How do you interpret the singular vectors and values in SVD?

1 What is SVD?

SVD is a method of factorizing a matrix A into three matrices U, S, and V, such that A = USV^T, where U and V are orthogonal matrices and S is a diagonal matrix. The columns of U are called the left singular vectors of A, the columns of V are called the right singular vectors of A, and the diagonal elements of S are called the singular values of A. The singular values are always non-negative and arranged in descending order.

2 How to compute SVD?

There are several algorithms to compute SVD, but one of the most common ones is based on the eigenvalue decomposition of A^TA or AA^T. For example, if you want to find the right singular vectors and values of A, you can compute the eigenvalues and eigenvectors of A^TA, and then normalize the eigenvectors to get the right singular vectors. The singular values are the square roots of the eigenvalues. Similarly, you can find the left singular vectors and values by computing the eigenvalues and eigenvectors of AA^T.

3 How to interpret SVD?

One way to interpret SVD is to think of it as a transformation of the original matrix A into a new coordinate system defined by the singular vectors. The singular values represent the scaling factors along each dimension of the new coordinate system. The larger the singular value, the more important the corresponding dimension is for capturing the variability of the data. The rank of A is equal to the number of non-zero singular values, and the null space of A is spanned by the right singular vectors corresponding to the zero singular values.

4 How to use SVD for dimensionality reduction?

One of the applications of SVD is to reduce the dimensionality of a data matrix by keeping only the most significant singular vectors and values. This can help you simplify the data, remove noise, and improve computational efficiency. To do this, you can truncate the SVD by setting some of the smallest singular values to zero, and then reconstruct a low-rank approximation of A by multiplying the truncated U, S, and V matrices. The error of this approximation is proportional to the sum of the squared singular values that are discarded.

5 How to use SVD for data compression?

Another application of SVD is to compress a data matrix by encoding it with fewer bits than the original matrix. This can help you save storage space and transmission bandwidth. To do this, you can quantize the singular values and vectors by rounding them to a fixed number of bits, and then store or transmit only the quantized values. The quality of the compression depends on the trade-off between the number of bits and the accuracy of the approximation.

6 How to use SVD for noise filtering?

A third application of SVD is to filter out noise from a data matrix by enhancing the signal and suppressing the noise. This can help you improve the clarity and reliability of the data. To do this, you can apply a threshold to the singular values and set those that are below the threshold to zero, and then reconstruct a denoised approximation of A by multiplying the thresholded U, S, and V matrices. The threshold can be chosen based on some criteria such as the signal-to-noise ratio or the percentage of variance explained.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?