ArrowFlow: Hierarchical Machine Learning in the Space of Permutations

Ozgur Yilmaz

cs.LGApr 5, 2026

ArrowFlow: Hierarchical Machine Learning in the Space of Permutations

Authors: Ozgur Yilmaz

Organizations: Department of Artificial Intelligence Adana Science and Technology University, Adana, Turkey

Abstract

We introduce ArrowFlow, a machine learning architecture that operates entirely in the space of permutations. Its computational units are ranking filters, learned orderings that compare inputs via Spearman's footrule distance and update through permutation-matrix accumulation, a non-gradient rule rooted in displacement evidence. Layers compose hierarchically: each layer's output ranking becomes the next layer's input, enabling deep ordinal representation learning without any floating-point parameters in the core computation. We connect the architecture to Arrow's impossibility theorem, showing that violations of social-choice fairness axioms (context dependence, specialization, symmetry breaking) serve as inductive biases for nonlinearity, sparsity, and stability. Experiments span UCI tabular benchmarks, MNIST, gene expression cancer classification (TCGA), and preference data, all against GridSearchCV-tuned baselines. ArrowFlow beats all baselines on Iris (2.7% vs. 3.3%) and is competitive on most UCI datasets. A single parameter, polynomial degree, acts as a master switch: degree 1 yields noise robustness (8-28% less degradation), privacy preservation (+0.5pp cost), and missing-feature resilience; higher degrees trade these for improved clean accuracy. ArrowFlow is not designed to surpass gradient-based methods. It is an existence proof that competitive classification is possible in a fundamentally different computational paradigm, one that elevates ordinal structure to a first-class citizen, with natural alignment to integer-only and neuromorphic hardware.

Figures & tables

Panel of Figure 1 — Figure 1: ArrowFlow pipeline. Real-valued input $x$ is encoded via polynomial expansion, standardization, random projection, and argsort. $K$ independent networks receive different projections $W_{k}$ . Each network (detail, right) is a stack of sort layers computing Spearman’s footrule distance to learned filters. Predictions are combined by majority vote.

Panel of Figure 2 — Figure 2: Ranking layer forward pass. Input $\pi_{x}=[C,A,E,B,D]$ is compared to three filters. The motion $m_{j}[p]=\mathrm{rank}(r_{j},\pi_{x}[p])-p$ measures signed displacement (red = forward, blue = backward). The footrule distance $D_{j}=\sum|m_{j}[p]|$ totals the displacements. Filters are ranked by proximity: $r_{2}$ (distance 2) is closest. The output $\pi^{\prime}$ becomes the input to the next layer.

Panel of Figure 3 — Figure 3: Permutation-matrix accumulation. (a) Accumulator starts as identity (encoding current filter $[A,B,C,D]$ ). (b) Training input $[C,A,D,B]$ : each item votes for its position in the input (Def. 2 ). (c) Votes sum additively (Eq. 4 ). (d) Weighted average and argsort yield the updated filter $[A,C,B,D]$ (Eq. 5 ), which has moved toward the training data. Convergence to the mean-position ranking is guaranteed (Proposition 9 ).

Panel of Figure 4 — Figure 4: Multi-view ensemble. A single preprocessed input is projected through $K$ different matrices using diverse strategies (target-aware, random, calibrated). Each produces a different permutation—a different ordinal “view.” Independent ArrowFlow networks are trained per view; predictions are combined by majority vote. Projection diversity approximates the independence condition of Theorem 10 .

Panel of Figure 5 — Figure 5: Permutation cones. The argsort encoding partitions $\mathbb{R}^{d}$ into $d!$ convex cones (Weyl chambers), separated by hyperplanes $\{x_{i}=x_{j}\}$ . Shown for $d=3$ : six orderings of three coordinates. All points within a cone map to the same permutation. The dotted circle illustrates the stability radius from Theorem 4 : perturbations smaller than $\delta_{\min}(x)/2$ cannot cross a boundary.

Dataset	$N$	Feat	Cls	ArrowFlow Config	AF Err	RF	SVM	MLP	KNN	XGB
Iris	150	4	3	[64,128] e=16 p=3	2.7 $\pm$ 0.6	3.3	3.3	3.3	3.3	3.3
Wine	178	13	3	[128] e=64 p=1	2.8 $\pm$ 0.0	0.0	2.8	2.8	0.0	2.8
Breast C.	569	30	2	[64,128] e=32 p=2	2.8 $\pm$ 0.3	4.4	1.8	4.4	2.6	4.4
Wine Q.	1599	11	3	[128] e=16 p=3	35.9 $\pm$ 1.3	25.9	30.6	30.3	29.7	24.1
Vehicle	846	18	4	[64] e=32 p=2	17.4 $\pm$ 0.6	27.1	14.1	12.4	27.1	21.8
Segment	2310	19	7	[128,256] e=32 p=2	5.2 $\pm$ 0.3	2.8	3.2	3.7	4.1	2.6

Table 1: Best ArrowFlow configuration vs. GridSearchCV-tuned baselines. Test error (%), mean

\pm

std over 5 simulations. Bold = best overall. ArrowFlow entries are the best configuration found over the sweep, selected on the test split (an optimistic estimate); baselines are CV-tuned on the training fold (see the model-selection caveats above).

Dataset	kNN 1-view	kNN 7-view	ArrowFlow	Gain
Iris	10.0%	9.3%	2.7%	3.4 $\times$
Wine	45.0%	31.1%	2.8%	11.1 $\times$
Breast C.	17.5%	12.1%	2.8%	4.3 $\times$
Wine Qual.	37.0%	33.4%	35.9%	0.9 $\times$
Vehicle	49.5%	42.8%	17.4%	2.5 $\times$
Segment	20.8%	14.4%	5.2%	2.8 $\times$

Table 2: ArrowFlow vs. kNN on the same encoded permutations (fixed config per dataset; Digits errors differ slightly from Table 1 due to different config/seed). Learning Gain = ratio of kNN ensemble error to ArrowFlow error (higher = more value from learning).

Dataset	$f$ =64	$f$ =128	$f$ =256	$f$ =512
Iris (e=16)	6.0	6.0	7.3	—
Wine (e=32)	4.4	3.3	4.4	—
Breast C. (e=32)	4.4	3.7	3.7	—
Segment (e=32)	7.1	7.1	6.8	—
Digits (e=32)	14.4	13.8	11.7	12.9

Table 3: Effect of filter count (width) on test error (%).

Dataset	$d$ =1 [128]	$d$ =2 [64,128]	$d$ =2 [128,256]
Iris	6.0	2.7	4.0
Wine	3.3	6.1	4.4
Breast C.	3.7	2.8	3.5
Segment	7.1	6.2	5.2
Digits	13.8	15.9	12.8

Table 4: Effect of network depth on test error (%).

Dataset	$e$ =8	$e$ =16	$e$ =32	$e$ =64
Iris	8.0	6.0	6.7	—
Wine	—	6.7	3.3	2.8
Breast C.	—	3.9	3.7	5.4
Digits	—	16.3	13.8	5.8

Table 5: Effect of embedding dimension on test error (%).

Dataset	$v$ =3	$v$ =5	$v$ =7	$v$ =11
Iris	6.0	6.0	6.0	5.3
Segment	9.9	—	7.1	7.0
Digits	18.7	—	13.8	9.9

Table 6: Effect of ensemble views on test error (%).

Dataset	lr=0.01	lr=0.05	lr=0.1	lr=0.2
Iris [64,128]	—	4.7	2.7	4.0
Wine [128]	—	3.3	3.3	3.3
Vehicle [128]	—	18.8	18.2	37.1
Digits [128] e=64	11.7	6.2	5.8	4.6

Table 7: Effect of learning rate on test error (%).

Dataset	llu=True	llu=False	Change
Iris	6.0%	4.0%	$-$ 33%
Wine	3.3%	3.3%	0%
Breast C.	3.7%	3.5%	$-$ 5%
Wine Qual.	39.2%	37.0%	$-$ 6%
Vehicle	18.2%	18.2%	0%
Segment	7.1%	6.6%	$-$ 7%

Table 8: Effect of freezing the output layer. Sort-only [128] config, 5 simulations.

Dataset	Sort-Only Best	Hybrid llu=F	Degradation
Iris	2.7%	11.3%	4 $\times$
Wine	2.8%	30.6%	11 $\times$
Breast C.	2.8%	14.6%	5 $\times$
Wine Qual.	35.9%	50.0%	1.4 $\times$
Vehicle	17.4%	51.2%	3 $\times$
Segment	5.2%	34.2%	7 $\times$

Table 9: Hybrid tensor

\to

sort vs. pure sort-only. Test error (%).

Component	Iris	Wine	Breast C.	Digits
Baseline (single net)	21.3	13.3	6.8	17.8
+ Polynomial expansion	14.7	8.3	5.2	17.8
+ Multi-view ensemble (7v)	7.3	5.6	4.0	5.4
+ Diverse projection cycling	7.3	5.6	3.3	5.4
+ Per-dataset tuning	2.7	2.8	2.8	4.6

Table 10: Cumulative error reduction (%) as components are added.

Method	$\sigma$ =0	$\sigma$ =0.1	$\sigma$ =0.25	$\sigma$ =0.5	$\sigma$ =1.0	$\sigma$ =2.0
ArrowFlow	4.6	5.0	5.7	7.8	13.9	29.1
SVM-RBF	2.8	2.8	5.6	6.7	29.4	59.4
RF	0.0	0.0	1.7	7.2	10.6	31.1
MLP	2.8	2.8	3.3	6.1	10.6	25.6
XGBoost	2.8	2.8	6.7	12.2	18.3	36.7
KNN	0.0	0.6	1.7	5.6	10.6	25.0

Table 11: Noise robustness on Wine (

\mathrm{pol\_deg}

=1). Error (%) at each noise level.

Method	$\sigma$ =0	$\sigma$ =0.1	$\sigma$ =0.25	$\sigma$ =0.5	$\sigma$ =1.0	$\sigma$ =2.0
ArrowFlow	4.8	4.4	5.8	10.0	25.8	54.6
SVM-RBF	1.7	1.7	1.9	3.9	17.2	87.3
RF	2.8	2.7	3.3	6.6	18.0	49.7
MLP	1.9	2.3	2.3	5.2	14.6	46.8
XGBoost	4.2	4.7	7.2	12.9	30.5	58.4
KNN	2.2	2.1	2.4	3.9	10.9	41.0

Table 12: Noise robustness on Digits (

\mathrm{pol\_deg}

=1). Error (%) at each noise level.

Dataset	pol_deg	AF	SVM	RF	MLP	XGB	KNN	Avg BL
Wine	1	24.5	56.6	31.1	22.8	33.9	25.0	33.9
Digits	1	49.8	85.6	46.9	44.9	54.2	38.8	54.1
Iris	3	46.7	48.7	47.4	48.7	47.4	43.4	47.1
Breast C.	2	20.0	18.9	18.8	18.8	26.5	16.5	19.9
Vehicle	2	49.1	58.7	38.6	52.2	42.8	29.6	44.4
Segment	2	74.1	55.4	61.1	59.7	63.6	50.6	58.1

Table 13: Noise degradation analysis: absolute error increase (pp) from

\sigma

=0 to

\sigma

=2.0. Lower = more robust.

\mathrm{pol\_deg}

=1 datasets highlighted.

Dataset	Method	Raw	Ranked	$\Delta$
Iris	ArrowFlow	4.4	2.2	$-$ 2.2
	RF	3.3	3.3	+0.0
	SVM-RBF	3.3	3.3	+0.0
	KNN	3.3	6.7	+3.3
Segment	ArrowFlow	4.8	5.3	+0.4
	SVM-RBF	3.2	3.0	$-$ 0.2

Table 14: Privacy via rank transform: error on raw vs. ranked features.

\Delta

= degradation in pp.

Method	$m$ =0%	$m$ =10%	$m$ =30%	$m$ =50%
ArrowFlow	4.4	11.8	16.0	24.7
RF	3.3	9.3	10.0	29.3
SVM-RBF	3.3	10.7	15.3	36.0
MLP	3.3	15.3	20.7	33.3
KNN	3.3	10.7	14.0	37.3
XGBoost	3.3	15.3	14.7	34.7

Table 15: Missing features on Iris (4 features). Error (%) at each masking level.

Method	$m$ =0%	$m$ =10%	$m$ =30%	$m$ =50%
ArrowFlow	4.8	6.5	13.1	23.7
SVM-RBF	1.7	2.3	7.3	25.0
RF	2.8	4.2	12.2	34.7
MLP	1.9	3.3	6.3	13.4
KNN	2.2	2.7	5.8	16.6
XGBoost	4.2	7.8	19.4	38.9

Table 16: Missing features on Digits (64 features). Error (%) at each masking level.

Dataset	Feat	AF	RF	SVM	MLP	XGB	Avg BL
Iris	4	20.3	26.0	32.7	30.0	31.4	30.8
Wine	13	6.5	7.2	8.9	9.4	7.2	8.8
Breast C.	30	3.8	1.6	3.3	1.4	5.3	2.6
Vehicle	18	42.2	27.0	36.8	35.7	27.5	29.4
Segment	19	46.5	37.6	45.9	36.7	36.8	38.1
Digits	64	18.9	31.9	23.3	11.5	34.7	23.2

Table 17: Missing-feature degradation: absolute error increase (pp) from

m

=0% to

m

=50%.

Method	$N$ =20	$N$ =50	$N$ =100	$N$ =200	$N$ =455
ArrowFlow	8.8	7.3	4.7	3.5	2.9
SVM-RBF	6.1	7.9	3.5	3.5	1.8
RF	7.0	8.8	8.8	8.8	4.4
MLP	9.7	7.0	2.6	7.9	4.4
XGBoost	7.9	7.0	7.9	11.4	4.4
KNN	7.9	6.1	5.3	6.1	1.8

Table 18: Sample efficiency on Breast Cancer (

N_{\mathrm{full}}

=455). Error (%).

Method	$N$ =20	$N$ =50	$N$ =100	$N$ =200	$N$ =1437
ArrowFlow	58.1	26.1	17.4	14.7	4.8
SVM-RBF	27.2	21.9	13.6	10.6	1.7
RF	29.2	26.7	13.3	10.6	3.6
MLP	31.1	25.0	13.9	15.0	2.5
XGBoost	62.8	56.1	28.9	19.2	3.6
KNN	26.9	22.5	14.4	14.4	3.3

Table 19: Sample efficiency on Digits (

N_{\mathrm{full}}

=1437). Error (%).

Method	Encoding	Error
Random Forest	numeric	35.2%
SVM-RBF	numeric	35.3%
XGBoost	numeric	35.9%
KNN	numeric	37.6%
ArrowFlow (native)	native	38.6 $\pm$ 0.4%
MLP	numeric	38.7%

Table 20: Sushi preference classification. Error (%). ArrowFlow native = rankings fed directly as permutations; ArrowFlow projection = standard pipeline.

Method	Encoding	10 genes	15 genes	20 genes
RF	raw	1.2	0.6	0.6
SVM-RBF	raw	1.9	0.6	0.0
MLP	raw	2.5	1.2	0.0
KNN	raw	1.9	1.2	0.0
XGBoost	raw	1.2	1.2	2.5
RF	ranked	3.7	1.9	0.0

Table 21: Gene expression cancer classification. Test error (%) on TCGA PANCAN (161 test samples). “Raw” = original expression values; “Ranked” = within-sample rank transform; “AF native” = rank-transform fed directly as permutations; “AF proj” = standard ArrowFlow projection pipeline.

Transform	SVM (raw)	RF (raw)	SVM (ranked)	AF native
None	0.6	0.6	2.5	2.5
$\log(1+x)$	82.6	45.3	2.5	2.5
$\sqrt{\|x\|}$	82.6	45.3	2.5	2.5
$x^{2}$ (signed)	82.6	20.5	2.5	2.5
$\times 0.01$ global	82.6	62.7	2.5	2.5
$\times 100$ global	82.6	55.9	2.5	2.5

Table 22: Batch effect Type 1: monotone per-sample transforms applied to test data. All methods trained on clean data (15 genes). Rank-based methods (ArrowFlow native, SVM-ranked) are perfectly invariant; raw-input methods catastrophically fail.

$\sigma$	SVM (raw)	RF (raw)	SVM (ranked)	RF (ranked)	AF native
0.0 (clean)	0.6	0.6	2.5	1.2	2.5
0.3 (mild)	0.0	1.2	1.2	1.2	2.9
0.5 (moderate)	16.8	9.3	0.6	1.9	2.5
0.7 (cross-pl.)	59.0	7.5	9.3	8.1	10.3
1.0 (severe)	71.4	16.8	9.3	20.5	12.7
1.5 (extreme)	82.6	51.5	43.5	34.2	60.8

Table 23: Batch effect Type 2: per-gene log-normal scaling. Error (%) on 15 genes.

\sigma=0

is clean;

\sigma=0.3

is routine batch variation;

\sigma=0.7

is cross-platform;

\sigma\geq 1.0

is severe/unrealistic. Rank-based methods degrade gracefully; raw-input methods fail early.

PCA	Method	$N{=}1{,}000$	$N{=}5{,}000$	$N{=}10{,}000$
16	Best baseline (SVM/MLP)	11.2	6.9	6.0
	ArrowFlow best	15.9 $\pm$ 0.3	12.2 $\pm$ 0.2	9.8 $\pm$ 0.1
	Gap	4.7pp	5.3pp	3.8pp
32	Best baseline (MLP/SVM)	10.5	5.7	4.6
	ArrowFlow best	15.6 $\pm$ 0.2	11.4 $\pm$ 0.0	9.3 $\pm$ 0.1
	Gap	5.1pp	5.7pp	4.7pp

Table 24: MNIST via PCA: test error (%). Best baseline and best ArrowFlow config per setting. ArrowFlow best = [1024,128] 3-layer, 7-view ensemble. Iterations scale with training size (100/200/400).

Architecture	Views	Error (%)	Params (filters)
[128] 2L	1	18.7 $\pm$ 0.4	128
[128] 2L	7	13.3 $\pm$ 0.1	$7\times 128$
[256] 2L	7	12.5 $\pm$ 0.1	$7\times 256$
[512] 2L	7	12.6 $\pm$ 0.1	$7\times 512$
[256,128] 3L	7	11.0 $\pm$ 0.1	$7\times 384$
[512,64] 3L	7	10.2 $\pm$ 0.1	$7\times 576$

Table 25: ArrowFlow architecture scaling on MNIST (PCA=64,

N

=10,000). Error (%)

\pm

SE.

Property	pol_deg=1	pol_deg $>$ 1
Clean accuracy	$-$ (magnitude lost)	$+$ (3 $\times$ on low-dim)
Noise robustness	$+$ (8–28% less degrad.)	$-$ (noise amplified)
Rank-only (rank transform)	$+$ ( $<$ 0.5pp cost)	$-$ (up to 7pp cost)
Missing features	$+$ (beats BL on Iris)	$-$ (poly amplifies zeros)
Batch-effect invariance	$+$ (perfect, monotone)	$-$ (poly breaks ranks)
Sample efficiency	$\circ$	$+$ (richer features)

Table 26: The polynomial trade-off. Each cell shows whether ArrowFlow has a structural advantage (

+

), disadvantage (

-

), or is neutral (

\circ

) relative to baselines.

Panel of Figure 6 — Figure 6: The polynomial trade-off. At degree 1 (left), features are well-separated: large gaps yield a wide stability region, but limited capacity. At degree $>1$ (right), cross-terms create small gaps: higher capacity but noise amplified by $B^{k-1}$ , shrinking stability. This parameter is a master switch between robustness and accuracy.

Operation	Energy (pJ)	Used by
8-bit integer ADD	0.03	ArrowFlow
32-bit integer ADD/CMP	0.1	ArrowFlow
32-bit integer MUL	3.1	—
32-bit FP ADD	0.9	MLP
32-bit FP MUL	3.7	MLP
32-bit FP MAC	4.6	MLP

Table 27: Energy per operation at 45nm CMOS ( Horowitz, 2014 ) . ArrowFlow uses only operations in the top group; standard neural networks use operations in both groups.

	Operations	Energy (pJ)	Op type
ArrowFlow sort layer
Displacement (3NV)	24,576	2,458	INT32 @ 0.1 pJ
Argsort ( $N\log_{2}N$ )	896	90	INT32 @ 0.1 pJ
Total	25,472	2,547
MLP layer
Matrix multiply ( $NV$ MACs)	8,192	37,683	FP32 @ 4.6 pJ

Table 28: Energy per layer, single data point. ArrowFlow [128] (

V{=}64

N{=}128

) vs. MLP (

64{\to}128

). All energies at 45nm CMOS.

Component	ArrowFlow (pJ)	MLP (pJ)
Hidden, dist.+sort ( $\times$ 7)	$7(49{,}152{+}2{,}048)(0.1)=35{,}840$	$8{,}192(4.6)=37{,}683$
Output, dist.+ $\arg\min$ ( $\times$ 7)	$7\times 7{,}690(0.1)=5{,}383$	$1{,}280(4.6)=5{,}888$
Majority vote	$\sim$ 70	—
Total arithmetic	41,293	43,571
Ratio	$\approx 1.05\times$ in favor of ArrowFlow

Table 29: Full inference energy comparison on Digits. ArrowFlow: 7-view [256] ,

V{=}64

. MLP:

64{\to}128{\to}10

. Arithmetic energy only (45nm CMOS).

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Open paper AskPDF arXiv

Learning permutations is fundamental to sorting, ranking, and matching, but existing differentiable methods based on entropy-regularized Sinkhorn produce a single softened solution and collapse under ambiguity. We present PermFlow, a conditional flow matching framework that operates directly on the affine subspace of matrices with unit row and column sums. A closed-form tangent-space projector preserves these constraints exactly along every trajectory, by construction rather than through iterative correction, and a nearest-target coupling routes distinct noisy initializations toward distinct valid permutations. The result is a model that captures multimodal permutation distributions rather than collapsing them to a single mode. On a range of problems, PermFlow achieves high accuracy on unambiguous inputs and recovers both valid permutations under ambiguity.

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