Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.6242 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

0.3446 Avg p-adic loss

1,338 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)

0.4312 Avg p-adic loss

3,574 Non-zero coefficients

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Zubarev Regression (Zeros init)

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

0.4980 Avg p-adic loss

3,800 Non-zero coefficients

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Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.4286 Avg p-adic loss

3,354 Non-zero coefficients

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Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.4257 Avg p-adic loss

3,361 Non-zero coefficients

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Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2611 Avg p-adic loss

6,484 Non-zero params

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Decision Tree

Unconstrained tree using ALL tags

0.2237 Avg p-adic loss

62,090 Effective params

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Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2339 Avg p-adic loss

24,486 Non-zero params

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Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.5538 Avg p-adic loss

864 Avg input weights

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Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.7010 Avg p-adic loss

21,056 Avg parameters

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ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

4,970 Tag battles

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Benchmark Comparisons

Dedicated `latest` and `paper` benchmark pages, including the average active parameters touched per classification for the importance-optimised p-adic linear regressor.

1.10 Latest active params / classification

1.11 Paper active params / classification

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Taxonomy ID	Name	Path	Samples	Share
gid://shopify/TaxonomyCategory/bt	Baby & Toddler	4	125	0.9%
gid://shopify/TaxonomyCategory/lb	Luggage & Bags	15	93	0.7%
gid://shopify/TaxonomyCategory/bu	Bundles	5	65	0.5%
gid://shopify/TaxonomyCategory/pa	Product Add-Ons	19	24	0.2%
gid://shopify/TaxonomyCategory/na	Uncategorized	25	19	0.1%
gid://shopify/TaxonomyCategory/gc	Gift Cards	11	15	0.1%
gid://shopify/TaxonomyCategory/os	Office Supplies	18	13	0.1%
gid://shopify/TaxonomyCategory/sg	Sporting Goods	23	13	0.1%
gid://shopify/TaxonomyCategory/me	Media	17	10	0.1%
gid://shopify/TaxonomyCategory/hg	Home & Garden	14	8	0.1%

Tag	Top taxonomy	Weight	Max \|weight\|
ZOX	5	3.4562	3.4562

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	0.000000	0.3452	0.0004	0.7998
PCLR	0.000028	0.4704	0.6323	5.87e-41
PCNN	0.000011	0.5064	0.1304	5.51e-07
ULR	0.000005	0.1906	0.7245	4.05e-44
UNN	0.000006	0.1669	0.5173	2.44e-25
Decision Tree	0.000005	0.1573	0.7320	3.63e-44
Zubarev (UMLLR)	0.000003	0.3886	0.3535	7.64e-15
Zubarev (zeros)	0.000008	0.3866	0.7730	1.36e-46
Zubarev (M1)	0.000001	0.4078	0.0998	1.36e-04
Zubarev (M2)	0.000002	0.4011	0.2120	9.18e-09
Dummy Baseline	-0.000029	0.8920	0.5931	4.90e-34

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)	33,170	27,496 - 43,835 (95% CI, σ=4,307)	>95%	2027-03-29 (±uncertain, R²=0.998, growth=63.6/product/day)
ULR (Unconstrained Logistic Regression)	30,269	25,529 - 39,404 (95% CI, σ=3,435)	>95%	2027-02-12 (±uncertain, R²=0.998, growth=63.6/product/day)
Decision Tree	40,649	33,232 - 54,542 (95% CI, σ=5,482)	>95%	2027-07-25 (±uncertain, R²=0.998, growth=63.6/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	0.000001	0.3345	0.0116	0.1475
PCLR	0.000041	0.2708	0.6891	1.53e-47
PCNN	0.000019	0.3974	0.2051	1.37e-10
ULR	0.000008	0.1513	0.7384	8.00e-46
UNN	0.000009	0.1177	0.6204	3.75e-33
Decision Tree	0.000007	0.1213	0.7360	1.22e-44
Zubarev (UMLLR)	0.000005	0.3590	0.4521	6.88e-20
Zubarev (zeros)	0.000013	0.3158	0.8357	2.23e-56
Zubarev (M1)	0.000002	0.3974	0.1506	1.98e-06
Zubarev (M2)	0.000003	0.3840	0.2882	6.78e-12
Dummy Baseline	-0.000044	1.1180	0.6850	3.03e-43

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)	27,832	23,754 - 35,243 (95% CI, σ=3,050)	>95%	2027-03-24 (±uncertain, R²=0.991, growth=44.9/tag/day)
ULR (Unconstrained Logistic Regression)	27,641	23,561 - 36,213 (95% CI, σ=3,149)	>95%	2027-03-20 (±uncertain, R²=0.991, growth=44.9/tag/day)
Decision Tree	35,977	29,343 - 49,804 (95% CI, σ=5,158)	>95%	2027-09-21 (±uncertain, R²=0.991, growth=44.9/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: 667
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	-0.5650	+0.4317
Dummy	1	0.6242	0.0000	-0.2047	-0.0528	+0.1519
ULR	6,484	0.2611	3.8118	-0.5831	-0.4339	+0.1492
UNN	24,486	0.2339	4.3889	-0.6309	-0.4917	+0.1392
Decision Tree	62,090	0.2237	4.7930	-0.6503	-0.5321	+0.1182
Importance-Optimised	1,338	0.3447	3.1265	-0.4626	-0.3654	+0.0972
Zubarev (M2)	3,361	0.4257	3.5265	-0.3709	-0.4054	-0.0345
Zubarev (M1)	3,354	0.4286	3.5255	-0.3680	-0.4053	-0.0373
Zubarev (UMLLR)	3,574	0.4312	3.5532	-0.3653	-0.4081	-0.0427
PCNN	864	0.5538	2.9365	-0.2566	-0.3464	-0.0898
Zubarev (zeros)	3,800	0.4980	3.5798	-0.3028	-0.4107	-0.1079
PCLR	21,056	0.7010	4.3234	-0.1543	-0.4851	-0.3308

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
Dummy Baseline	167	+0.0481	0.1221	0.3347	+0.1519	13,299
Unconstrained Logistic Regression with L1	153	+0.1653	0.0195	0.1083	+0.1492	13,299
Unconstrained Neural Network with L1	151	+0.1167	0.0319	0.1631	+0.1392	13,299
Decision Tree	120	+0.1461	0.0182	0.0782	+0.1182	13,299
Importance-Optimised $p$-adic Linear Regression	120	+0.0566	0.0485	0.1273	+0.0972	13,299
Zubarev (UMLLR init)	141	-0.0642	0.0201	0.0810	-0.0428	13,299
PCNN	151	-0.2004	0.0691	0.1992	-0.0898	13,299
PCLR	151	-0.3789	0.0233	0.1880	-0.3308	13,299

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
-0.0956	-0.1980	0.9862	0.0007	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

Benchmark Comparisons

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?