Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.5825 Avg p-adic loss

1 Parameter

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Importance-Optimised p-adic Linear Regression

P-adic coefficients assigned to tags to predict taxonomy

0.3778 Avg p-adic loss

1,103 Avg non-zero coefficients

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Level-wise Logistic Regression

Hierarchy-aware top-down classifier that always emits a valid taxonomy path

0.1008 Avg p-adic loss

83.01% Prefix-2 accuracy

132,415 Non-zero params

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Zubarev Regression (UMLLR init)

Stochastic p-adic optimization starting from UMLLR (arXiv:2503.23488)

0.4273 Avg p-adic loss

2,901 Non-zero coefficients

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Zubarev Regression (Zeros init)

Stochastic p-adic optimization starting from zeros (arXiv:2503.23488)

0.4586 Avg p-adic loss

3,048 Non-zero coefficients

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Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.4253 Avg p-adic loss

2,727 Non-zero coefficients

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Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.4261 Avg p-adic loss

2,732 Non-zero coefficients

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Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2416 Avg p-adic loss

4,552 Non-zero params

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Decision Tree

Unconstrained tree using ALL tags

0.2081 Avg p-adic loss

40,902 Effective params

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Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2279 Avg p-adic loss

27,154 Non-zero params

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Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.6923 Avg p-adic loss

864 Avg input weights

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Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.6650 Avg p-adic loss

15,661 Avg parameters

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ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

3,753 Tag battles

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Taxonomy ID	Name	Path	Samples	Share
gid://shopify/TaxonomyCategory/bt	Baby & Toddler	4	94	1.0%
gid://shopify/TaxonomyCategory/lb	Luggage & Bags	15	82	0.9%
gid://shopify/TaxonomyCategory/bu	Bundles	5	30	0.3%
gid://shopify/TaxonomyCategory/na	Uncategorized	25	17	0.2%
gid://shopify/TaxonomyCategory/sg	Sporting Goods	23	13	0.1%
gid://shopify/TaxonomyCategory/os	Office Supplies	18	12	0.1%
gid://shopify/TaxonomyCategory/gc	Gift Cards	11	11	0.1%
gid://shopify/TaxonomyCategory/hg	Home & Garden	14	8	0.1%
gid://shopify/TaxonomyCategory/ma	Mature	16	7	0.1%
gid://shopify/TaxonomyCategory/fb	Food, Beverages & Tobacco	9	6	0.1%

Tag	Top taxonomy	Weight	Max \|weight\|
FPM	23	3.9876	3.9876

Historical Performance Trends

Tracking model performance and dataset growth over time. Lower p-adic loss indicates better predictions.

Model	Slope (per product)	Intercept	R²	p-value
Importance-Optimised p-adic LR	0.000009	0.3081	0.4992	1.70e-20
PCLR	0.000044	0.4026	0.6522	1.90e-30
PCNN	0.000044	0.3658	0.7196	2.55e-36
ULR	0.000006	0.1871	0.4820	2.25e-15
UNN	0.000011	0.1349	0.6477	5.11e-23
Decision Tree	0.000005	0.1568	0.4341	3.92e-13
Zubarev (UMLLR)	0.000008	0.3606	0.6758	2.99e-22
Zubarev (zeros)	0.000013	0.3589	0.7864	6.74e-30
Zubarev (M1)	0.000003	0.3936	0.4531	1.25e-12
Zubarev (M2)	0.000005	0.3805	0.6238	1.61e-19
Dummy Baseline	-0.000056	1.0247	0.7712	5.03e-37

Extrapolation Analysis: When Will Importance-Optimised p-adic LR Outperform Other Models?

Based on current regression trends, we can extrapolate when Importance-Optimised p-adic LR will achieve better performance (lower p-adic loss) than other models as the dataset grows. The confidence intervals are calculated using bootstrap resampling (n=1000).

Model	Crossover Point (products)	95% Confidence Interval	Probability	Estimated Date
UNN (Unconstrained Neural Networks)	74,492	37,619 - 469,203 (95% CI, σ=1,106,117)	>95%	2029-02-21 (±uncertain, R²=0.996, growth=61.1/product/day)

Statistical Notes: The crossover points are calculated by finding where the regression lines intersect. The 95% confidence intervals are derived from bootstrap resampling of the regression parameters. The probability estimates indicate the likelihood that the crossover will occur given the current trends. Date predictions are based on linear extrapolation of dataset growth and should be interpreted with caution.

Model performance vs number of distinct tags

Model	Slope (per tag)	Intercept	R²	p-value
Importance-Optimised p-adic LR	0.000012	0.2579	0.5296	3.27e-22
PCLR	0.000055	0.1639	0.6829	5.74e-33
PCNN	0.000055	0.1279	0.7510	1.49e-39
ULR	0.000008	0.1539	0.5057	2.31e-16
UNN	0.000015	0.0652	0.7251	4.20e-28
Decision Tree	0.000006	0.1281	0.4661	2.54e-14
Zubarev (UMLLR)	0.000011	0.3092	0.7451	1.18e-26
Zubarev (zeros)	0.000018	0.2758	0.8393	4.24e-35
Zubarev (M1)	0.000005	0.3719	0.4851	9.59e-14
Zubarev (M2)	0.000007	0.3476	0.6590	2.54e-21
Dummy Baseline	-0.000070	1.3238	0.8032	1.25e-40

Extrapolation Analysis: When Will Importance-Optimised p-adic LR Outperform Other Models?

Model	Crossover Point (tags)	95% Confidence Interval	Probability	Estimated Date
UNN (Unconstrained Neural Networks)	53,151	32,507 - 249,219 (95% CI, σ=217,546)	>95%	2028-07-03 (±uncertain, R²=0.997, growth=49.6/tag/day)

Model complexity vs performance (parameter count vs p-adic loss) — Both axes use log scale. The red line is the fixed parsimoniousness baseline rather than a fitted regression.

Why parsimony matters. The question here is not just which model has the lowest loss, but which model gets good p-adic loss with the fewest effective parameters. That is exactly where the smaller p-adic models are interesting.

Where this baseline came from. The original score came from a log-log regression on model size versus loss, rounded to -0.1 × log₁₀(params) - 0.2. Looking across historical snapshots, those scores drifted as the dataset covered more taxonomies, so the current baseline adds + 0.3 × log₁₀(taxonomies / 1,000) to keep comparisons stable as the benchmark grows. For readability, we also re-centre the displayed score by dropping the old constant offset; that keeps the current tables mostly positive without changing the relative comparisons.

Parsimoniousness baseline: log₁₀(loss) = -0.1 × log₁₀(params) + 0.3 × log₁₀(taxonomies / 1,000)
Current snapshot taxonomies: 493
Parsimony score = baseline log₁₀(loss) − observed log₁₀(loss). Positive means better than baseline.

Model	Params	Loss	log₁₀(params)	log₁₀(loss)	Baseline log₁₀(loss)	Parsimony score
Level-wise Logistic	132,415	0.1008	5.1219	-0.9966	-0.6043	+0.3923
ULR	4,552	0.2416	3.6582	-0.6170	-0.4580	+0.1590
Dummy	1	0.5825	0.0000	-0.2347	-0.0921	+0.1426
Decision Tree	40,902	0.2081	4.6117	-0.6818	-0.5533	+0.1285
UNN	27,154	0.2279	4.4338	-0.6423	-0.5355	+0.1067
Importance-Optimised	1,103	0.3778	3.0427	-0.4227	-0.3964	+0.0263
Zubarev (M1)	2,727	0.4253	3.4357	-0.3714	-0.4357	-0.0644
Zubarev (M2)	2,732	0.4261	3.4365	-0.3705	-0.4358	-0.0653
Zubarev (UMLLR)	2,901	0.4273	3.4626	-0.3693	-0.4384	-0.0691
Zubarev (zeros)	3,048	0.4586	3.4840	-0.3385	-0.4406	-0.1020
PCNN	864	0.6923	2.9365	-0.1597	-0.3858	-0.2261
PCLR	15,661	0.6650	4.1948	-0.1772	-0.5116	-0.3344

Historical parsimony score stability — Left: parsimony score versus dataset size. Right: score distribution across historical snapshots. Positive means better than the taxonomy-adjusted baseline.

Model	Snapshots	Mean score	Std dev	Span	Latest score	Latest products
Unconstrained Logistic Regression with L1	98	+0.1689	0.0223	0.1083	+0.1590	9,110
Dummy Baseline	112	-0.0013	0.1216	0.3180	+0.1426	9,110
Decision Tree	65	+0.1555	0.0165	0.0720	+0.1285	9,110
Unconstrained Neural Network with L1	96	+0.1042	0.0332	0.1631	+0.1067	9,110
Importance-Optimised $p$-adic Linear Regression	65	+0.0144	0.0091	0.0402	+0.0263	9,110
Zubarev (UMLLR init)	86	-0.0781	0.0095	0.0465	-0.0692	9,110
PCNN	96	-0.2460	0.0154	0.0646	-0.2261	9,110
PCLR	96	-0.3806	0.0257	0.1880	-0.3344	9,110

Smaller standard deviation and span mean a model’s parsimoniousness is more stable as the dataset grows.

Unconstrained models: complexity vs performance (log-log scale) — Unconstrained models only (no PCLR/PCNN). Both axes on log scale.

Regression: log₁₀(loss) = slope × log₁₀(params) + intercept

Slope	Intercept	R²	p-value	Significant?	n
-0.0967	-0.2151	0.9248	0.0090	Yes	5

Model performance trajectory over time — Arrows show how each model's complexity and performance have changed over time.

Padjective Tag Hierarchy

Dataset coverage

Dummy Baseline

Importance-Optimised p-adic Linear Regression

Level-wise Logistic Regression

Zubarev Regression (UMLLR init)

Zubarev Regression (Zeros init)

Zubarev Mahler-1 (UMLLR init)

Zubarev Mahler-2 (UMLLR init)

Unconstrained Logistic Regression

Decision Tree

Unconstrained Neural Network

Parameter Constrained Neural Network

Parameter Constrained Logistic Regression

ELO-Inspired Rankings

Taxonomy distribution

Top 10 taxonomy classes

Tags with strongest signal

Historical Performance Trends

Extrapolation Analysis: When Will Importance-Optimised p-adic LR Outperform Other Models?

Extrapolation Analysis: When Will Importance-Optimised p-adic LR Outperform Other Models?