Multidimensional Scaling By Marc Sobel

The Goal  We observe (possibly non-euclidean) proximity data. For each pair of objects number ‘i’ and ‘j’ we observe their ‘proximity’ δ[i,j]. We would like to ‘project’ the data into a low-dimensional space. As an example, suppose we observe the correlations between the times at which different people go to work. We would like to create a map of the people in e.g., 2 dimensions which correctly position people relative to one another.

Methodology  To achieve this map, we assign unknown parametric points to i=2,…,n to each data point (except for the first). We regress the proximities on the parametric points via,

Simplest Possible Framework  The simplest possible framework makes the proximities ‘copies’ of the distances themselves. This is:  This leads to a Newton Raphson update of:

A Newton-Raphson based update

Controlled Newton Raphson:  Suppose we update with Newton Raphson but only move if the ‘stress’ is made smaller. That is, after each update we move only if,

Controlled MDS (better by a factor of 10)

Reducing the dimensionality of manifolds  Identifying and drawing inferences about high dimensional data sets play a fundamental role in statistical inference. Typically, what we would like to ascertain is elements of similarity between different (parts of the) data set(s). [For example, satellite data]. A famous example is the swiss roll data (pictured in the next slide). Here we would like to find neighboring points in the swiss roll manifold.

The Swiss Roll

For the Swiss Roll data sets (see A global geometric framework for Dimensionality Reduction by Tennenbaum, et al)  We represent X points lieing in a high dimensional space by trying to predict them from their neighbors (with proper weights W).  Having chosen the W’s optimally, we think of them as fixed and optimize: 

Applications of the Swiss Roll Algorithm  For high dimensional data, we’d like to learn the global structure or ‘data shape’. Using the swiss roll algorithm we can approximate geodesic distances between points on the underlying data manifold. (see A global Geometric Framework for Dimensionality Reduction by Tennenbaum, et al.)

Scaled Framework  The scaled framework makes the proximities ‘copies’ of the distances themselves but changes the error to a ‘relative error’:  This leads to a Newton Raphson update of:

Hybrid Bayes MDS  We can substitute Newton-Raphson updates with Hybrid Bayes MDS:  Where ‘p’ is kinetic energy (see MCMC part II).

A Full Bayesian Treatment  Assume,

Full Bayes Treatment  Form the posterior distribution by multiplying the terms in the last slide. Simulate using Hybrid Sampling (for X) and Gibbs Sampling for

Globally Scaled Framework  The scaled framework makes the proximities ‘copies’ of the distances themselves but changes the error to a ‘relative error’: