2.3. Singular Value Decomposition - Geometric interpretation

We introduce the geometric interpretation of the svd by using a toy example.

1 Iris dataset

The iris dataset is a dataset on iris flowers.

• Three species (setosa, virginica and versicolor)
• Length and width of Sepal leafs
• Length and width of Petal Leafs

For didactical purposes we will use a subset of the data.

• Virginica Species
• 3 Variables: Sepal Length, Sepal Width, Petal Length
• This allows us to visualise the data in 3D plots
• Illustrate the data compression of the SVD from 3 to two dimensions.

1.1 Subset the data

library(tidyverse)
library(plotly)

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irisSub <- iris %>%
  filter(Species == "virginica") %>%
  dplyr::select("Sepal.Length","Sepal.Width","Petal.Length")

1.2 Center the data

X <- irisSub %>% scale(scale=FALSE)

The data is translated to a mean of [0, 0, 0].

We zoom in and add the original axis in grey in the origin.

2 SVD

1. We adopt the SVD on the centered data

irisSvd <- svd(X)

2. We extract

• the right singular vectors \(\mathbf{V}\) and
• the projections \(\mathbf{Z}\)

V <- irisSvd$v  
Z <- irisSvd$u %*% diag(irisSvd$d)

Note, that

• the SVD is essentially a rotation to a new coordinate system.
• we plotted \(\mathbf{V}_3\) with dots because we will use the SVD for dimension reduction \[\text{3D} \rightarrow \text{2D}\]

Rotate the plot

• Note, that

  • V1 points in the direction of the largest variability in the data
  • V2 points in a direction orthogal on V1 pointing in the direction of the second largest variability in the data.

3 Geometric Interpretation?

Write the truncated SVD as \[ \mathbf{X}_k = \mathbf{U}_k \boldsymbol{\Delta}_k \mathbf{V}_k^T = \mathbf{Z}_k \mathbf{V}_k^T \] with \[ \mathbf{Z}_k = \mathbf{U}_k \boldsymbol{\Delta}_k \] an \(n \times k\) matrix.

Each of the \(n\) rows of \(\mathbf{Z}_k\), say \(\mathbf{z}^T_{k,i}\), represents a point in a \(k\)-dimensional space.

V2 <- V[,1:2]
Z2 <- Z[,1:2]
X2 <- Z2 %*% t(V2)

Because of the orthonormality of the singular vectors, we also have \[\begin{eqnarray*} \mathbf{X}_k\mathbf{V}_k &=& \mathbf{Z}_k \mathbf{V}_k^T\mathbf{V}_k \\ \mathbf{X}_k\mathbf{V}_k &=& \mathbf{Z}_k. \end{eqnarray*}\]

Thus the matrix \(\mathbf{V}_k\) is a transformation matrix that may be used to transform \(\mathbf{X}_k\) into \(\mathbf{Z}_k\), and \(\mathbf{Z}_k\) into \(\mathbf{X}_k\).

---

More importantly, it can be shown that (thanks to orthonormality of \(\mathbf{V}\)) \[ \mathbf{X}\mathbf{V}_k = \mathbf{Z}_k. \]

This follows from (w.l.g. rank(\(\mathbf{X}\))=\(r\)) \[\begin{eqnarray*} \mathbf{X}\mathbf{V}_k &=& \mathbf{UDV}^T\mathbf{V}_k = \mathbf{UD}\begin{pmatrix} \mathbf{v}_1^T \\ \vdots \\ \mathbf{v}_r^T \end{pmatrix} \begin{pmatrix} \mathbf{v}_1 \ldots \mathbf{v}_k \end{pmatrix} \\ &=& \mathbf{UDV}^T\mathbf{V}_k = \mathbf{UD}\begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \ldots & 1 \\ 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \ldots & 0 \end{pmatrix} \ = \mathbf{U}_k\boldsymbol{\Delta}_k = \mathbf{Z}_k \end{eqnarray*}\]

The \(p \times k\) matrix \(\mathbf{V}_k\) acts as a transformation matrix: transforming \(n\) points in a \(p\) dimensional space to \(n\) points in a \(k\) dimensional space.

Z2proj <- X %*% V2
range(Z2 - Z2proj)

## [1] -8.881784e-16  1.082467e-15

3.1 Projection of a single data point

• Zoom in to see projection.
• The projection is indicated for the blue point \(X_{44}\) to the red point \(X_{2,44}\) in the plane spaned by V2.

3.2 Projection of all datapoints: project all rows of X on V2

• Zoom in first look orthonal via direction V2 (rotate until text V2 is viewed in the origin)
• Zoom in first look orthonal via direction V1 (rotate until text V1 is viewed in the origin)
• Note, that
  • V1 points in the direction of the largest variability in the data
  • V2 points in a direction orthogal on V1 pointing in the direction of the second largest variability in the data.

• Projection only.
• This clearly shows that the projected points X2 (X projected on V2) live in a two dimensional space