Truncated SVD

Truncated SVD computes only the largest singular components of a matrix. It is useful for compression, latent semantic indexing, and baseline recommender representations, but it inherits the input-matrix assumptions of classical SVD.

The rank-k approximation

The rank-$k$ approximation is

$$A\approx U_k{\Sigma }_k{V}_k^{\top },$$

where $k\ll \min (m,n)$. In information retrieval, this can compress a TF-IDF matrix. In recommenders, it can compress a deliberately prepared item or interaction matrix, but sparse utility matrices still require careful missing-value semantics.

Worked example

The matrix below can be read as four users or documents over four item/term features:

Row	Feature pattern
0	Features 0 and 1 active.
1	Features 0 and 2 active.
2	Features 2 and 3 active.
3	Features 1 and 3 active.

This snippet applies TruncatedSVD to sparse user-item rows and reports explained variance plus the first row embedding.

import numpy as np
from sklearn.decomposition import TruncatedSVD
X = np.array([[1, 1, 0, 0], [1, 0, 1, 0],
              [0, 0, 1, 1], [0, 1, 0, 1]], dtype=float)
svd = TruncatedSVD(n_components=2, random_state=3).fit(X)
Z = svd.transform(X)
print("explained_variance_ratio", np.round(svd.explained_variance_ratio_, 3).tolist())
print("row0_embedding", np.round(Z[0], 3).tolist())

Observed output:

explained_variance_ratio [0.0, 0.5]
row0_embedding [1.0, 0.707]

The two-dimensional embedding is a compact representation of row co-occurrence. The first component captures the shared overall activity scale, while the second separates rows by which feature pair they contain. A recommender would still need ranking, filters, and evaluation around this representation.

Caveats

The choice of $k$ controls underfitting versus noise retention. Randomized truncated SVD is approximate, so set random_state when results must be reproducible. Directly applying it to zero-filled feedback is the same missing-data problem described in SVD versus matrix factorization.

References

• scikit-learn documentation: TruncatedSVD

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