Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.6172 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

0.3601 Avg p-adic loss

2,266 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)

0.4223 Avg p-adic loss

4,768 Non-zero coefficients

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Zubarev Regression (Zeros init)

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

0.4156 Avg p-adic loss

3,858 Non-zero coefficients

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Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.4225 Avg p-adic loss

4,470 Non-zero coefficients

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Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.4222 Avg p-adic loss

4,471 Non-zero coefficients

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Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2823 Avg p-adic loss

10,619 Non-zero params

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Decision Tree

Unconstrained tree using ALL tags

0.2322 Avg p-adic loss

120,392 Effective params

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Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2427 Avg p-adic loss

22,390 Non-zero params

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Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.5980 Avg p-adic loss

864 Avg input weights

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Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.8334 Avg p-adic loss

31,795 Avg parameters

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ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

9,067 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.

1.12 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	339	1.5%
gid://shopify/TaxonomyCategory/lb	Luggage & Bags	15	106	0.5%
gid://shopify/TaxonomyCategory/bu	Bundles	5	90	0.4%
gid://shopify/TaxonomyCategory/gc	Gift Cards	11	42	0.2%
gid://shopify/TaxonomyCategory/pa	Product Add-Ons	19	38	0.2%
gid://shopify/TaxonomyCategory/os	Office Supplies	18	22	0.1%
gid://shopify/TaxonomyCategory/na	Uncategorized	25	19	0.1%
gid://shopify/TaxonomyCategory/hg	Home & Garden	14	18	0.1%
gid://shopify/TaxonomyCategory/sg	Sporting Goods	23	18	0.1%
gid://shopify/TaxonomyCategory/bi	Business & Industrial	6	17	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	0.000001	0.3426	0.0163	0.0223
PCLR	0.000012	0.6003	0.4026	1.88e-37
PCNN	0.000000	0.6066	0.0005	0.6946
ULR	0.000003	0.2189	0.4627	7.20e-41
UNN	0.000001	0.2196	0.0438	3.38e-04
Decision Tree	0.000001	0.1904	0.2228	2.21e-17
Zubarev (UMLLR)	0.000002	0.3906	0.3242	2.25e-25
Zubarev (zeros)	0.000000	0.4490	0.0036	0.3163
Zubarev (M1)	0.000001	0.3962	0.2168	2.04e-16
Zubarev (M2)	0.000002	0.3941	0.2481	6.65e-19
Dummy Baseline	-0.000013	0.8372	0.3130	1.61e-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)	199,373	96,129 - 1,552,872 (95% CI, σ=633,146)	>95%	2038-08-25 (±uncertain, R²=0.495, growth=42.1/product/day)
ULR (Unconstrained Logistic Regression)	54,914	46,527 - 66,591 (95% CI, σ=5,097)	>95%	2029-04-04 (±uncertain, R²=0.495, growth=42.1/product/day)
Decision Tree	158,015	104,897 - 320,519 (95% CI, σ=69,251)	>95%	2035-12-17 (±uncertain, R²=0.495, growth=42.1/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	0.000001	0.3395	0.0147	0.0303
PCLR	0.000019	0.4699	0.5959	1.59e-64
PCNN	0.000009	0.4868	0.1970	6.97e-17
ULR	0.000005	0.1811	0.7924	1.19e-100
UNN	0.000006	0.1514	0.6399	1.32e-65
Decision Tree	0.000003	0.1581	0.6723	3.02e-71
Zubarev (UMLLR)	-0.000000	0.4145	0.0065	0.1803
Zubarev (zeros)	-0.000001	0.4665	0.0148	0.0422
Zubarev (M1)	-0.000001	0.4261	0.0712	6.20e-06
Zubarev (M2)	-0.000001	0.4226	0.0464	2.89e-04
Dummy Baseline	-0.000001	0.7213	0.0009	0.6043

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)	36,355	33,050 - 41,004 (95% CI, σ=2,087)	>95%	2027-09-01 (±uncertain, R²=0.997, growth=47.1/tag/day)
ULR (Unconstrained Logistic Regression)	37,636	34,321 - 41,836 (95% CI, σ=1,945)	>95%	2027-09-28 (±uncertain, R²=0.997, growth=47.1/tag/day)
Decision Tree	64,139	54,255 - 79,155 (95% CI, σ=6,470)	>95%	2029-04-13 (±uncertain, R²=0.997, 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: 1,001
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	-0.5121	+0.4846
Dummy	1	0.6172	0.0000	-0.2095	0.0001	+0.2097
UNN	22,390	0.2427	4.3501	-0.6150	-0.4349	+0.1801
ULR	10,619	0.2823	4.0261	-0.5493	-0.4025	+0.1469
Decision Tree	120,392	0.2322	5.0806	-0.6342	-0.5079	+0.1262
Importance-Optimised	2,266	0.3601	3.3552	-0.4435	-0.3354	+0.1081
Zubarev (zeros)	3,858	0.4156	3.5864	-0.3813	-0.3585	+0.0228
Zubarev (M2)	4,471	0.4222	3.6504	-0.3745	-0.3649	+0.0096
Zubarev (M1)	4,470	0.4225	3.6503	-0.3742	-0.3649	+0.0093
Zubarev (UMLLR)	4,768	0.4223	3.6784	-0.3744	-0.3677	+0.0067
PCNN	864	0.5980	2.9365	-0.2233	-0.2935	-0.0702
PCLR	31,795	0.8334	4.5024	-0.0792	-0.4501	-0.3710

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	305	+0.0618	0.1206	0.3846	+0.2097	23,097
Unconstrained Neural Network with L1	289	+0.1078	0.0595	0.2763	+0.1801	23,097
Unconstrained Logistic Regression with L1	291	+0.1550	0.0252	0.1836	+0.1469	23,097
Decision Tree	258	+0.1394	0.0252	0.1372	+0.1262	23,097
Importance-Optimised $p$-adic Linear Regression	258	+0.0741	0.0408	0.1295	+0.1081	23,097
Zubarev (UMLLR init)	279	-0.0422	0.0296	0.1192	+0.0067	23,097
PCNN	289	-0.1808	0.0808	0.2611	-0.0702	23,097
PCLR	289	-0.3824	0.0256	0.2073	-0.3710	23,097

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
-0.0864	-0.1999	0.9685	0.0024	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?