SVD (SVD와 Latent Factor 모형)

import numpy as np
from numpy import linalg as la
np.random.seed(42)


def flip_signs(A, B):
    """
    utility function for resolving the sign ambiguity in SVD
    http://stats.stackexchange.com/q/34396/115202
    """
    signs = np.sign(A) * np.sign(B)
    return A, B * signs


# Let the data matrix X be of n x p size,
# where n is the number of samples and p is the number of variables
n, p = 5, 3
X = np.random.rand(n, p)
# Let us assume that it is centered
# 평균이 0으로 맞춘 후에 (centering 작업 수행) 공분산 행렬을 계산함
X -= np.mean(X, axis=0)

# the p x p covariance matrix
C = np.cov(X, rowvar=False)
print ("covariance matrix = \n", C)
# C is a symmetric matrix and so it can be diagonalized:
l, principal_axes = la.eig(C)
# sort results wrt. eigenvalues
idx = l.argsort()[::-1]
l, principal_axes = l[idx], principal_axes[:, idx]
# the eigenvalues in decreasing order
print ("eigenvalues = \n", l)
# a matrix of eigenvectors (each column is an eigenvector)
print ("eigenvectors = \n", principal_axes)
# projections of X on the principal axes are called principal components
principal_components = X.dot(principal_axes)
print ("principal_components = \n", principal_components)

# we now perform singular value decomposition of X
# "economy size" (or "thin") SVD
U, s, Vt = la.svd(X, full_matrices=False)
V = Vt.T
S = np.diag(s)

# 1) then columns of V are principal directions/axes.
print ("V = \n", V)
assert np.allclose(*flip_signs(V, principal_axes))

# 2) columns of US are principal components
print ("US = \n", U.dot(S))
assert np.allclose(*flip_signs(U.dot(S), principal_components))

# 3) singular values are related to the eigenvalues of covariance matrix
print ((s ** 2) / (n - 1))
assert np.allclose((s ** 2) / (n - 1), l)

# 8) dimensionality reduction
k = 2
PC_k = principal_components[:, 0:k]
US_k = U[:, 0:k].dot(S[0:k, 0:k])
print (US_k)
assert np.allclose(*flip_signs(PC_k, US_k))

print (U)
print (S)
print (V)
# 10) we used "economy size" (or "thin") SVD
assert U.shape == (n, p)
assert S.shape == (p, p)
assert V.shape == (p, p)

import numpy as np
from pprint import pprint
import math
M = np.array([[math.sqrt(3), 2], [0, math.sqrt(3)]])
U, s, Vt = numpy.linalg.svd(M, full_matrices=True)

M

U

Vt

np.diag(s)

U.dot(np.diag(s)).dot(Vt)

import numpy as np
from pprint import pprint
M = np.array([[1,0,0,0,0],[0,0,2,0,3],[0,0,0,0,0],[0,2,0,0,0]])

U, S0, V0 = np.linalg.svd(M, full_matrices=True)

M

U

S0

S = np.hstack([np.diag(S0), np.zeros(M.shape[0])[:, np.newaxis]])

S

V = V0.T

V

print("\nU.dot(S).dot(V.T):"); pprint(U.dot(S).dot(V0))

S

from sklearn.decomposition import TruncatedSVD
from sklearn.random_projection import sparse_random_matrix
X = sparse_random_matrix(5, 5)
svd = TruncatedSVD(n_components=2, n_iter=7)  
U = svd.fit_transform(X)
Sigma = svd.explained_variance_ratio_
VT = svd.components_
svd.fit(X)

U

Sigma

VT

import numpy as np
import surprise  # run 'pip install scikit-surprise' to install surprise

class SVD_SGD(surprise.AlgoBase):
    
    def __init__(self, learning_rate, n_epochs, n_factors):
        
        self.lr = learning_rate  # learning rate for SGD
        self.n_epochs = n_epochs  # number of iterations of SGD
        self.n_factors = n_factors  # number of factors (몇 개의 Latent Factor로 요인 벡터를 만들지를 정함)
        
    def train(self, trainset):
        '''Learn the vectors p_u and q_i with SGD'''
        
        print('Fitting data with SGD...')
        
        # Randomly initialize the user and item factors.
        p = np.random.normal(0, .1, (trainset.n_users, self.n_factors))
        q = np.random.normal(0, .1, (trainset.n_items, self.n_factors))
        
        # SGD procedure
        for _ in range(self.n_epochs):
            for u, i, r_ui in trainset.all_ratings():
                err = r_ui - np.dot(p[u], q[i])
                # Update vectors p_u and q_i
                p[u] += self.lr * err * q[i]
                q[i] += self.lr * err * p[u]
                # Note: in the update of q_i, we should actually use the previous (non-updated) value of p_u.
                # In practice it makes almost no difference.
        
        self.p, self.q = p, q
        self.trainset = trainset

    def estimate(self, u, i):
        '''Return the estmimated rating of user u for item i.'''
        
        # return scalar product between p_u and q_i if user and item are known,
        # else return the average of all ratings
        if self.trainset.knows_user(u) and self.trainset.knows_item(i):
            return np.dot(self.p[u], self.q[i])
        else:
            return self.trainset.global_mean

# data loading. We'll use the movielens dataset (https://grouplens.org/datasets/movielens/100k/)
# it will be downloaded automatically.
data = surprise.Dataset.load_builtin('ml-100k')
data.split(2)  # split data for 2-folds cross validation

algo = SVD_SGD(learning_rate=.01, n_epochs=10, n_factors=10)
surprise.evaluate(algo, data, measures=['RMSE'])