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REVIEW 3 major objections 6 minor 89 references

Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that the Hadamard Product's power in click-through rate prediction comes from the quadratic feature space it induces, and uses that insight to build QNN-alpha, a multi-head Khatri-Rao product neuron with self-ensemble…

desk verdict A solid empirical study of QNN formats for CTR whose central mechanistic claim about the Khatri–Rao product collapses on inspection; the empirical results may still hold, but the paper overstates what its KRP actually does. read the letter →

arxiv 2505.17999 v2 pith:QAJOBZND submitted 2025-05-23 cs.IR cs.LG

Honghao Li , Yiwen Zhang , Yi Zhang , Lei Sang , Jieming Zhu This is my paper · ORCID
classification cs.IRcs.LG
keywords QuadraticNeuralNetworksClick-throughRatePredictionFeatureInteractionsHadamardProductKhatri-RaoSelf-EnsembleLossRecommenderSystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the Hadamard Product, the standard element-wise multiplication used to combine feature embeddings in click-through rate (CTR) models, works for a deeper reason than usually assumed: it turns a linear layer into a quadratic neuron, expanding the feature space and giving smooth nonlinear decision boundaries without any activation function. On that diagnosis the paper builds a new neuron format, QNN-alpha, whose core is a Multi-Head Khatri-Rao Product that generalizes the Hadamard Product, plus a Self-Ensemble Loss that averages two dropout-perturbed forward passes at training time. The paper reports that QNN-alpha outperforms existing models on all six public datasets tested while keeping inference latency low enough for production use. If correct, the result gives CTR practitioners a drop-in interaction block that is both more accurate and no slower than components already deployed.

What carries the argument

The central object is the quadratic neuron, a unit whose output contains linearly independent quadratic polynomials of its inputs, not just a linear map followed by an activation. It carries the argument because it unifies the Hadamard Product, MLP, and existing CTR interaction layers into one design space. The improved machinery is the Multi-Head Khatri-Rao Product, a column-wise Kronecker product with sum pooling that reduces to the Hadamard Product at rank $M=1$, scales capacity through the rank $M$, and cuts parameter and compute cost by a factor of $H$ through head splitting. The other load-bearing piece is the Self-Ensemble Loss, which runs two dropout-perturbed forward passes and adds a consistency term between their predictions, giving ensemble regularization without extra sub-networks.

What would settle it

Train QNN-alpha and the strongest baseline ten times with different random seeds on Tenrec and Criteo; if the reported gains of 0.04 and 0.08 AUC points fall within the run-to-run spread, the central state-of-the-art claim fails. Independently, replace the Khatri-Rao product with a linear layer of equal parameter count while keeping the residual and mid-activation structure; if accuracy matches QNN-alpha, the expressiveness mechanism is falsified.

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Extended reading notes

Core claim

The central claim is that the Hadamard Product's effectiveness is not a property of element-wise multiplication itself but of the quadratic-neural-network structure it induces: each layer generates linearly independent second-order polynomials, which expands the feature space and supplies smooth nonlinear approximation without activation functions. From experiments on 25 QNN neuron formats, the paper concludes that placing ReLU after the whole quadratic neuron (post-activation) is unnecessary and often hurts, while placing it inside the layer before the product (mid-activation) helps; the strongest basic format is T19, written $\Phi(X)=X\odot \mathrm{ReLU}(W_a X)+X$. QNN-$\alpha$ improves T19 by replacing the Hadamard Product with a sum-pooled Multi-Head Khatri-Rao Product, $\Psi(X_h)=X_h\bullet \mathrm{ReLU}(W^h_a X)+X_h$, and by adding a Self-Ensemble Loss that enforces consistency between two stochastic forward passes of the same network. The paper claims this single network achieves new state-of-the-art results on six public datasets and that it can replace the interaction block or MLP of existing models with consistent gains.

Load-bearing premise

The whole result depends on a new feature-combination operation being genuinely more expressive than the old element-wise multiplication, not just bigger or better tuned, and on the small accuracy gains on the largest datasets surviving repeated runs from different random seeds.

Editorial extensions

If this is right

  • Existing Hadamard-product layers can be understood as quadratic neurons, which gives a principled explanation for their success and a design vocabulary (mid-activation, feature-space expansion) for improving them.
  • Removing post-activation and using mid-activation helps or matches in most of the 25 neuron formats compared, so QNN-style interaction layers do not need a nonlinearity on the output.
  • The Multi-Head Khatri-Rao Product is a drop-in replacement for the Hadamard Product that scales capacity through $M$ and reduces cost through $H$, with $M=1$ degenerating to the original operation.
  • Self-Ensemble Loss improves accuracy using only two forward passes during training and no extra sub-networks, so the inference cost stays that of a single network.
  • QNN-alpha can replace the MLP or interaction block of models such as xDeepFM, DeepFM, and DCNv2 and improves them, so the architecture can be adopted incrementally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's smooth-versus-sharp boundary argument is qualitative; a direct test would measure decision-boundary curvature or local Lipschitz constants for HP-based versus activation-based layers on toy data to check whether smoothness is the actual mechanism.
  • If feature-space expansion is the true source of the gains, the same Multi-Head Khatri-Rao substitution should transfer to product-based interaction models outside CTR, such as matching or ranking towers, which the paper does not test.
  • Because Self-Ensemble Loss is a consistency regularizer over dropout stochasticity, a cheaper variant using two dropout masks inside one forward pass rather than two full forward passes is a natural untested simplification.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes QNN-alpha, a CTR prediction architecture built on a quadratic-neural-network view of feature interactions. It revisits Hadamard-product interaction layers (e.g., DCNv2, ECN-like T9) and derives equivalence formulations showing that these layers expand the feature space through quadratic polynomials. After evaluating 25 QNN neuron formats on Criteo, the authors conclude that post-activation is unnecessary for QNN while mid-activation is beneficial, and they identify the T19 format as a strong performer. They then introduce two enhancements: a Multi-Head Khatri-Rao Product (MH-KRP) that is claimed to be a more expressive alternative to the Hadamard product, and a Self-Ensemble Loss (SE Loss) that ensembles two dropout views of the same network during training. Experiments on six public datasets report state-of-the-art AUC/Logloss, plus ablation, scalability, compatibility, and latency studies.

Significance. If the empirical results hold, QNN-alpha is a strong and deployable CTR architecture, and the paper provides a useful taxonomic perspective that connects existing CTR interaction layers to the QNN literature. The paper ships code, running logs, and hyperparameter configurations; the theoretical rewrites in Eqs. (2) and (4) are simple and largely correct; and the 25-format investigation plus the compatibility and latency analyses are valuable. However, the central mechanistic claim about MH-KRP is compromised by the sum-pooling collapse identified below, and the statistical evidence on two of the six datasets is weaker than the paper's own significance threshold. The SE Loss is also presented as a new contribution but is very close to existing consistency/self-distillation losses. These issues do not necessarily invalidate the reported results, but they must be addressed before the SOTA and mechanism claims can be accepted.

major comments (3)
  1. [§3.1–3.2, Eqs. (8)–(10), Fig. 7] The sum-pooled 'Khatri-Rao Product' is not a rank-M generalization of the Hadamard product. In Eq. (8), C_j = sum_p A_j B_{p,j} = A_j (sum_p B_{p,j}), so the M dimension is summed away before the elementwise interaction with A_j. Consequently, Eq. (10) is algebraically Psi(X_h) = X_h ⊙ (sum_{p=1}^M ReLU((W_h)_p X_h)) + X_h, i.e., a Hadamard product with a wider ReLU projection, not a columnwise Kronecker rank expansion. The claim in Eq. (9) that KRP expands the feature space to D^2 interaction items is therefore not supported for a single layer, and the phrase 'rank expansion' is misleading. The comparison M=1 vs M>1 in Fig. 7 varies the parameter count of W roughly M-fold without a matched-capacity HP baseline, so it does not isolate a KRP structural advantage. The 'w/o KRP' ablation in Table 4 replaces KRP with a linear transformation, which also fails to distinguish KRP from HP. Please either reframe the contribution as a wider nonlinear projection with capacity-matched baselines, or modify the operation so that the M dimension participates in the interaction before pooling.
  2. [§4.2, Table 3] The paper reports a two-tailed T-test (star: p < 1e-3) but gives no standard deviations, no number of seeds, and no statement of whether the test is over samples or over repeated runs. If the test is over test samples, it does not address sensitivity to initialization. On Tenrec and Criteo, the AUC gains over the best baseline are +0.04 and +0.08 points (0.0004 and 0.0008 absolute), and the Logloss gains are 0.0006 on both datasets, which is below the 0.001 (0.1%) threshold that the paper itself cites as practically significant. The 'new state-of-the-art' claim on these two datasets is therefore not established without per-seed results. Please report mean plus/minus standard deviation over multiple seeds and exact p-values, and clarify the test procedure.
  3. [§3.3, Eq. (11)] The Self-Ensemble Loss is presented as a new dynamic ensemble method, but as written it is the sum of two cross-entropy terms between a shared soft target y_pred_tilde = stopgrad((y_pred1 + y_pred2)/2) and the two dropout views: L_SE = CE(y_pred_tilde, y_pred1) + CE(y_pred_tilde, y_pred2). This is very close to R-Drop's consistency regularization and to the authors' own Tri-BCE, which are cited but not formally differentiated. Since SE Loss is one of the three named contributions, please provide an explicit comparison (e.g., show how the loss decomposes relative to R-Drop's KL terms and hard-label CE) or temper the novelty claim accordingly.
minor comments (6)
  1. [Table 3] The 'Abs.Imp' row is signed for Logloss (negative means improvement) but unsigned for AUC; this is confusing and should be clarified with a sign convention or by using absolute differences with a direction indicator.
  2. [Eq. (8)] The pooling operation is written as 'sum' without specifying the summation dimension; please define it explicitly and avoid calling the result a Khatri-Rao product without qualification, since the standard KRP's columnwise Kronecker structure is destroyed by the summation.
  3. [§3.3] The text says additional forward passes 'exponentially increase' training cost; the cost increases linearly with the number of forward passes, not exponentially.
  4. [Table 1] The 'Time x Epochs' column (e.g., '2min x 11') is ambiguous; please specify whether this is total wall-clock time, per-epoch time, or the product of per-epoch time and epoch count.
  5. [Table 4, footnote 12] The w/o Res entries show backslashes with only a footnote saying the network fails to train; this is easy to miss and should be stated in the table caption or in the main text.
  6. [References] Reference [42] has a stray period and duplicate '(2023)'; the baseline list in §4.1.3 also has inconsistent year formatting (e.g., xDeepFM appears as '(2018)' twice).

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the core derivation is self-contained, with only minor non-load-bearing self-citations; the sum-pooled-KRP concern is an evidence gap, not circularity.

full rationale

The claimed derivation chain does not reduce to its inputs. Section 2.1's QNN analysis is an algebraic reinterpretation of DCNv2 (Eq. 1-2) and ECN-like T9 (Eq. 4-5), showing quadratic polynomial terms; it does not assume HP's effectiveness. The 25-format screen (Table 1) is an empirical search, and QNN-alpha's later benchmark (Table 3) is an evaluation, not a fitted parameter relabeled as a prediction. Equations (8)-(10) explicitly define the multi-head KRP neuron; the assertion that KRP is 'more expressive' than HP is an empirical hypothesis tested by M sweeps (Figure 7) and ablations (Table 4), not a conclusion derived from the definition. It is true that sum-pooling algebraically collapses the Khatri-Rao rank so that Ψ(X_h)=X_h⊙Σ_p ReLU((W_h)_pX_h)+X_h, which weakens the paper's mechanistic claim; however, this is an evidentiary/interpretation defect, not a circular reduction. The Self-Ensemble Loss (Eq. 11) uses the network's own frozen outputs as soft targets; this is a self-distillation training regularizer evaluated against other losses (Figure 6), so it is not circular evidence for model quality. Self-citations, principally ECN/Tri-BCE [32], are used for context and inspiration but are not load-bearing: T19's selection and SE Loss's effectiveness are demonstrated in this paper's own experiments, and the central SOTA claim is benchmarked against external baselines. Accordingly the circularity score is at the floor: minor self-citation overlap without any load-bearing reduction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central empirical claim depends on tuned hyperparameters (embedding size, M, H, L, dropout, learning rate) and on the benchmark assumption that 0.1% metric shifts matter. No new physical entities are introduced; the KRP and SE Loss are architectural and training constructs. The core algebraic rewrites are standard math, so the main listed axioms concern design choices and empirical conventions.

free parameters (6)
  • Embedding dimension d = 16 for most datasets, 128 for KKBox
    Follows prior benchmarks [79,81]; controls input capacity.
  • Khatri-Rao expansion size M = Tuned per dataset; larger M improves Tenrec and Frappe in Figure 7
    Controls model capacity in Equation (9).
  • Number of heads H = H=4 for ML-1M and iPinYou, H=2 for KKBox (Table 6)
    Multi-head split in Equation (10); affects both performance and latency.
  • Network depth L = Optimal L=2 for Tenrec, L=4 for Criteo, L=1 for ML-1M (Figure 4)
    Depth of KRP interaction layers.
  • Dropout rate = 0.1 recommended
    Stochastic source for SE Loss; discussed in Section 4.3.4.
  • Mid-activation choice = ReLU
    Compared in Figure 5; no single activation wins across all datasets, yet ReLU is used in QNN-alpha.
assumptions (5)
  • standard math Quadratic neural networks are universal approximators and provide smooth nonlinear approximation capability.
    Invoked in Sections 1 and 2.1 via [13,74] to justify QNN's approximation power.
  • domain assumption ReLU's non-smoothness at zero explains sharp MLP decision boundaries.
    Used in Section 2.1.2 to interpret decision-boundary plots; not proven for this model class.
  • ad hoc to paper Sum-pooled Khatri-Rao product preserves interaction information and is a generalized Hadamard product.
    Equations (8)-(10) define the architecture; no theorem guarantees that sum pooling keeps all useful interaction structure.
  • domain assumption Two dropout forward passes form a beneficial ensemble.
    Basis of SE Loss (Equation 11), justified empirically with reference to R-Drop [69] and Tri-BCE [32].
  • domain assumption A 0.1% improvement in AUC or logloss is statistically and practically significant in CTR benchmarks.
    Stated in Table 1 and Table 3 captions and used to declare SOTA; no variance analysis is provided to support it.

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Cite this review

Pith. "Pith review of Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction." pith.science (2026). https://pith.science/paper/QAJOBZND

@misc{pith2026250517999,
  author       = {Pith},
  title        = {Pith review of: Revisiting Feature Interactions from the Perspective of Quadratic Neural Networks for Click-through Rate Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QAJOBZND}},
  note         = {Machine review of arXiv:2505.17999}
}
read the original abstract

Hadamard Product (HP) has long been a cornerstone in click-through rate (CTR) prediction tasks due to its simplicity, effectiveness, and ability to capture feature interactions without additional parameters. However, the underlying reasons for its effectiveness remain unclear. In this paper, we revisit HP from the perspective of Quadratic Neural Networks (QNN), which leverage quadratic interaction terms to model complex feature relationships. We further reveal QNN's ability to expand the feature space and provide smooth nonlinear approximations without relying on activation functions. Meanwhile, we find that traditional post-activation does not further improve the performance of the QNN. Instead, mid-activation is a more suitable alternative. Through theoretical analysis and empirical evaluation of 25 QNN neuron formats, we identify a good-performing variant and make further enhancements on it. Specifically, we propose the Multi-Head Khatri-Rao Product as a superior alternative to HP and a Self-Ensemble Loss with dynamic ensemble capability within the same network to enhance computational efficiency and performance. Ultimately, we propose a novel neuron format, QNN-alpha, which is tailored for CTR prediction tasks. Experimental results show that QNN-alpha achieves new state-of-the-art performance on six public datasets while maintaining low inference latency, good scalability, and excellent compatibility. The code, running logs, and detailed hyperparameter configurations are available at: https://github.com/salmon1802/QNN.

Figures

Figures reproduced from arXiv: 2505.17999 by the authors.

Figure 1
Figure 1. Comparison of MLP and QNN. question worthy of further exploration: Are Activation Functions Necessary for HP-based CTR Prediction? To answer this question, we empirically analyze 25 different neuron formats of QNN, as discussed in Section 2.2 and 2.3, claim the connections between existing CTR models and QNN, and iden￾tify the T19 variant as a strong performer. Moreover, we find that, in most cases, introducing post… view at source ↗
Figure 2
Figure 2. Comparison of decision boundaries for 2D moon (line 1) and circle (line 2) datasets in different neuron formats. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Architecture of the QNN-𝛼. future research. We recommend readers to refer to [3, 8, 10, 12, 13] for a broader and deeper understanding of QNN. 3 A Better QNN Design for CTR prediction Based on the conclusions in Section 2, we believe that QNN holds significant potential for improving CTR model performance but remains underexplored. Therefore, in this section, we aim to further understand the underlying mechanisms of… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Performance of different 𝐿 in QNN-𝛼. ReLU Mish GELU ReLU6 Sigmoid 89.0 89.5 90.0 90.5 91.0 91.5 AUC (%) 90.87 90.88 90.52 90.45 89.35 AUC 0.295 0.300 0.305 0.310 0.315 Logloss Logloss (a) ML-1M ReLU Mish GELU ReLU6 Sigmoid 98.0 98.2 98.4 98.6 98.8 99.0 AUC (%) 98.62 98…
Figure 7
Figure 7. Figure 7: Performance of different 𝑀 in QNN-𝛼 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.