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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?

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