Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.8485 Avg p-adic loss

1 Parameter

View model →

Importance-Optimised p-adic Linear Regression

P-adic coefficients assigned to tags to predict taxonomy

0.3185 Avg p-adic loss

628 Avg non-zero coefficients

View model →

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

View model →

Zubarev Regression (UMLLR init)

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

0.3579 Avg p-adic loss

1,476 Non-zero coefficients

View fold details →

Zubarev Regression (Zeros init)

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

0.4143 Avg p-adic loss

2,710 Non-zero coefficients

View fold details →

Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.3567 Avg p-adic loss

1,428 Non-zero coefficients

View fold details →

Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.3559 Avg p-adic loss

1,433 Non-zero coefficients

View fold details →

Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2747 Avg p-adic loss

2,663 Non-zero params

View model →

Decision Tree

Unconstrained tree using ALL tags

0.2318 Avg p-adic loss

19,494 Effective params

View model →

Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2768 Avg p-adic loss

31,140 Non-zero params

View model →

Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.7919 Avg p-adic loss

864 Avg input weights

View model →

Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.8178 Avg p-adic loss

11,168 Avg parameters

View model →

ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

8,438 Tag battles

View rankings →

Benchmark Comparisons

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

0.84 Latest active params / classification

1.11 Paper active params / classification

Open benchmark pages →

Taxonomy ID	Name	Path	Samples	Share
gid://shopify/TaxonomyCategory/bt	Baby & Toddler	Baby & Toddler	294	6.4%
gid://shopify/TaxonomyCategory/lb	Luggage & Bags	Luggage & Bags	102	2.2%
gid://shopify/TaxonomyCategory/bu	Bundles	Bundles	83	1.8%
gid://shopify/TaxonomyCategory/pa	Product Add-Ons	Product Add-Ons	35	0.8%
gid://shopify/TaxonomyCategory/gc	Gift Cards	Gift Cards	31	0.7%
gid://shopify/TaxonomyCategory/os	Office Supplies	Office Supplies	21	0.5%
gid://shopify/TaxonomyCategory/na	Uncategorized	Uncategorized	19	0.4%
gid://shopify/TaxonomyCategory/hg	Home & Garden	Home & Garden	18	0.4%
gid://shopify/TaxonomyCategory/sg	Sporting Goods	Sporting Goods	18	0.4%
gid://shopify/TaxonomyCategory/bi	Business & Industrial	Business & Industrial	17	0.4%

Tag	Top taxonomy	Weight	Max \|weight\|
HLAČE	Baby & Toddler	2.7227	2.7227

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	0.3473	0.0003	0.7662
PCLR	0.000019	0.5507	0.4542	3.65e-38
PCNN	0.000003	0.5844	0.0155	0.0382
ULR	0.000003	0.2138	0.3536	3.36e-25
UNN	0.000002	0.2128	0.0448	8.12e-04
Decision Tree	0.000002	0.1856	0.1785	4.52e-12
Zubarev (UMLLR)	0.000003	0.3791	0.4656	8.08e-34
Zubarev (zeros)	0.000007	0.3931	0.8252	5.81e-91
Zubarev (M1)	0.000002	0.3891	0.2914	2.51e-19
Zubarev (M2)	0.000003	0.3853	0.3485	1.18e-23
Dummy Baseline	-0.000021	0.8943	0.3530	1.73e-26

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)	70,400	47,020 - 181,110 (95% CI, σ=298,057)	>95%	2032-07-31 (±uncertain, R²=0.318, growth=27.0/product/day)
ULR (Unconstrained Logistic Regression)	39,726	33,180 - 50,129 (95% CI, σ=4,352)	>95%	2029-06-20 (±uncertain, R²=0.318, growth=27.0/product/day)
Decision Tree	78,024	56,463 - 133,274 (95% CI, σ=19,869)	>95%	2033-05-10 (±uncertain, R²=0.318, growth=27.0/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	0.000000	0.3424	0.0029	0.3747
PCLR	0.000023	0.4319	0.6008	5.74e-57
PCNN	0.000015	0.4297	0.3263	1.77e-25
ULR	0.000005	0.1811	0.7023	6.15e-67
UNN	0.000008	0.1252	0.8196	4.46e-93
Decision Tree	0.000004	0.1519	0.6623	1.89e-59
Zubarev (UMLLR)	-0.000001	0.4226	0.0389	0.0023
Zubarev (zeros)	0.000003	0.4186	0.1315	8.93e-09
Zubarev (M1)	-0.000002	0.4371	0.1784	1.14e-11
Zubarev (M2)	-0.000002	0.4330	0.1281	1.44e-08
Dummy Baseline	0.000005	0.6640	0.0151	0.0463

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)	27,826	26,099 - 29,909 (95% CI, σ=977)	>95%	2027-03-12 (±uncertain, R²=0.997, growth=46.2/tag/day)
ULR (Unconstrained Logistic Regression)	35,924	31,804 - 42,098 (95% CI, σ=2,531)	>95%	2027-09-03 (±uncertain, R²=0.997, growth=46.2/tag/day)
Decision Tree	51,955	43,666 - 63,761 (95% CI, σ=5,207)	>95%	2028-08-15 (±uncertain, R²=0.997, growth=46.2/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: 353
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	-0.6479	+0.3488
ULR	2,663	0.2747	3.4254	-0.5612	-0.4782	+0.0830
Importance-Optimised	628	0.3186	2.7977	-0.4967	-0.4154	+0.0813
Decision Tree	19,494	0.2318	4.2899	-0.6349	-0.5647	+0.0703
Zubarev (M2)	1,433	0.3559	3.1563	-0.4486	-0.4513	-0.0027
Zubarev (M1)	1,428	0.3567	3.1546	-0.4477	-0.4511	-0.0034
Zubarev (UMLLR)	1,476	0.3579	3.1690	-0.4462	-0.4526	-0.0063
UNN	31,140	0.2768	4.4933	-0.5578	-0.5850	-0.0272
Dummy	1	0.8485	0.0000	-0.0713	-0.1357	-0.0643
Zubarev (zeros)	2,710	0.4143	3.4330	-0.3827	-0.4790	-0.0963
PCNN	864	0.7919	2.9365	-0.1013	-0.4293	-0.3280
PCLR	11,168	0.8178	4.0480	-0.0873	-0.5405	-0.4531

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	249	+0.1575	0.0238	0.1658	+0.0830	4,580
Importance-Optimised $p$-adic Linear Regression	216	+0.0685	0.0420	0.1295	+0.0813	4,580
Decision Tree	216	+0.1401	0.0240	0.1296	+0.0703	4,580
Zubarev (UMLLR init)	237	-0.0501	0.0239	0.1079	-0.0060	4,580
Unconstrained Neural Network with L1	247	+0.1007	0.0538	0.2416	-0.0272	4,580
Dummy Baseline	263	+0.0443	0.1160	0.3436	-0.0643	4,580
PCNN	247	-0.1948	0.0730	0.2528	-0.3280	4,580
PCLR	247	-0.3842	0.0248	0.2073	-0.4531	4,580

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
-0.1200	-0.1044	0.9310	0.0079	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?