Multidimensional Scaling and PCA are the Same Thing

There is a very simple argument that shows that MDS and PCA achieve the same results.

This argument has 2 important components. The first of these shows that an eigendecomposition of a gram matrix can be used for dimension-reduction.

PCA leverages the singular value decomposition (SVD). Given a matrix $X$, the SVD is $X = USV^{T}$.

We drop columns from X by using $US_{t}$ where we drop some rows and columns from S.

This is also conviently obtained using an eigendecomposition of the covariance matrix.

Working with the gram matrix, we have $XX^{T}$ and when expressed in terms of $U$, $S$ and $V$, we have $XX^{T}$ = $(USV^{T})(VSU^{T})$.

Simple algebra tells us that this is equal to $US^{2}U^{T}$. The spectral theorem tells us that this the eigendecomposition of the gram matrix will return this decomposition. $U$ and $S$ can be retrieved and a dataset with fewer dimensions can be obtained.

The second part of the argument involves proving that a matrix of distances is indeed a gram matrix. This argument was discussed in a previous post.

Powerful Ideas in Manifold Learning

In a previous post, I described the MDS (multidimensional scaling) algorithm. This algorithm operates on a proximity matrix which is a matrix of distances between the points in a dataset. From this matrix, a configuration of points is retrieved in a lower dimension.

The MDS strategy is:

• We have a matrix $D$ for distances between points in the data. This matrix is symmetric.
• We express distances as dot-products (using a proof from Schonberg). This means that $D$ is expressed as $X^T X$. (Observe that $X^T X$ is a matrix of dot-products).
• Once we have $X^T X$, dimension reduction is trivial. Running an eigendecomposition on this matrix will produced centered coordinates. The low-dimension embedding is recovered by discarding eigenvalues (eigenvectors).

Thus, assuming that we work with euclidean distances between points, we retrieve an embedding that PCA itself would produce. Thus, MDS with euclidean distances is identical to PCA.

Then what exactly is the value of running MDS on a dataset?

First, the PCA is not the most powerful approach. For certain datasets, euclidean distances do not capture the shape of the underlying manifold. Running the steps of the MDS on a different distance matrix (at least one that doesn’t contain euclidean distances) can lead to better results - a technique that the Isomap algorithm exploits.

Second, the PCA requires a vector-representation for points. In several situations, the objects in the dataset are not points in a metric space (like strings). We can retrieve distances between objects (say edit-distance for strings) and then obtain a vector-representation for the objects using MDS.

In the next blog post, I will describe and implement the Isomap algorithm that leverages the ideas in the MDS strategy. Isomap constructs a distance matrix that attempts to do a better job at recovering the underlying manifold.

PROOFS: