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Landscape-Dependent Performance of Photonic Quantum Solvers in QUBO Feature Selection for Financial Risk Detection

Nirvik Sahoo, Paul Robert Griffin

arXiv:2610.03161v1quant-phcs.LGq-fin.RM

TL;DR

Feature selection for imbalanced financial data must balance predictive relevance, redundancy, and search cost, and it is unclear whether approaches suited to compact ULB also apply to larger AmEx. The paper benchmarks solver-matched feature-selection methods across both datasets; ULB tolerates moderate compression, whereas AmEx reaches the full-model level only near full feature budgets.

  • Problem

    It is unclear whether solver and method choices and compression-performance trade-offs that work on compact ULB also hold for the larger, structurally different AmEx dataset.

    Source: §1 · p. 2

  • Method

    The paper benchmarks 13 feature-selection methods across Gurobi, Dirac-3, and Piquasso, routing each method to a solver matched to its mathematical structure.

    Source: §1 · p. 2, §4 · p. 6

  • Results

    ULB tolerates moderate compression, whereas AmEx reaches the full-model level only near full feature budgets; most same-method Gurobi–Dirac-3 differences fall within run-to-run variation.

    Source: §9 · p. 26

  • Takeaways & Limitations

    At matched budgets, F1 varies about ten times more across methods on ULB than on AmEx, alongside concentrated ULB signal and signal spread across many AmEx features.

    Source: §9 · p. 26

    AmEx Tier 3 runs use a 135-feature pool pre-selected by a classical MI filter, so cross-dataset hardware-native comparisons are not on identical footing.

    Source: §8.4 · p. 26

Abstract

from arXiv · show

Feature selection for imbalanced classification tasks such as credit card fraud and consumer default detection requires balancing predictive relevance, inter-feature redundancy, and computational feasibility. We benchmark three computing paradigms, classical branch-and-bound optimization (Gurobi), photonic entropy computing (QCI Dirac-3), and simulated photonic boson sampling (Piquasso), across thirteen feature-selection methods on two datasets: ULB Credit Card Fraud (30 features) and AmEx consumer default (159 features). Each method is routed to the solver matched to its mathematical structure. On ULB, Dirac-3 MI-Spearman matches the all-features model using 13 of 30 features (mean F1 0.873 +/- 0.023 over five runs, best run 0.896), and Piquasso is the best method at k=5. On AmEx, performance rises steadily with the feature budget and every paradigm approaches F1 = 0.80 only near the full feature set. Most differences between Gurobi and Dirac-3 on identical methods fall within run-to-run variation; the large gaps occur where the certified optimum generalizes poorly, most sharply for distance correlation on AmEx at k=25 (Gurobi F1 = 0.422 vs. a Dirac-3 mean of 0.746). At matched budgets, F1 varies about ten times more across methods on ULB than on AmEx, which we trace to how concentrated the predictive signal is in each feature space.

1 Introduction

Feature selection in imbalanced financial data must balance predictive relevance, redundancy, and search cost, while minority-class imbalance makes accuracy misleading. The paper tests whether solver-aware feature-selection methods and compression trade-offs generalize from ULB fraud detection to the larger, structurally different AmEx default dataset.

1 Introduction

  • Because fraud accounts for only 0.17% of ULB transactions, accuracy is misleading and evaluation must reflect minority-class performance.Feature selection also bears on faster scoring, reduced overfitting to noise, and model governance and auditability.Source: §1 · pp. 1–2
  • ULB has 30 features and largely uncorrelated PCA components, while AmEx has 159 features with signal spread across partly redundant features, leaving transfer of ULB findings uncertain.The paper asks whether solver and method choices and compression-performance trade-offs carry over to AmEx.Source: §1 · p. 2
  • Prior QUBO results from D-Wave are reference points only because they use a different solver and, on AmEx, a different feature working set.The paper instead compares its three solver paradigms and an all-features model trained on the same data.Source: §1 · p. 2
  • The benchmark evaluates 13 methods across three solver paradigms and two datasets, tracing compression-performance frontiers and selected-subset overlap.It compares ULB budgets k ∈ {5, 10, 13, 15, 20, 25} with AmEx budgets k ∈ {25, 50, 75, 100, 125, 150}.Source: §1 · p. 2

2 Related Work

Prior work supplies information-theoretic and graph-based feature-selection foundations, along with quantum optimization and photonic sampling approaches. This paper adapts these ideas to feature graphs, hardware-native objectives, and permanent-weighted sampling.

2.1 Information-Theoretic and QUBO Feature Selection

  • QUBO feature selection builds on mRMR’s relevance-redundancy trade-off; earlier work includes mutual-information criteria and a D-Wave benchmark on ULB and AmEx.Brown et al. unify mutual-information criteria, while Samanta et al. report reference results on both datasets.Source: §2.1 · p. 3

2.2 Graph-Theoretic Feature Selection

  • Graph-based feature-selection work motivates adapting spectral leverage, community detection, and attention-weighted redundancy to graphs whose nodes are features and edges are correlations.Spectral feature selection, Louvain community detection [12], and graph attention networks inform these adaptations.Source: §2.2 · p. 3

2.3 Higher-Order and Hardware-Native Quantum Methods

  • Conditional mutual information motivates HUBO’s cubic redundancy terms, while native cubic hardware can avoid auxiliary-variable overhead from reducing higher-order terms to quadratic form.Dirac-3’s multilevel variables and cubic objectives are described in [17].Source: §2.3 · p. 3

2.4 Boson Sampling

  • The paper uses standard boson sampling with single-photon inputs, so sampled subsets are weighted by squared matrix permanents rather than using the more common Gaussian variant.The sampler is implemented in Piquasso [20].Source: §2.4 · p. 3

3 Dataset and Evaluation

The benchmark contrasts ULB’s compact fraud dataset with AmEx’s larger consumer-default dataset, whose feature spaces distribute predictive signal differently. It evaluates minority-class performance alongside ranking quality and situates results against prior D-Wave benchmarks.

3.1 ULB Credit Card Fraud Dataset

  • ULB contains 284,807 transactions, 492 confirmed fraud cases (0.17%), and 30 features: Time, Amount, and 28 anonymized PCA components.The PCA feature names do not support domain-based selection, so statistical relationships must guide it.Source: §3.1 · p. 4

3.2 AmEx Consumer Default Dataset

  • After preprocessing, AmEx has 159 numeric features across Payment, Balance, Spend, Delinquency, and Risk categories, with partly redundant features.Unlike ULB’s anonymized PCA components, AmEx feature names carry categorical meaning.Source: §3.2 · p. 4

3.3 Feature Dependence Structure

  • ULB’s fraud signal is concentrated in a few components, whereas AmEx’s default signal is spread across many features, with redundancy concentrated in a small number of near-duplicate pairs.On ULB, only V17 and V14 exceed |ry| = 0.3; on AmEx, 29 features exceed |ry| = 0.3.Source: §3.3 · p. 5

3.4 Evaluation Metrics

  • F1 is the primary metric because both targets are minorities, while Gini measures threshold-independent ranking quality and can favor a different subset.A majority-class-only classifier would have misleadingly high accuracy while catching none of the target class.Source: §3.4 · pp. 5–6

3.5 Reference Results from Prior Work

  • The prior D-Wave reference reports ULB best Gini of 97.678% for MI-Pearson at k=13 and AmEx best Gini of 85.07098% for MIQUBO at k=150.That work used a 174-feature AmEx working set, compared with 159 features here, because of different preprocessing.Source: §3.5 · p. 6

4 Methods

The methods follow a solver-aware design: each feature-selection formulation is routed to a solver suited to its mathematical structure, with standard QUBOs also compared directly across solvers. The approaches range from pairwise and graph-based objectives to hardware-native objectives and boson sampling.

4.1 Paradigm 1: Classical Exact Optimization (Gurobi)

  • Gurobi solves standard QUBO and graph-theoretic QUBO or ILP formulations to certified global optimality with deterministic, reproducible output.The benchmark assigns it standard pairwise QUBO methods and graph-theoretic methods, including an optional ILP diversity constraint.Source: §4.1 · p. 6

4.1.1 Tier 1: Standard QUBO Methods

  • Six standard QUBO variants compute feature relevance and redundancy using Pearson correlation, mutual information, parametric MI, mixed MI-correlation measures, Spearman correlation, or distance correlation.MI-Spearman captures monotone nonlinear relationships that Pearson correlation misses; distance correlation detects arbitrary statistical dependence.Source: §4.1.1 · p. 7

4.1.2 Tier 2: Graph-Theoretic Methods

  • Graph-theoretic methods add global structure to pairwise correlations: spectral leverage rewards peripheral features, while Graph-Attention weights redundancy penalties by feature relevance.Spectral leverage is computed from the feature-correlation graph’s Laplacian; attention penalizes redundancy more for high-relevance pairs.Source: §4.1.2 · pp. 7–8
  • Community-ILP detects feature communities and requires at least one selected feature from each, enforcing structural diversity through Gurobi’s ILP solver.The constraint prevents selection from being concentrated in a single correlated cluster.Source: §4.1.2 · p. 8

4.2 Paradigm 2: Quantum-Entropic Annealing (Dirac-3)

  • Dirac-3 supports multilevel variables and higher-order objectives; standard and graph-theoretic QUBOs also run on it for direct comparisons with Gurobi.Its native capabilities allow Tier 3 methods to avoid reformulation overhead.Source: §4.2 · p. 8

4.2.1 Tier 3: Dirac-3-Native Methods

  • Dirac-3-native methods include HUBO’s cubic conditional-redundancy penalty, graded integer feature weights, and Community-Poly’s polynomial approximation of a community-diversity constraint.HUBO models three-way redundancy that pairwise QUBO can only approximate by collapsing triplets to pairs; integer weights range from none to strong.Source: §4.2.1 · pp. 8–9

4.3 Paradigm 3: Photonic Boson Sampling (Piquasso)

  • Piquasso samples size-three feature subsets with probability proportional to squared permanents of relevance-weighted matrices, then ranks features by selection frequency.The permanent weighting rewards features that score well jointly across multiple relevance interactions, especially at small k.Source: §4.3 · p. 9

5 Experimental Setup

The experiments vary feature budgets by dataset, evaluate selected subsets with a class-weighted XGBoost classifier, and account for Dirac-3 capacity limits on AmEx. They also characterize run-to-run variation and compare selection overlap with a random-overlap baseline.

5.1 Feature Budget Sweep

  • ULB used k={5, 10, 13, 15, 20, 25}, while AmEx used k={25, 50, 75, 100, 125, 150}, with budgets reflecting dataset size and reference configurations.Source: §5.1 · p. 9

5.2 Downstream Classifier

  • Each selected feature subset was evaluated with XGBoost on a held-out test split, using class-weight balancing for each dataset’s class imbalance.Performance was measured with F1 and Gini.Source: §5.2 · p. 10

5.3 Classical Pre-Filtering for Dirac-3-Native Methods on AmEx

  • AmEx Tier 3 methods were prefiltered to 135 features, retaining the MI pool only when its overlap with the mRMR pool exceeded 120/135 features.Their reported results therefore use a 135-feature candidate pool and budgets through k=130; the all-features result on that pool is F1 = 0.8036 and Gini = 91.545.Source: §5.3 · p. 10

5.4 Dirac-3 Configuration

  • Each Dirac-3 configuration ran five times, reporting each run’s best of 20 samples on ULB and 5 on AmEx.Within-run 95% bootstrap confidence intervals used 1,000 resamples; run-to-run variation was reported as mean ± 95% CI.Source: §5.4 · p. 10

5.5 Jaccard Similarity Analysis

  • Jaccard similarity was analyzed within two Dirac-3 method groups and separately across directly comparable Gurobi–Dirac-3 pairs, against a random-subset expectation.Agreement above the random expectation indicates shared selection structure; agreement at that level indicates overlap no greater than chance.Source: §5.5 · pp. 10–11

6 Results

ULB supports strong compression: Dirac-3 MI-Spearman reaches the all-features F1 with 13 features, while AmEx methods approach that baseline only near the full feature set. Cross-solver differences are usually small relative to run-to-run variation, but certified QUBO optima can generalize poorly.

6.1 Overview of Per-Method Results

  • On ULB, Dirac-3 MI-Spearman’s best configuration at k=13 scores F1 = 0.873 ± 0.023 versus all-features F1 = 0.871; on AmEx, every method converges to F1 ≈0.800–0.804 by k=150.Table 4 identifies highest-F1 configurations across budgets and at the smallest budget; Tables 5 and 6 report Dirac-3 QUBO F1 by k for ULB and AmEx, respectively.Source: §6.1 · p. 11, Table 4 · p. 12, Table 5 · p. 12
  • Tier 3 Community-Poly is strongest on average on ULB at F1 = 0.875 ± 0.010 at k=25; AmEx Tier 3 results use a 135-feature pre-filtered pool.On AmEx, the three hardware-native methods rise from F1 = 0.747–0.754 at k=25 to 0.800–0.803 at k=130, conditional on that pre-filter.Source: §6.1 · p. 14, Table 7 · p. 14

6.2.1 ULB

  • On ULB, Dirac-3 leads for four of six methods at k=5, but most differences fall within run-to-run variation; some certified optima select none of the four most fraud-correlated features.At k=5, Gurobi MIQUBO selects all four, while its MI-Pearson and MI-Spearman optima select none; Dirac-3 MI-Spearman's best run at k=13 reaches F1 = 0.896.Source: §6.2.1 · p. 15, Table 8 · p. 15

6.2.2 AmEx

  • On AmEx, the largest cross-solver gap is +0.324 F1 for Dist-Corr at k=25: Dirac-3 scores 0.746 versus Gurobi’s 0.422.Most cross-solver differences are small relative to Dirac-3 run-to-run variation; the QUBO optimum is not guaranteed to maximize classifier F1.Source: §6.2 · pp. 14–15, §6.2.2 · pp. 15–16, Table 9 · p. 16

6.3.1 ULB

  • F1 = 0.896 is the best single-run result: Dirac-3 MI-Spearman at k=13 exceeds the all-features model’s 0.871 using 13 of 30 features.This is also the strongest configuration on average, although individual reruns vary within the reported confidence interval.Source: §6.3.1 · p. 16, Table 10 · p. 16

6.3.2 AmEx

  • On AmEx, the best individual-run F1 is 0.8043 for Parametric-MI at k=150, only marginally above the all-features model’s 0.8022.Both the best individual-run F1 and Gini occur at the largest tested budget, consistent with the near-monotonic pattern in the mean results.Source: §6.3.2 · p. 17, Table 11 · p. 17

6.4.1 ULB

  • At k=5, Piquasso scores F1 = 0.874, above the all-features model’s 0.871 and every other method; it dips to 0.840 at k=10 before recovering by k=13–15.Permanent weighting reflects joint relevance across triplets, while the current marginal-frequency aggregation can fail to preserve pairwise complementarity; at k=15, F1 = 0.880.Source: §6.4.1 · pp. 17–18, Table 12 · p. 17

6.4.2 AmEx

  • On AmEx, Piquasso rises from F1 = 0.767 at k=25 to 0.803 at k=150, matching the all-features baseline only at the largest tested budget.The paper relates this pattern to predictive signal spread across many AmEx features, unlike the more concentrated ULB signal.Source: §6.4.2 · p. 18, Table 13 · p. 18

7 Subset Diversity: Jaccard Similarity Analysis

The section compares feature-subset overlap within and between solvers against random-subset expectations, then relates overlap to differences in F1 across methods. Across datasets, subset overlap alone does not explain when solver choice matters; the spread of method performance is about ten times larger on ULB than AmEx.

7 Subset Diversity: Jaccard Similarity Analysis

  • Across both datasets and method groups, within-Dirac-3 overlap rises with feature budget and stays near random-subset expectations; the section measures overlap against that baseline across runs.The analysis uses subsets selected in Section 6; Dirac-3 similarities are averaged across independent runs.Source: §7 · pp. 18–19, Table 14 · p. 19, Figure 3 · p. 21

7.1.1 ULB

  • Across ULB’s nine Dirac-3 QUBO/Graph methods, mean overlap rises from 0.096 ± 0.022 at k=5 to 0.719 ± 0.008 at k=25, tracking random-subset expectation.The tight confidence intervals indicate a stable trend across runs.Source: Table 15 · p. 19, §7.1.1 · p. 19

7.1.2 AmEx

  • On AmEx, Dirac-3 QUBO/Graph overlap likewise follows random-subset expectation; at k=25, Gurobi’s Tier 1 solutions agree above chance while Dirac-3 re-runs do not.Gurobi’s mean overlap is 0.181 versus E[J_rand]=0.087 at k=25; overlap alone cannot distinguish a near-degenerate objective landscape from incomplete convergence.Source: Table 16 · p. 20, §7.1.2 · p. 20

7.2.1 ULB

  • For ULB Tier 3 methods, mean overlap rises from 0.116 ± 0.051 at k=5 to 0.731 ± 0.023 at k=25, remaining within one confidence interval of random expectation.Confidence intervals are wider than for the nine-method QUBO/Graph group, consistent with fewer method pairs contributing to the mean.Source: Table 17 · p. 20, §7.2.1 · p. 21

7.2.2 AmEx

  • For AmEx Tier 3 methods on the 135-feature pre-filtered pool, overlap rises from 0.105 ± 0.035 at k=25 to 0.927 ± 0.002 at k=130, tracking random-subset expectation.There is no Gurobi counterpart for Tier 3 methods; low overlap at small k concerns choices within the pre-filtered pool.Source: Table 18 · p. 21, §7.2.2 · p. 22

7.3.1 ULB

  • On ULB, Gurobi–Dirac-3 overlap rises from 0.195 at k=5 to 0.798 at k=25, exceeding chance and within-Dirac-3 overlap at every budget.For MIQUBO at k=5, the Dirac-3 run with the highest F1, 0.832, shares only two features with Gurobi’s certified subset, whose F1 is 0.823.Source: Table 19 · p. 22, §7.3.1 · p. 22

7.3.2 AmEx

  • On AmEx, cross-solver overlap rises from 0.112 at k=25 to 0.900 at k=150, but is only slightly above chance and does not predict the winner.At k=25, Gurobi’s exact solution wins on four of six methods; cross-solver agreement is lower relative to chance than ULB’s at a similar budget.Source: Table 20 · p. 23, §7.3.2 · p. 23

7.4 Overlap Versus Performance Sensitivity

  • At matched budgets near 16% and 45% of features, method-to-method F1 spread is about ten times larger on ULB than AmEx, where signal is more dispersed.AmEx’s default signal exceeds |r_y| > 0.3 for 29 features, versus two fraud-signal components on ULB; the spread describes typical subsets, not every method.Source: §7.4 · pp. 23–24, Table 21 · p. 24

8 Discussion

The discussion finds that solver choice must be considered alongside method and dataset structure: ULB supports strong compression, while AmEx generally needs a large feature budget. It also identifies limits on what the benchmark establishes, including temporal generalization and hardware-capacity constraints.

8.1 The Solver-Method Alignment Principle

  • Certified Gurobi optima can perform poorly when the QUBO objective is only an approximation of downstream classifier performance, while AmEx Tier 3 methods require classical pre-filtering to fit Dirac-3.For Dist-Corr at k=25 on AmEx, Gurobi gives F1 = 0.422 versus Dirac-3’s mean of 0.746; AmEx Tier 3 uses a 135-feature pool from a classical MI filter.Source: §8.1 · p. 24

8.2.1 ULB

  • On ULB, the F1- and Gini-best configurations differ, so the preferred metric depends on whether the model supports threshold-based alerts or risk ranking.On AmEx, the best single-run F1 and Gini occur at k=150, and the metric choice matters much less.Source: §8.2.1 · p. 25

8.2.2 AmEx

  • On AmEx, the best single-run F1 and Gini occur at k=150, making the choice between these metrics matter less than on ULB.The reported best single-run results differ by less than 0.001 F1 and about 0.05 Gini points.Source: §8.2.2 · p. 25

8.3.1 ULB: The k=13 Sweet Spot

  • At k=13, ULB retains near-best performance with 57% fewer features than the full 30-feature input.Reducing from 25 to 13 features costs less than 0.01 F1 on average.Source: §8.3.1 · p. 25

8.3.2 AmEx: No Comparable Sweet Spot

  • AmEx has no comparable compression sweet spot: its signal is spread across many features, and performance approaches the full model only at large budgets.Piquasso reaches F1 = 0.874 at k=5 on ULB but improves steadily with k on AmEx; whether its ULB advantage persists across repeated runs remains untested.Source: §8.3.2 · pp. 25–26

8.4 Limitations

  • The ULB results cover 48 hours of European card transactions from 2013, and temporal generalization is not assessed.AmEx Tier 3 uses a 135-feature pool pre-selected by a classical MI filter, so those results are conditional on that pre-filter rather than the full 159-feature set.Source: §8.4 · p. 26

9 Conclusion

The benchmark compares three solver paradigms and thirteen feature-selection methods across ULB and AmEx. Results show that compression and solver differences depend on the dataset and method, with feature-signal concentration distinguishing the datasets.

9 Conclusion

  • The study benchmarks Gurobi, QCI Dirac-3, and Piquasso across thirteen feature-selection methods on two financial datasets with different scale and structure..Source: §9 · p. 26
  • On ULB, Dirac-3 MI-Spearman at k=13 matches the all-features model with mean F1 = 0.873 ± 0.023, while AmEx approaches the full-model F1 only at k=125–150.Piquasso has the best ULB result at k=5, but that advantage is not seen on AmEx.Source: §9 · p. 26
  • Most Gurobi–Dirac-3 differences on identical methods fall within run-to-run variation, but AmEx Dist-Corr at k=25 gives F1 = 0.422 for Gurobi versus a Dirac-3 mean of 0.746.Certified QUBO optimality alone does not guarantee downstream performance.Source: §9 · p. 26
  • At matched budgets, F1 varies about ten times more across methods on ULB than on AmEx, whose default signal is spread across many features.ULB’s fraud signal is concentrated in a few uncorrelated components.Source: §9 · p. 26
  • The benchmark leaves generality unresolved: it does not compare Dirac-3 solutions with Gurobi’s certified QUBO energies, and further datasets are needed to test whether the observed pattern generalizes.Tier 3 on AmEx also relies on a classical pre-filter to fit Dirac-3’s capacity.Source: §9 · pp. 26–27

Appendices

The appendices provide supporting material for the paper’s results, including detailed per-method discussion and additional result tables.

Appendices

  • The appendices support Sections 6 and 7 with per-method discussion, F1 and Gini tables, and single-run pairwise Jaccard matrices.Appendix B contains tables not shown in the main text; Appendix C provides single-run matrices illustrating Section 7’s multi-run averages.Source: Appendices · p. 29

A Detailed Per-Method Results

A.1 Gurobi: Standard QUBO Methods

The Gurobi standard-QUBO results show different patterns across the datasets: MI-Spearman leads on ULB, while AmEx improves gradually with larger feature budgets and has a severe small-budget Dist-Corr failure.

ULB

  • On ULB, MI-Spearman at k=25 achieves F1 = 0.885, the benchmark’s highest F1, while its k=20 result has Gini = 97.0%.The passage attributes its strength to Spearman redundancy capturing monotone non-linear relationships that Pearson misses.Source: §A.1 · p. 29
  • At k=5 on ULB, MI-Pearson reaches F1 = 0.459 and Dist-Corr reaches F1 = 0.448, while MIQUBO reaches F1 = 0.823.The passage attributes the weaker results to unreliable redundancy metrics at very tight budgets.Source: §A.1 · p. 29
  • On ULB, Parametric-MI reaches Gini = 96.4% at k=13 and Gini = 96.8% at k=25, but its F1 at k=10 is 0.701.The passage suggests its density estimator needs approximately 13 features for reliable scores.Source: §A.1 · p. 29

AmEx

  • On AmEx, MIQUBO at k=25 reaches F1 = 0.789, only 0.013 below the all-features baseline of 0.802; no method exceeds that baseline at any tested k.Pearson and MI-Pearson at k=150 come closest, with F1 = 0.804.Source: §A.1 · p. 29
  • On AmEx, Dist-Corr at k=25 reaches F1 = 0.422 and Gini = 39.6%, a severe failure of the certified QUBO optimum at that budget.The passage characterizes the Gini as indicating a near-random risk ranking.Source: §A.1 · p. 29
  • On AmEx, all six Tier 1 methods rise steadily with k and reach F1 = 0.801–0.804 at k=150, with no sharp small-budget compression advantage.Parametric-MI and Dist-Corr, the weakest at k=25, recover by k=150.Source: §A.1 · p. 30

A.2 Gurobi: Graph-Theoretic Methods

Graph-based methods show dataset-dependent strengths: Community-ILP leads on AmEx, while Dirac-3 QUBO and Tier 3 results vary with feature budget and solver stochasticity. On AmEx, methods generally need large feature budgets to approach the all-features baseline.

A.2 Gurobi: Graph-Theoretic Methods

  • Spectral-MI leads ULB graph methods at k=5 (F1 = 0.819), but on AmEx it falls to F1 = 0.644 at k=25; Graph-Attention reaches F1 = 0.879 at ULB k=25.On AmEx, Spectral-MI recovers to F1 = 0.800 by k=150, while Graph-Attention improves steadily to F1 = 0.801.Source: §A.2 · p. 30
  • On AmEx, Community-ILP leads every budget and nearly matches the all-features F1 baseline at k=75 (0.801 vs. 0.802); on ULB it yields graph methods’ best Gini, 96.9% at k=15.The ULB result reflects its hard diversity constraint, which requires selections from multiple Louvain communities.Source: §A.2 · p. 30

A.3 Dirac-3: QUBO Methods

  • Dirac-3 MI-Spearman leads ULB QUBO methods at k=13 (mean F1 = 0.873 ± 0.023), while Dist-Corr at k=5 ranges from F1 = 0.448 to 0.836 across runs.At k=25, Dirac-3 MI-Spearman is statistically indistinguishable from Gurobi’s exact result; the Dist-Corr spread cautions against relying on a single run.Source: §A.3 · p. 31
  • On AmEx, Dirac-3 QUBO methods show no small-budget advantage: mean F1 rises with k, reaching 0.803–0.804 by k=150, while the best k=25 result is 0.759.Parametric-MI reaches mean F1 = 0.804 ± 0.001 at k=150; MIQUBO’s relative small-k weakness differs from its Gurobi result.Source: §A.3 · p. 31

A.4 Dirac-3: Graph-Theoretic Methods

  • Community-ILP is deterministic and leads Dirac-3 graph methods at every AmEx budget and five of six ULB budgets; no AmEx graph method beats the all-features baseline before k=100.Graph-Attention narrowly leads Community-ILP at ULB k=13, within its run-to-run variation.Source: §A.4 · p. 32

A.5 Dirac-3: Advanced Methods (Tier 3)

  • On ULB, Community-Poly leads Tier 3 mean F1 (0.875 ± 0.010 at k=25); HUBO has the lowest Tier 3 variance, while Integer-Weighted has the highest.The proposed explanation for HUBO’s lower variance remains a hypothesis because the cubic-term contribution was not isolated in an ablation; Integer-Weighted’s instability is attributed to sensitivity to its weight budget.Source: §A.5 · pp. 32–33
  • On AmEx, Tier 3 methods approach F1 = 0.800–0.803 only near the 135-feature pool ceiling; the comparison with Gurobi at k=25 uses a classically pre-filtered pool, not identical inputs.Community-Poly’s best single-run F1 = 0.8044 at k=130 is the only Tier 3 cell to exceed the all-features baseline.Source: §A.5 · p. 33

B.1 Per-Method F1 Sweeps

The appendix provides per-method F1 sweeps and Gini tables for Gurobi and Dirac-3, across QUBO, graph, and Tier 3 methods. These tables supply detailed metric-by-budget results rather than additional prose findings.

B.1 Per-Method F1 Sweeps

B.2 Gini by Method

C Additional Jaccard Results

The appendix presents representative-run pairwise Jaccard matrices at selected feature budgets for ULB and AmEx. The AmEx matrices illustrate low similarity at k=25 and uniformly high similarity at k=150, with single-run identities not treated as stable findings.

C Additional Jaccard Results

C.2 AmEx

  • At AmEx k=25, the run mean similarity is 0.102, near the random-subset expectation of 0.087; at k=150, pair similarities are 0.89–0.92.Because these are single-run values, the identity of the most distinct or similar method pair is not a stable property.Source: §C.2 · p. 39, Table 40 · p. 39, Table 41 · p. 39
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