NOTE
Nomen et Locus names the two decisions that sit inside every label space. Locus is the place: where the class points lie relative to one another. Nomen is the name: which class occupies which point. Every design that structures the output of a classifier makes both decisions at once, and almost no design distinguishes them.
Context
A classifier has to put its classes somewhere. The routine answer is a one-hot vector per class, and the answer is so automatic that it rarely registers as an answer to anything. But it is a map from labels into a space, and it is a specific map: centred, the one-hot vectors are the vertices of a regular simplex, points at equal distance from one another and from the centroid.
Nothing forces that equidistance. Classes a model already confuses could sit close together at no cost. Classes that are semantically unrelated could sit far apart. This is not a hypothetical design: zero-shot classifiers embed class names with a text encoder and use the result as the output space; frozen classifier heads fix the prototype matrix in advance; hierarchical methods inject a declared taxonomy. All three put information into the labels before any training happens.
What none of them has is a unit. Ask “what did that label space buy me?” and the available answer is a difference in accuracy between two conditions. That difference is a property of the pair (label space, everything else), not a property of the label space. It has no declared zero, it does not add, and it does not compare across tasks.
This series is about the question, the objects it needs, and the comparisons that would answer it.
The question
A label space is two decisions at once: where the points lie, and which label sits where. What does each decision buy, against what reference, and can either be read before training?
Scope
Five articles, each self-contained, in order:
- Why one-hot is already a geometry. The centred one-hot code is the regular simplex, and three separate literatures arrive at the same object for different reasons.
- The two decisions. Arrangement and assignment, why the simplex is the case where the second does not exist, and why practical comparisons confound them.
- Why a comparison needs a declared zero. Proper scores, the uniform forecast, and why “above chance” is not a portable statement.
- What a permutation test concedes. The label-permutation null as a group action, what restricting the group means, and an audit it leaves open.
- What the spectrum cannot see. The log-determinant and its cousins are functions of eigenvalues alone, and a configuration is not determined by its spectrum.
The series is expository. It states the problem and surveys what is published; it does not report measurements.
Articles
| File | Date | Status |
|---|---|---|
| 01 - One-Hot Is a Choice | 2026-09-18 | done |
| 02 - The Two Decisions | 2026-09-18 | done |
References
- Dietterich, T. G., & Bakiri, G. (1995). Solving Multiclass Learning Problems via Error-Correcting Output Codes. JAIR, 2, 263–286.
- Evron, I., Onn, O., Weiss Orzech, T., Azeroual, H., & Soudry, D. (2023). The Role of Codeword-to-Class Assignments in Error-Correcting Codes: An Empirical Study. AISTATS, PMLR 206, 8053–8077.
- Good, I. J. (1952). Rational Decisions. Journal of the Royal Statistical Society B, 14(1), 107–114.
- Ojala, M., & Garriga, G. C. (2010). Permutation Tests for Studying Classifier Performance. JMLR, 11, 1833–1863.
- Palatucci, M., Pomerleau, D., Hinton, G., & Mitchell, T. (2009). Zero-shot Learning with Semantic Output Codes. NeurIPS.
- Papyan, V., Han, X. Y., & Donoho, D. L. (2020). Prevalence of neural collapse during the terminal phase of deep learning training. PNAS, 117(40), 24652–24663.
- Xu, Y., Zhao, S., Song, J., Stewart, R., & Ermon, S. (2020). A Theory of Usable Information Under Computational Constraints. ICLR.
Note
Each article carries its own references. The list above is the entry point, not the bibliography.