Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.6100 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

0.3231 Avg p-adic loss

1,083 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)

0.4147 Avg p-adic loss

3,124 Non-zero coefficients

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Zubarev Regression (Zeros init)

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

0.4624 Avg p-adic loss

3,309 Non-zero coefficients

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Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.4072 Avg p-adic loss

2,925 Non-zero coefficients

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Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.4089 Avg p-adic loss

2,950 Non-zero coefficients

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Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2355 Avg p-adic loss

5,130 Non-zero params

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Decision Tree

Unconstrained tree using ALL tags

0.2055 Avg p-adic loss

48,072 Effective params

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Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2184 Avg p-adic loss

25,164 Non-zero params

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Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.5254 Avg p-adic loss

864 Avg input weights

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Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.7584 Avg p-adic loss

17,696 Avg parameters

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ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

3,876 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.

1.09 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	106	1.0%
gid://shopify/TaxonomyCategory/lb	Luggage & Bags	15	88	0.8%
gid://shopify/TaxonomyCategory/bu	Bundles	5	47	0.4%
gid://shopify/TaxonomyCategory/pa	Product Add-Ons	19	20	0.2%
gid://shopify/TaxonomyCategory/na	Uncategorized	25	18	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/el	Electronics	8	7	0.1%

Tag	Top taxonomy	Weight	Max \|weight\|
No tag signal data available

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	0.000003	0.3327	0.0750	6.65e-04
PCLR	0.000035	0.4396	0.6220	2.77e-33
PCNN	0.000026	0.4367	0.4134	5.56e-19
ULR	0.000005	0.1908	0.5690	1.13e-23
UNN	0.000007	0.1556	0.5136	3.45e-20
Decision Tree	0.000004	0.1590	0.5495	5.40e-22
Zubarev (UMLLR)	0.000004	0.3803	0.3861	4.40e-13
Zubarev (zeros)	0.000009	0.3785	0.7127	5.07e-31
Zubarev (M1)	0.000001	0.4049	0.1228	1.74e-04
Zubarev (M2)	0.000002	0.3956	0.2626	1.05e-08
Dummy Baseline	-0.000042	0.9623	0.7086	1.08e-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)	38,945	26,990 - 82,965 (95% CI, σ=18,101)	>95%	2027-07-12 (±uncertain, R²=0.997, growth=62.0/product/day)
ULR (Unconstrained Logistic Regression)	61,594	33,382 - 479,451 (95% CI, σ=1,412,463)	>95%	2028-07-11 (±uncertain, R²=0.997, growth=62.0/product/day)
Decision Tree	111,547	49,916 - 1,045,870 (95% CI, σ=7,188,610)	>95%	2030-09-25 (±uncertain, R²=0.997, growth=62.0/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	0.000005	0.3064	0.1218	1.12e-05
PCLR	0.000047	0.2227	0.6726	5.94e-38
PCNN	0.000038	0.2554	0.4954	6.83e-24
ULR	0.000007	0.1556	0.6065	4.61e-26
UNN	0.000011	0.0973	0.6262	5.60e-27
Decision Tree	0.000006	0.1283	0.5817	6.95e-24
Zubarev (UMLLR)	0.000007	0.3414	0.5038	3.94e-18
Zubarev (zeros)	0.000014	0.3039	0.7970	3.44e-39
Zubarev (M1)	0.000002	0.3903	0.1903	1.90e-06
Zubarev (M2)	0.000004	0.3720	0.3551	6.52e-12
Dummy Baseline	-0.000059	1.2376	0.7787	1.02e-45

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)	32,465	24,661 - 57,226 (95% CI, σ=8,675)	>95%	2027-06-10 (±uncertain, R²=0.993, growth=47.1/tag/day)
ULR (Unconstrained Logistic Regression)	61,230	31,900 - 472,989 (95% CI, σ=318,162)	>95%	2029-02-10 (±uncertain, R²=0.993, growth=47.1/tag/day)
Decision Tree	121,921	46,409 - 1,240,786 (95% CI, σ=7,811,809)	>95%	2032-08-23 (±uncertain, R²=0.993, growth=47.1/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: 563
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	-0.5870	+0.4096
ULR	5,130	0.2355	3.7101	-0.6280	-0.4459	+0.1822
UNN	25,164	0.2184	4.4008	-0.6607	-0.5149	+0.1457
Decision Tree	48,072	0.2055	4.6819	-0.6871	-0.5430	+0.1441
Dummy	1	0.6100	0.0000	-0.2147	-0.0748	+0.1398
Importance-Optimised	1,083	0.3232	3.0345	-0.4905	-0.3783	+0.1123
Zubarev (M1)	2,925	0.4072	3.4662	-0.3902	-0.4215	-0.0313
Zubarev (M2)	2,950	0.4089	3.4698	-0.3884	-0.4218	-0.0334
Zubarev (UMLLR)	3,124	0.4147	3.4947	-0.3823	-0.4243	-0.0420
PCNN	864	0.5254	2.9365	-0.2795	-0.3685	-0.0890
Zubarev (zeros)	3,309	0.4624	3.5197	-0.3350	-0.4268	-0.0919
PCLR	17,696	0.7584	4.2479	-0.1201	-0.4996	-0.3795

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	122	+0.1680	0.0205	0.1083	+0.1817	10,968
Unconstrained Neural Network with L1	120	+0.1102	0.0327	0.1631	+0.1453	10,968
Decision Tree	89	+0.1523	0.0158	0.0720	+0.1436	10,968
Dummy Baseline	136	+0.0241	0.1233	0.3268	+0.1394	10,968
Importance-Optimised $p$-adic Linear Regression	89	+0.0374	0.0418	0.1273	+0.1118	10,968
Zubarev (UMLLR init)	110	-0.0719	0.0158	0.0810	-0.0425	10,968
PCNN	120	-0.2281	0.0473	0.1992	-0.0895	10,968
PCLR	120	-0.3788	0.0248	0.1880	-0.3800	10,968

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
-0.1029	-0.2106	0.9844	0.0008	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?