Auditing what was learned
The two things a Q-space plot tells you, what the four experiments looked like, and how this differs from PCA, t-SNE or a SOM.
What the picture tells you, what the four experiments looked like, and why this is not just another scatter plot.
Two questions the picture answers
First, what is the rule? The way the examples cluster shows how the model organises the problem and where the borders between classes lie. Second, did it learn properly? Any dot on the wrong colour is a mistake, any lonely dot is an outlier, and you can see both without any statistics.
What the four experiments looked like
- Parity
- Perfect order: nine parallel streaks, one per count of switches, alternating answers. A crisp rule with no exceptions.
- Mice
- Clear groups with ragged edges and a few strays. A biological rule with exceptions, and a prompt to ask what causes them.
- Nucleosynthesis
- A tight island per nucleus, each covered by its own vortex, like cells in a mosaic.
- Digits
- Ten territories filling the whole map with irregular borders, two islands for the digit 1, a few stray 4s and 9s.
Why this is not just another scatter plot
Other methods can also squeeze data into two dimensions, but their pictures are separate from the model that makes the predictions. The replicant's picture is the model: the territories are the ones used to classify new cases. Choosing two dimensions gives you the picture; choosing more can give higher accuracy. A practical habit is to train one replicant for accuracy and a two-dimensional one to look at the rule.
Two questions the plot answers
When interpretability is a key factor, the two-dimensional \(Q\) makes a logic replicant completely interpretable in two respects. First, understanding the problem's logic: how the learnt instances are distributed in \(Q\), revealing a class-based coherent segmentation and where the vortices place the limits of each class. Second, verifying the learning: whether all instances were properly learnt, or whether there are outliers or misclassified instances, simply by checking that every low-entropy isolation contains elements of one class only. This is relevant whenever the interest goes beyond black-box prediction and the problem requires deeper analysis, for instance in clinical diagnosis, where the logic that explains a prediction can be as important as the prediction itself.
What the four experiments looked like
- Parity function
- Regularity and symmetry: nine parallel streaks grouped by the number of ones, alternating class, with vortices on each streak. A compact logic with no exceptions.
- Down syndrome in mice
- An aggregation logic that is not perfectly regular, with subgroups within a class and a few exceptions outside the main groups. Exactly what a primary logic affected by other factors (different mice, different responses to the drug) should look like, and a prompt to ask which factors cause the irregularities.
- Nucleosynthesis
- Dense groups for almost every isotope and one vortex per class covering each group: the Voronoi picture of a deterministic logic.
- Handwritten digits
- A large logic distributed over the whole plane with irregular borders, two groups for digit 1 and a few outliers of 4 and 9: sub-logics nested in the main one.
How this differs from other 2-D views
PCA. Taking the two principal components also represents a problem in two dimensions, but PCA is not intended to find low-entropy isolations and split the classes into isolated areas as the leietanic function does. The replicant's plot shows the classification logic applied to all the given instances and, in addition, the distribution of the vortices that explains where the limits of the classes are and how new instances will be classified.
t-SNE and SOM. The colourful plots may resemble those methods, but they are connected to the replicant's learned logic rather than to a separate embedding, so they say how good or bad the learning was and how precise the learned logic is. A single SOM offers a similar map but not competitive accuracy; SOM ensembles recover accuracy at the cost of interpretability. The replicant covers both at once.
Choosing the dimension of Q
Two dimensions give a picture; more dimensions can give more room for the logic. In the reported results the most accurate replicants used \(D=1\) for parity and nucleosynthesis, \(D=5\) for the mice and \(D=8\) for the digits, while the plots on the experiment pages come from separate replicants trained with \(D=2\) for visualisation. A practical workflow is to train the replicant with the \(D\) that gives the best accuracy and a second one with \(D=2\) to inspect the logic, or to inspect the \(D=2\) replicant when accuracy differences are small.