10.8-dars: SVD — singulyar yoyilma

A (m × n) = U (m × m) @ Sigma (m × n) @ Vt (n × n)

U   — ortogonal (U.T @ U = I): ustunlari — "chap singulyar vektorlar" (qatorlar fazosi uchun)
s   — singulyar qiymatlar: s1 >= s2 >= ... >= 0 (kamayish tartibida, doim manfiy emas)
Vt  — ortogonal: qatorlari — "o'ng singulyar vektorlar" (ustunlar/belgilar fazosi uchun)

"Iqtisodiy" SVD (full_matrices=False): U (m × r), s (r), Vt (r × n),  r = min(m, n)
U, s, Vt = np.linalg.svd(A, full_matrices=False)
A == U @ np.diag(s) @ Vt

A @ x = U @ (s × (Vt @ x))

1. Vt @ x    — x ni burish (yangi o'qlarga o'tkazish)
2. s × ...   — har o'q bo'yicha s_i marta cho'zish (o'lcham o'zgarishi ham mumkin)
3. U @ ...   — natijani chiqish fazosida burish

Birlik aylana → A → ellips: yarim o'qlari uzunligi = s1, s2; yo'nalishlari = U ustunlari

A_k = U[:, :k] @ diag(s[:k]) @ Vt[:k, :]       — faqat eng katta k ta singulyar qiymat

Eckart-Young teoremasi: rank-k matritsalar orasida A ga ENG YAQINI — A_k
  xato ||A - A_k||_F = sqrt(s_{k+1}^2 + ... + s_r^2)
  "saqlangan energiya" = sum(s[:k]^2) / sum(s^2)

Saqlash: m × n  →  k × (m + n + 1) son
  1000 × 1000 rasm, k = 50:  1 000 000 → 100 050 (≈ 10%)

X = U S Vt   →   X.T @ X = V S^2 Vt      (belgilar, p × p)
                 X @ X.T = U S^2 U.T      (kuzatuvlar, n × n)

X.T X ning xos vektorlari = V (o'ng singulyar vektorlar)
X.T X ning xos qiymatlari = s^2
Markazlashtirilgan X uchun: kovariatsiya xos qiymatlari = s^2 / (n - 1)   → PCA (10.9)

import numpy as np

U, s, Vt = np.linalg.svd(A, full_matrices=False)     # iqtisodiy SVD
np.allclose(A, U @ np.diag(s) @ Vt)                  # tiklash

k = 10
A_k = U[:, :k] @ np.diag(s[:k]) @ Vt[:k]             # past rankli yaqinlashtirish
energiya = (s[:k] ** 2).sum() / (s ** 2).sum()

np.sum(s > 1e-10)                                    # rank
s[0] / s[-1]                                         # shart soni
Vt[0]                                                # 1-o'ng singulyar vektor (QATOR)

# katta/siyrak — faqat k ta
from sklearn.decomposition import TruncatedSVD
TruncatedSVD(n_components=k, random_state=0).fit_transform(X)
QOIDA: full_matrices=False · Vt qatorlari · s kamayish tartibida · PCA — avval markazlashtir

A = U diag(s) Vt — har qanday matritsa; U, V ortogonal, s >= 0 kamayuvchi
Geometriya: burish → cho'zish → burish; aylana → ellips
rank = nolmas s soni; cond = s1 / s_min
A_k — eng yaxshi rank-k yaqinlashtirish (Eckart-Young)
X.T X: xos vektorlar = V, xos qiymatlar = s^2
Siqish, shovqin, tavsiya, LSA, PCA, lstsq

"""SVD: U, s, Vt; ortogonallik, tiklash, rank, cond (real numpy)."""

import numpy as np


def main() -> None:
    rng = np.random.default_rng(0)
    A = rng.normal(size=(5, 3))

    print("=== 1. Shakllar (iqtisodiy SVD) ===")
    U, s, Vt = np.linalg.svd(A, full_matrices=False)
    print(f"  U {U.shape}, s {s.shape}, Vt {Vt.shape}")
    print(f"  s = {np.round(s, 3)}  (kamayish tartibida)")

    print("\n=== 2. Ortogonallik ===")
    print(f"  U.T @ U = I: {np.allclose(U.T @ U, np.eye(3))}")
    print(f"  Vt @ Vt.T = I: {np.allclose(Vt @ Vt.T, np.eye(3))}")

    print("\n=== 3. Tiklash ===")
    print(f"  A == U diag(s) Vt: {np.allclose(A, U @ np.diag(s) @ Vt)}")

    print("\n=== 4. Rank va shart soni ===")
    B = np.column_stack([A[:, 0], A[:, 1], A[:, 0] + A[:, 1]])     # bog'liq ustun
    sB = np.linalg.svd(B, compute_uv=False)
    print(f"  bog'liq matritsa s = {np.round(sB, 6)}")
    print(f"  rank = {np.sum(sB > 1e-10)}, matrix_rank = {np.linalg.matrix_rank(B)}")
    print(f"  cond(A) = s1/s3 = {s[0] / s[-1]:.3f}, np.linalg.cond = {np.linalg.cond(A):.3f}")

    print("\n=== 5. full_matrices=True ===")
    U_full = np.linalg.svd(A)[0]
    print(f"  U shakli: {U_full.shape}  (katta m da ulkan)")
    print("  ⭐ Har qanday matritsa = burish × cho'zish × burish")


if __name__ == "__main__":
    main()

=== 1. Shakllar (iqtisodiy SVD) ===
  U (5, 3), s (3,), Vt (3, 3)
  s = [3.151 1.709 0.335]  (kamayish tartibida)

=== 2. Ortogonallik ===
  U.T @ U = I: True
  Vt @ Vt.T = I: True

=== 3. Tiklash ===
  A == U diag(s) Vt: True

=== 4. Rank va shart soni ===
  bog'liq matritsa s = [4.965774 1.002119 0.      ]
  rank = 2, matrix_rank = 2
  cond(A) = s1/s3 = 9.410, np.linalg.cond = 9.410

=== 5. full_matrices=True ===
  U shakli: (5, 5)  (katta m da ulkan)
  ⭐ Har qanday matritsa = burish × cho'zish × burish

"""SVD geometriyasi: birlik aylana ellipsga, yarim o'qlar = singulyar qiymatlar (real numpy)."""

import numpy as np


def main() -> None:
    A = np.array([[3.0, 1.0], [1.0, 2.0]])
    U, s, Vt = np.linalg.svd(A)

    t = np.linspace(0, 2 * np.pi, 3601)
    aylana = np.vstack([np.cos(t), np.sin(t)])
    ellips = A @ aylana
    uzunlik = np.linalg.norm(ellips, axis=0)

    print("=== 1. Ellips yarim o'qlari ===")
    print(f"  eng uzun:  {uzunlik.max():.4f},  s1 = {s[0]:.4f}")
    print(f"  eng qisqa: {uzunlik.min():.4f},  s2 = {s[1]:.4f}")

    print("\n=== 2. Eng uzun yo'nalish = U ning 1-ustuni ===")
    i = uzunlik.argmax()
    yo = ellips[:, i] / uzunlik[i]
    print(f"  ellipsdan: {np.round(yo * np.sign(yo[0]), 4)}, U[:, 0]: {np.round(U[:, 0] * np.sign(U[0, 0]), 4)}")

    print("\n=== 3. Uch qadam ===")
    x = np.array([1.0, 0.0])
    q1 = Vt @ x
    q2 = s * q1
    q3 = U @ q2
    print(f"  Vt x = {np.round(q1, 3)} → s × ... = {np.round(q2, 3)} → U ... = {np.round(q3, 3)}")
    print(f"  A x = {A @ x}")
    print("  ⭐ Burish → cho'zish → burish")


if __name__ == "__main__":
    main()

=== 1. Ellips yarim o'qlari ===
  eng uzun:  3.6180,  s1 = 3.6180
  eng qisqa: 1.3820,  s2 = 1.3820

=== 2. Eng uzun yo'nalish = U ning 1-ustuni ===
  ellipsdan: [0.8507 0.5256], U[:, 0]: [0.8507 0.5257]

=== 3. Uch qadam ===
  Vt x = [-0.851 -0.526] → s × ... = [-3.078 -0.727] → U ... = [3. 1.]
  A x = [3. 1.]
  ⭐ Burish → cho'zish → burish

"""Sintetik 'rasm' ni SVD bilan siqish: k, energiya, xato, saqlash (real numpy)."""

import numpy as np


def main() -> None:
    # 120 × 160 sintetik rasm: gradient + doira + chiziqlar + shovqin
    y, x = np.mgrid[0:120, 0:160]
    rasm = 100 + 0.5 * x
    rasm += 80 * ((x - 80) ** 2 + (y - 60) ** 2 < 30 ** 2)
    rasm += 40 * (np.sin(x / 6) > 0.9)
    rasm += np.random.default_rng(0).normal(0, 5, rasm.shape)

    U, s, Vt = np.linalg.svd(rasm, full_matrices=False)

    print("=== 1. Singulyar qiymatlar tez so'nadi ===")
    print(f"  s[:6] = {np.round(s[:6], 0)}")
    print(f"  s[50] = {s[50]:.1f}, s[-1] = {s[-1]:.1f}")

    print("\n=== 2. k bo'yicha sifat va hajm ===")
    m, n = rasm.shape
    for k in [1, 5, 20, 50]:
        A_k = U[:, :k] @ np.diag(s[:k]) @ Vt[:k]
        xato = np.linalg.norm(rasm - A_k) / np.linalg.norm(rasm)
        energiya = (s[:k] ** 2).sum() / (s ** 2).sum()
        hajm = k * (m + n + 1) / (m * n)
        print(f"  k={k:>2}: energiya {energiya:.4f}, nisbiy xato {xato:.3f}, hajm {hajm:.0%}")

    print("\n=== 3. Eckart-Young: xato = qolgan s lar ===")
    k = 20
    A_k = U[:, :k] @ np.diag(s[:k]) @ Vt[:k]
    print(f"  ||A - A_k||_F = {np.linalg.norm(rasm - A_k):.2f}, sqrt(sum(s[k:]^2)) = {np.sqrt((s[k:] ** 2).sum()):.2f}")
    print("  ⭐ Kam qatlam — asosiy tuzilma; qolgani — tafsilot va shovqin")


if __name__ == "__main__":
    main()

=== 1. Singulyar qiymatlar tez so'nadi ===
  s[:6] = [22313.  2206.   954.   556.   388.   313.]
  s[50] = 66.2, s[-1] = 9.2

=== 2. k bo'yicha sifat va hajm ===
  k= 1: energiya 0.9861, nisbiy xato 0.118, hajm 1%
  k= 5: energiya 0.9984, nisbiy xato 0.040, hajm 7%
  k=20: energiya 0.9993, nisbiy xato 0.026, hajm 29%
  k=50: energiya 0.9998, nisbiy xato 0.015, hajm 73%

=== 3. Eckart-Young: xato = qolgan s lar ===
  ||A - A_k||_F = 573.29, sqrt(sum(s[k:]^2)) = 573.29
  ⭐ Kam qatlam — asosiy tuzilma; qolgani — tafsilot va shovqin

"""Tavsiya tizimi: past rankli SVD bilan bo'sh baholarni bashorat qilish (real numpy)."""

import numpy as np


def main() -> None:
    rng = np.random.default_rng(1)
    n_user, n_film, k_haqiqiy = 300, 80, 3
    P = rng.normal(size=(n_user, k_haqiqiy))           # foydalanuvchi moyilliklari
    Q = rng.normal(size=(n_film, k_haqiqiy))           # film profillari
    toliq = np.clip(3 + P @ Q.T * 0.6 + rng.normal(0, 0.3, (n_user, n_film)), 1, 5)

    maska = rng.random(toliq.shape) < 0.25              # 25% baho ma'lum
    test = maska & (rng.random(toliq.shape) < 0.2)      # ma'lumlarning bir qismi — test
    oquv = maska & ~test

    print("=== 1. Ma'lumot ===")
    print(f"  ma'lum baholar: {oquv.sum()} o'quv, {test.sum()} test ({oquv.mean():.0%} to'la)")

    ortacha = toliq[oquv].mean()
    R = np.where(oquv, toliq, ortacha)                  # bo'shlar — o'rtacha bilan

    def rmse(bashorat: np.ndarray) -> float:
        return float(np.sqrt(np.mean((bashorat[test] - toliq[test]) ** 2)))

    print("\n=== 2. Baza: umumiy o'rtacha ===")
    print(f"  RMSE = {rmse(np.full_like(R, ortacha)):.3f}")

    print("\n=== 3. Past rankli SVD (iterativ to'ldirish) ===")
    for k in [1, 3, 10, 30]:
        X = R.copy()
        for _ in range(30):
            U, s, Vt = np.linalg.svd(X, full_matrices=False)
            X_k = U[:, :k] @ np.diag(s[:k]) @ Vt[:k]
            X = np.where(oquv, toliq, X_k)             # ma'lumlar joyida, bo'shlar — bashorat
        print(f"  k={k:>2}: test RMSE = {rmse(np.clip(X_k, 1, 5)):.3f}")
    print("  ⭐ Juda katta k — overfitting (k ni validatsiya bilan tanlang)")


if __name__ == "__main__":
    main()

=== 1. Ma'lumot ===
  ma'lum baholar: 4729 o'quv, 1208 test (20% to'la)

=== 2. Baza: umumiy o'rtacha ===
  RMSE = 0.908

=== 3. Past rankli SVD (iterativ to'ldirish) ===
  k= 1: test RMSE = 0.942
  k= 3: test RMSE = 0.621
  k=10: test RMSE = 0.620
  k=30: test RMSE = 0.862
  ⭐ Juda katta k — overfitting (k ni validatsiya bilan tanlang)

v1 = Vt[:, 0]                                                     # ⚠️
v1 = Vt[0]                                                        # ✅

U, s, Vt = np.linalg.svd(X)          # X: 100000 × 50 → U 100000^2  # ⚠️
U, s, Vt = np.linalg.svd(X, full_matrices=False)                  # ✅

A_k = U[:, :k] @ s[:k] @ Vt[:k]      # shakl xatosi                 # ⚠️
A_k = U[:, :k] @ np.diag(s[:k]) @ Vt[:k]                          # ✅

np.linalg.svd(tfidf.toarray())                                    # ⚠️
TruncatedSVD(n_components=100, random_state=0).fit_transform(tfidf)  # ✅

R = np.where(maska, baholar, 0)      # 0 — "eng yomon baho"         # ⚠️
R = np.where(maska, baholar, baholar[maska].mean())               # ✅

k = argmin(oquv_xato)                # har doim eng katta k          # ⚠️
k = argmin(test_xato)                # yoki cross-validation         # ✅

U, s, Vt = np.linalg.svd(X)          # 1-komponenta ≈ o'rtacha       # ⚠️
U, s, Vt = np.linalg.svd(X - X.mean(0), full_matrices=False)      # ✅

1.  # SVD qaysi matritsalar uchun?
2.  # s tartibi?
3.  # s manfiy bo'la oladimi?
4.  # rank = ?
5.  # cond = ?
6.  # 1000 × 1000, k = 50 saqlash hajmi?
7.  # A_k qanday yaqinlashtirish?
8.  # X.T X xos qiymatlari?
9.  # X.T X xos vektorlari?
10. # aylana → ?
11. # Vt[0] nima?
12. # tavsiyada bo'sh katak = 0 to'g'rimi?

1.  U, s, Vt = np.linalg.svd(X)   # X: 50000 × 20

2.  v1 = Vt[:, 0]

3.  A_k = U[:, :k] * s[:k] @ Vt   # k qatlam

4.  R = np.nan_to_num(baholar)   # NaN → 0, keyin SVD

5.  U, s, Vt = np.linalg.svd(X, full_matrices=False)   # PCA uchun