차원 라벨링 자동화.

❌ 안티패턴

• Implicit feedback에 MSE 그대로 — 0(미상호작용)을 negative로 학습 → confidence weighting 필요.
• k 무작정 크게 — 과적합·메모리 폭발. validation으로 elbow 찾기.
• 정규화 0 — train RMSE만 떨어지고 test 폭발.
• 유저/아이템 ID 인코딩 누락 — sparse matrix 인덱스 mismatch 흔함.
• Cold-start에 MF 단독 — 신규 유저/아이템은 hybrid/content 필수.
• 시간 무시 — 추천에선 최근성 큼. session-aware 모델 또는 time-decay weighting.

🧪 검증 / 중복

• 중복: SVD, NMF, ALS 별도 — 본 문서는 우산.
• 검증: held-out RMSE/MAE (explicit), NDCG@k·Recall@k·MAP (implicit). bootstrap CI 권장.

🕓 Changelog

• 2026-05-08 | Phase 1 — 자동 시드.
• 2026-05-10 | Manual cleanup — SVD/NMF/ALS/Surprise/FunkSVD 코드, cold-start, 안티패턴 정리.