19 Singular Value Decomposition

Matrix Manipulation

SVD is a factorization method that decomposes a rectangular matrix X into the product of three matrices:

Description of the illustration x_eq_usv.png


The U matrix consists of a set of 'left' orthonormal bases
The S matrix is a diagonal matrix
The V matrix consists of set of 'right' orthonormal bases

The values in S are called singular values. They are non-negative, and their magnitudes indicate the importance of the corresponding bases (components). The singular values reflect the amount of data variance captured by the bases. The first basis (the one with largest singular value) lies in the direction of the greatest data variance. The second basis captures the orthogonal direction with the second greatest variance, and so on.

SVD essentially performs a coordinate rotation that aligns the transformed axes with the directions of maximum variance in the data. This is a useful procedure under the assumption that the observed data has a high signal-to-noise ratio and that a large variance corresponds to interesting data content while a lower variance corresponds to noise.

SVD makes the assumption that the underlying data is Gaussian distributed and can be well described in terms of means and covariances.

Low Rank Decomposition

To reduce dimensionality, SVD keeps lower-order bases (the ones with the largest singular values) and ignores higher-order bases (the ones with the smallest singular values). The rationale behind this strategy is that the low-order bases retain the characteristics of the data that contribute most to its variance and are likely to capture the most important aspects of the data.

Given a data set X (nxm), where n is the number of rows and m is the number of attributes, a low-rank SVD will use only k components (k ≤ min(m, n)). In typical implementations of SVD, the value of k requires a visual inspection of the ranked singular values associated with the individual components. In Oracle Data Mining, SVD automatically estimates the cutoff point, which corresponds to a significant drop in the explained variance.

SVD produces two sets of orthonormal bases (U and V). Either of these bases could be used as a new coordinate system. In Oracle Data Mining SVD, V is the new coordinate system, and U represents the projection of X in this coordinate system. The algorithm computes the projection of new data as follows:

Description of the illustration x_eq_xvs.png

where X (nxk) is the projected data in the reduced data space, defined by the first k components, and V_k and S_k define the reduced component set.

Scalability

In Oracle Data Mining, SVD can process data sets with millions of rows and thousands of attributes. Oracle Data Mining automatically recommends an appropriate number of features, based on the data, for dimensionality reduction.

SVD has linear scalability with the number of rows and cubic scalability with the number of attributes when a full decomposition is computed. A low-rank decomposition is typically linear with the number of rows and linear with the number of columns. The scalability with the reduced rank depends on how the rank compares to the number of rows and columns. It can be linear when the rank is significantly smaller or cubic when it is on the same scale.

Model Size

The U matrix in SVD has as many rows as the number of rows in the build data. To avoid creating a large model, the U matrix is persisted only when an algorithm-specific setting is enabled. By default the U matrix is not persisted.

Performance

SVD can use approximate computations to improve performance. Approximation may be appropriate for data sets with many columns. An approximate low-rank decomposition provides good solutions at a reasonable computational cost. The quality of the approximation is dependent on the characteristics of the data. For data sets with more than 2500 attributes (the maximum number of features allowed) only approximate decomposition is possible.

PCA scoring

SVD models can be configured to perform PCA projections. PCA is closely related to SVD. PCA computes a set of orthonormal bases (principal components) that are ranked by their corresponding explained variance. The main difference between SVD and PCA is that the PCA projection is not scaled by the singular values. The PCA projection to the new coordinate system is given by:

Description of the illustration x_eq_vx.png

where (nxk) is the projected data in the reduced data space, defined by the first k components, and V_k defines the reduced component set.