Padjective Tag Hierarchy

Dummy Baseline

Always predicts most common taxonomy (baseline for comparison)

0.6244 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

0.3398 Avg p-adic loss

1,375 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)

0.4302 Avg p-adic loss

3,643 Non-zero coefficients

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Zubarev Regression (Zeros init)

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

0.5029 Avg p-adic loss

3,845 Non-zero coefficients

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Zubarev Mahler-1 (UMLLR init)

Mahler affine basis (degree 1) with UMLLR initialization

0.4273 Avg p-adic loss

3,414 Non-zero coefficients

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Zubarev Mahler-2 (UMLLR init)

Mahler quadratic basis (degree 2) with UMLLR initialization

0.4243 Avg p-adic loss

3,409 Non-zero coefficients

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Unconstrained Logistic Regression

L1-regularized model using ALL tags

0.2571 Avg p-adic loss

6,672 Non-zero params

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Decision Tree

Unconstrained tree using ALL tags

0.2172 Avg p-adic loss

64,105 Effective params

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Unconstrained Neural Network

L1-regularized NN with weight pruning

0.2301 Avg p-adic loss

25,503 Non-zero params

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Parameter Constrained Neural Network

Neural network predicting taxonomy from tags

0.5541 Avg p-adic loss

864 Avg input weights

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Parameter Constrained Logistic Regression

Logistic regression model predicting Shopify taxonomy from tags

0.7804 Avg p-adic loss

21,421 Avg parameters

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ELO-Inspired Rankings

Battle-tested tag hierarchy from product title positions

5,032 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.

1.11 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	66	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	14	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	9	0.1%

Tag	Top taxonomy	Weight	Max \|weight\|
ZOX	5	3.4427	3.4427

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	0.3457	0.0000	0.9247
PCLR	0.000027	0.4754	0.6294	9.84e-42
PCNN	0.000010	0.5127	0.1113	3.07e-06
ULR	0.000005	0.1909	0.7406	1.46e-47
UNN	0.000005	0.1679	0.5245	1.21e-26
Decision Tree	0.000005	0.1574	0.7488	9.34e-48
Zubarev (UMLLR)	0.000003	0.3887	0.3737	2.51e-16
Zubarev (zeros)	0.000008	0.3866	0.7908	8.95e-51
Zubarev (M1)	0.000001	0.4073	0.1276	9.58e-06
Zubarev (M2)	0.000002	0.4010	0.2346	5.87e-10
Dummy Baseline	-0.000027	0.8820	0.5700	5.62e-33

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)	33,514	27,886 - 43,880 (95% CI, σ=4,170)	>95%	2027-04-03 (±uncertain, R²=0.998, growth=63.7/product/day)
ULR (Unconstrained Logistic Regression)	29,980	25,460 - 37,253 (95% CI, σ=3,066)	>95%	2027-02-06 (±uncertain, R²=0.998, growth=63.7/product/day)
Decision Tree	40,084	33,173 - 52,783 (95% CI, σ=4,940)	>95%	2027-07-15 (±uncertain, R²=0.998, growth=63.7/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	0.3364	0.0083	0.2148
PCLR	0.000039	0.2854	0.6810	9.02e-48
PCNN	0.000017	0.4160	0.1759	2.30e-09
ULR	0.000008	0.1526	0.7520	4.28e-49
UNN	0.000009	0.1218	0.6172	6.29e-34
Decision Tree	0.000007	0.1223	0.7512	4.56e-48
Zubarev (UMLLR)	0.000005	0.3608	0.4645	2.85e-21
Zubarev (zeros)	0.000013	0.3181	0.8463	1.96e-60
Zubarev (M1)	0.000002	0.3965	0.1818	8.08e-08
Zubarev (M2)	0.000003	0.3845	0.3078	3.69e-13
Dummy Baseline	-0.000041	1.0899	0.6508	1.10e-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)	28,300	24,263 - 35,853 (95% CI, σ=3,016)	>95%	2027-04-01 (±uncertain, R²=0.992, growth=45.1/tag/day)
ULR (Unconstrained Logistic Regression)	27,347	23,525 - 34,497 (95% CI, σ=2,758)	>95%	2027-03-11 (±uncertain, R²=0.992, growth=45.1/tag/day)
Decision Tree	35,390	29,226 - 47,556 (95% CI, σ=4,542)	>95%	2027-09-05 (±uncertain, R²=0.992, growth=45.1/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: 681
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	-0.5622	+0.4344
ULR	6,672	0.2571	3.8243	-0.5899	-0.4325	+0.1574
Dummy	1	0.6244	0.0000	-0.2045	-0.0501	+0.1545
UNN	25,503	0.2301	4.4066	-0.6381	-0.4907	+0.1474
Decision Tree	64,105	0.2172	4.8069	-0.6632	-0.5307	+0.1325
Importance-Optimised	1,375	0.3398	3.1384	-0.4688	-0.3639	+0.1049
Zubarev (M2)	3,409	0.4243	3.5326	-0.3723	-0.4033	-0.0310
Zubarev (M1)	3,414	0.4273	3.5333	-0.3693	-0.4034	-0.0341
Zubarev (UMLLR)	3,643	0.4302	3.5615	-0.3663	-0.4062	-0.0399
PCNN	864	0.5541	2.9365	-0.2564	-0.3437	-0.0873
Zubarev (zeros)	3,845	0.5029	3.5849	-0.2985	-0.4085	-0.1100
PCLR	21,421	0.7804	4.3308	-0.1077	-0.4831	-0.3754

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	158	+0.1650	0.0193	0.1083	+0.1574	13,661
Dummy Baseline	172	+0.0511	0.1215	0.3347	+0.1545	13,661
Unconstrained Neural Network with L1	156	+0.1176	0.0318	0.1631	+0.1474	13,661
Decision Tree	125	+0.1452	0.0183	0.0782	+0.1325	13,661
Importance-Optimised $p$-adic Linear Regression	125	+0.0585	0.0485	0.1273	+0.1049	13,661
Zubarev (UMLLR init)	146	-0.0633	0.0204	0.0810	-0.0399	13,661
PCNN	156	-0.1968	0.0707	0.1992	-0.0873	13,661
PCLR	156	-0.3786	0.0231	0.1880	-0.3754	13,661

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
-0.0975	-0.1974	0.9875	0.0006	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?