Get U, Sigma, V* matrix from Truncated SVD in scikit-learn

numpy svd
singular value decomposition
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singular value decomposition vs eigenvalue decomposition
matrix factorization svd

I am using truncated SVD from scikit-learn package.

In the definition of SVD, an original matrix A is approxmated as a product AUΣV* where U and V have orthonormal columns, and Σ is non-negative diagonal.

I need to get the U, Σ and V* matrices.

Looking at the source code here I found out that V* is stored in self.components_ field after calling fit_transform.

Is it possible to get U and Σ matrices?

My code:

import sklearn.decomposition as skd
import numpy as np

matrix = np.random.random((20,20))
trsvd = skd.TruncatedSVD(n_components=15)
transformed = trsvd.fit_transform(matrix)
VT = trsvd.components_

Looking into the source via the link you provided, TruncatedSVD is basically a wrapper around sklearn.utils.extmath.randomized_svd; you can manually call this yourself like this:

from sklearn.utils.extmath import randomized_svd

U, Sigma, VT = randomized_svd(X, 
                              n_components=15,
                              n_iter=5,
                              random_state=None)

sklearn.decomposition.TruncatedSVD, In particular, truncated SVD works on term count/tf-idf matrices as returned by the default in randomized_svd to handle sparse matrices that may have large  Dimensionality reduction using truncated SVD (aka LSA). This transformer performs linear dimensionality reduction by means of truncated singular value decomposition (SVD). Contrary to PCA, this estimator does not center the data before computing the singular value decomposition. This means it can work with sparse matrices efficiently.

One can use scipy.sparse.svds (for dense matrices you can use svd).

import numpy as np
from scipy.sparse.linalg import svds

matrix = np.random.random((20, 20))
num_components = 2
u, s, v = svds(matrix, k=num_components)
X = u.dot(np.diag(s))  # output of TruncatedSVD

If you're working with really big sparse matrices (perhaps your working with natural text), even scipy.sparse.svds might blow up your computer's RAM. In such cases, consider the sparsesvd package which uses SVDLIBC, and what gensim uses under-the-hood.

import numpy as np
from sparsesvd import sparsesvd


X = np.random.random((30, 30))
ut, s, vt = sparsesvd(X.tocsc(), k)
projected = (X * ut.T)/s

Using truncated SVD to reduce dimensionality, is done on the data matrix, whereas for PCA, the factorization is done on the covariance matrix. Obtenir U, Sigma, V* matrice de Tronqués SVD dans scikit-learn j'utilise SVD tronqué de scikit-learn paquet. dans la définition de SVD, une matrice originale est approxmated comme un produit ≈ UΣV* où U et V ont des colonnes orthonormales, et Σ est diagonale non négative.

Just as a note:

svd.transform(X)

and

svd.fit_transform(X)

generate U * Sigma.

svd.singular_values_

generates Sigma in vector form.

svd.components_

generates VT. Maybe we can use

svd.transform(X).dot(np.linalg.inv(np.diag(svd.singular_values_)))

to get U because U * Sigma * Sigma ^ -1 = U * I = U.

Singular value decomposition, How do you find the SVD of a matrix in python? Recall that SVD (X) decomposes X into three matrices, U, Sigma, and V^t. In scikit-learn, TruncatedSVD treats.fit ().transform () differently from.fit_transform (). On the one hand,.fit (X).transform (X) will return X @ V. On the other hand,.fit_transform (X) will return U * Sigma.

I know this is an older question but the correct version is-

U = svd.fit_transform(X)
Sigma = svd.singular_values_
VT = svd.components_

However, one thing to keep in mind is that U and VT are truncated hence without the rest of the values it not possible to recreate X.

Proof of the Singular Value Decomposition, Singular-Value Decomposition; Calculate Singular-Value Decomposition SVD can also be used in least squares linear regression, image The function takes a matrix and returns the U, Sigma and V^T elements. The scikit-learn provides a TruncatedSVD class that implements this capability directly. 2.5.2. Truncated singular value decomposition and latent semantic analysis¶ TruncatedSVD implements a variant of singular value decomposition (SVD) that only computes the \(k\) largest singular values, where \(k\) is a user-specified parameter.

From the source code, we can see X_transformed which is U * Sigma (Here Sigma is a vector) is returned from the fit_transform method. So we can get

svd = TruncatedSVD(k)
X_transformed = svd.fit_transform(X)

U = X_transformed / svd.singular_values_
Sigma_matrix = np.diag(svd.singular_values_)
VT = svd.components_

Remark

Truncated SVD is an approximation. X ≈ X' = UΣV*. We have X'V = UΣ. But what about XV? An interesting fact is XV = X'V. This can be proved by comparing the full SVD form of X and the truncated SVD form of X'. Note XV is just transform(X), so we can also get U by

U = svd.transform(X) / svd.singular_values_

How to Calculate the SVD from Scratch with Python, i using truncated svd scikit-learn package. in definition of svd, original matrix a approxmated product a ≈ uΣv* u , v have orthonormal columns,  Matrix decomposition, also known as matrix factorization, involves describing a given matrix using its constituent elements. Perhaps the most known and widely used matrix decomposition method is the Singular-Value Decomposition, or SVD. All matrices have an SVD, which makes it more stable than other methods, such as the eigendecomposition. As such, it is often used […]

python, Singular Value Decomposition, or SVD, has a wide array of In order to determine the associated U matrix, we must first find the matrix A. After solving, we obtain the expression for V. from sklearn.decomposition import TruncatedSVD Fortunately, the scikit-learn library provides a wrapper function for  Using Singular Value Decomposition (SVD) for PCA¶ SVD is a decomposition of the data matrix \(X = U S V^T\) where \(U\) and \(V\) are orthogonal matrices and \(S\) is a diagnonal matrix. Compared to the rst two techniques, this is a superior technique in reducing the image RMSE. 04 Thrusty Tahr machine for use also with Python.

Singular Value Decomposition Example In Python, Recall that SVD(X) decomposes X into three matrices, U, Sigma, and V^t. In scikit​-learn, TruncatedSVD treats .fit().transform() differently from . The singular value decomposition can be used for computing the pseudoinverse of a matrix. (Various authors use different notation for the pseudoinverse; here we use †.) Indeed, the pseudoinverse of the matrix M with singular value decomposition M = U Σ V * is M † = V Σ † U *

TruncatedSVD.fit(X).transform(X) is not the same as .fit_transform(X , Truncated Singular Value Decomposition (SVD) is a matrix factorization technique that factors a matrix M into the three matrices U, Σ, and V. Typically, SVD is used under the hood to find the principle components of a matrix. It has the scikit-learn API, so you can put it in a sklearn.Pipeline object and call transform on a new matrix instead of having to figure out the matrix multiplications yourself. It offers two algorithms: either a fast randomized SVD solver (the default), or scipy.sparse.svds. (Full disclosure: I wrote TruncatedSVD.)

Comments
  • This is true but for the regular numpy.linalg.svd method you can't pass the number of components as a parameter so you have to extract the top K yourself. Minor inconvenience.
  • U is definitely not svd.fit_transform(X) . This is wrong.
  • I believe this answer is not correct: SVD.fit_transform(X) = U*np.diag(Sigma) != U and SVD.explained_variance_ratio_ = np.var(X_transformed, axis=0) / np.var(X, axis=0).sum() != Sigma
  • This answer is not correct, as mentioned by rth as well.