SVD decomposition of a data matrix $A$ of order $n \times d$ and rank $r$ can be expressed as follows

$$A_{n\times d} = U_{n\times r}D_{r \times r}V^{T}_{r \times d}$$

The rows of the data matrix $A$ are the data points in $d$ dimensional space. Thus, there are $n$ points in $d$ dimensional space.

The matrix $V$ contains $r$ right singular vectors as columns. Right singular vectors are orthonormal and forms a $r-$dimensional subspace that best fits the given $n$ data points.

The matrix $D$ is a diagonal matrix that contains singular values. Singular values signify the least squares (loss) of n-data points on the subspace $r$ right singular vectors.

The matrix $U$ contains left singular vectors as columns. Left singular vectors are also orthonormal.

But what does the $r$ left singular vectors signify?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.