1.6 Phase transitions, a measure of learning

Phase transitions, a measure of learning

https://rumble.com/vcg8gw-1.6-phase-transitions-a-measure-of-learning.html

Phase transitions demonstrate a loose coupling.  A tight coupling like in KMeans mirror the distortion at each level.  A loose coupling enables the higher level to move at a different pace, disjoint, from the lower level.  This separation between levels, indicates that the levels represent different descriptions of the data, they speak different languages.

Important to differentiate between the model, the heirarchy, the grammar, and the content.  So next series I will do that.  The learning is in the model not the content.  The content can be memorized, its the relationships between the content that are learnt.

Here is a slightly different dataset, there are at least two apparent scales.  Lets see what KMeans does as we increase 'k':

Neat right, KMeans spreads its representations equally across the entire dataset, minimising the global loss of information.  Its a smooth transition from case to case, as we increase the number of centroids, KMeans shifts all of the centroids, such that it spreads out the representation.

Now take a look at faddc:

The thing to notice is that the representation is very stable up to a point, at a certain point there is a dramatic shift in the representation.   The faddc approach has created a new descriptive space, decoupled from the original data space.

What this means is that faddc preserves the representation even as the number of centroids changes.  This is like creating a word, 'citrus' for a category and then preserving the use of the word even as more data and more memory is allocated, as long as the concept of the category is stable, the word is stable.  

On the other hand, KMeans will change words each time, so if I start a conversation with you utilising the word 'citrus', mid-conversation I might start using a different word.  In effect words have no inherent meaning, they just constantly move to represent the global data.

Again this is because KMeans does not learn anything, it tries to memorise the original data as best it can.  While faddc has created a new descriptive space, decoupled from the original data space.

Here is a subtle point critical point, there are actually two inherent descriptive levels in the data.  The first is obvious, the feature space, for example the color of the objects, a red apple an orange orange.  The second descriptive space is less obvious.  The quantity of observations provides a second descriptive space.  This is more significant when we talk about the co-occurrence of features.  

So when multiple observation have the same relative features it is not that we learn only from the features.  We learn from the number of observations.  

This idea will become more significant when we talk about sequence analysis.  And the ability to factor out the sequence information independent of the content information.