WPGRec: Wavelet Packet Guided Graph Enhanced Sequential Recommendation

Abstract: Sequential recommendation aims to model users' evolving interests from noisy and non-stationary interaction streams, where long-term preferences, short-term intents, and localized behavioral fluctuations may coexist across temporal scales. Existing frequency-domain methods mainly rely on either global spectral operations or filter-based wavelet processing. However, global spectral operations tend to entangle local transients with long-range dependencies, while filter-based wavelet pipelines may suffer from temporal misalignment and boundary artifacts during multi-scale decomposition and reconstruction. Moreover, collaborative signals from the user-item interaction graph are often injected through scale-inconsistent auxiliary modules, limiting the benefit of jointly modeling temporal dynamics and structural dependencies. To address these issues, we propose Wavelet Packet Guided Graph Enhanced Sequential Recommendation (WPGRec), a unified time-frequency and graph-enhanced framework that aligns multi-resolution temporal modeling with graph propagation at matching scales. WPGRec first applies a full-tree undecimated stationary wavelet packet transform to generate equal-length, shift-invariant subband sequences. It then performs subband-wise interaction-graph propagation to inject high-order collaborative information while preserving temporal alignment across resolutions. Finally, an energy- and spectral-flatness-aware gated fusion module adaptively aggregates informative subbands and suppresses noise-like components. Extensive experiments on four public benchmarks show that WPGRec consistently outperforms sequential and graph-based baselines, with particularly clear gains on sparse and behaviorally complex datasets, highlighting the effectiveness of band-consistent structure injection and adaptive subband fusion for sequential recommendation.

• The paper introduces WPGRec, which combines stationary wavelet packet decomposition with subband-wise Chebyshev graph propagation to align temporal and collaborative signals across multiple resolutions.
• WPGRec outperforms seven baselines on all four benchmarks under full-ranking evaluation, achieving relative HR@10 gains of 10.3% on Beauty, 9.5% on Sports, 7.8% on LastFM, and 2.5% on ML-1M.
• The results show that energy–spectral-flatness gating and boundary handling improve robustness, while the method’s computational cost grows with the number of wavelet subbands and may limit large-scale deployment.

Motivation and problem statement

Sequential recommendation requires modeling user preference evolution from interaction streams that mix long-term preferences, medium-range transition patterns, and short-term fluctuations. The authors of WPGRec identify two structural deficiencies in prior work. First, time-domain sequential models (GRU4Rec, SASRec, BERT4Rec, BSARec) encode heterogeneous temporal factors in a single feature stream, so when behavior is sparse or noisy, low-frequency long-term signals dominate and short-term localized patterns are attenuated. Second, frequency-domain methods such as FEARec and WaveRec, while better suited to non-stationary behavior, remain sequence-centric: collaborative graph signals are injected only after temporal encoding, through scale-inconsistent auxiliary modules. This creates a scale mismatch — graph propagation operates on representations in which different temporal resolutions have already been entangled, weakening both multi-resolution modeling and boundary-noise control.

The central design principle of the paper is that multi-resolution temporal modeling and graph propagation should be aligned at the same temporal scale. WPGRec operationalizes this by decomposing sequences into equal-length, shift-invariant subbands via a full-tree undecimated Stationary Wavelet Packet Transform (SWPT), propagating collaborative information within each subband over the user–item bipartite graph, and adaptively fusing subbands with an energy- and spectral-flatness-aware gate.

Method

WPGRec proceeds in five stages.

Boundary handling and SWPT decomposition. Each user's item-embedding matrix $\mathbf X_u \in \mathbb R^{T_u \times d}$ is extended with symmetric reflection plus $\tau$ learnable boundary tokens on both sides to mitigate boundary artifacts at sequence endpoints. A full-tree undecimated SWPT of depth $\ell$ then produces $B = 2^\ell$ subbands $\{Z_u^{(b)}\}$, each of length $T_u' = T_u + 2\tau$. The undecimated (stationary) construction is critical: unlike downsampled wavelet pipelines, all subbands retain full temporal resolution and shift invariance, which is what makes temporally aligned propagation and fusion possible without resampling or interpolation.

Subband statistics and attention summarization. For each subband, the model computes per-user energy $E_u^{(b)}$ (average squared magnitude) and spectral flatness $\mathrm{SFM}_u^{(b)}$ along the temporal axis. These descriptors are not used as features directly; they later drive gating. Additive attention aggregates token-level subband features into a compact user summary $p_u^{(b)}$, stacked across users into a per-subband user feature matrix.

Subband-wise Chebyshev graph propagation. For each subband $b$, node features are initialized by concatenating the user-side summaries with the shared item embedding matrix, and $\tau$0 layers of Chebyshev polynomial filtering are applied on the normalized graph Laplacian:

$\tau$1

This is the key mechanism for band-consistent collaborative enhancement: high-order collaborative structure is injected within each frequency band rather than after cross-scale mixing. The paper uses $\tau$2 and $\tau$3 throughout.

Energy–SFM gated fusion. A per-user descriptor $\tau$4 is passed through an MLP followed by softmax to produce gate weights $\tau$5, which weight the fused user representation. Spectral flatness serves as a noise-likeness indicator, allowing the model to suppress noise-dominated subbands — a rationale borrowed from audio fingerprinting literature. Item representations are fused by uniform averaging across subbands, since no user-specific signal is available on the item side; this asymmetry is a deliberate simplification.

Training. Scoring is inner-product based, trained with full-softmax cross-entropy over the entire item set plus a gating regularization term ($\tau$6) and $\tau$7 weight decay. Notably, no negative sampling is used in either training or evaluation.

The reported complexity is $\tau$8 per epoch. The linear dependence on $\tau$9 means decomposition depth directly multiplies both SWPT/attention cost and graph-propagation cost — a nontrivial overhead relative to single-scale baselines, though bounded in practice by small $\ell$0.

Experimental results

Experiments use four benchmarks under a strict full-ranking protocol (ground-truth item ranked against the entire catalog, excluding seen items), with leave-one-out chronological splits following BSARec preprocessing. Baselines span both paradigms — sequence models (BERT4Rec, BSARec, FEARec, WaveRec) and graph-based models (SGL, TGSRec, DGRec) — re-implemented under identical splits, training objectives, and tuning budgets.

Dataset	Best baseline	WPGRec HR@10	Rel. gain (HR@10)	Largest metric gain
Beauty	DGRec (0.1085)	0.1197	+10.3%	NDCG@10 +8.0%
Sports	DGRec (0.0643)	0.0704	+9.5%	HR@20 +10.7%
LastFM	DGRec (0.0785)	0.0847	+7.8%	HR@20 +15.4%
ML-1M	DGRec (0.3455)	0.3542	+2.5%	NDCG@20 +0.6%

Two patterns stand out. First, WPGRec wins every metric on every dataset against seven baselines, with the strongest baseline consistently being DGRec among graph methods. Second, gains correlate inversely with density: improvements exceed 8% on the sparse Amazon datasets (sparsity >99.9%) and LastFM, but shrink to under 3% on dense ML-1M (95.16% sparsity, average length 165.5). The authors attribute this to sparse, temporally heterogeneous behavior benefiting most from explicit scale separation and band-consistent structure injection. An implication worth noting: the marginal value of multi-resolution decomposition appears to diminish once interactions are dense enough that a single representation suffices, suggesting the framework's advantage is conditional on data regime rather than universal.

Ablations confirm that each component contributes. Removing the SWPT module (collapsing to $\ell$1, pure time domain) causes the largest degradation, establishing multi-resolution separation as the foundational element. Removing subband-wise GNN propagation yields the next largest drop, supporting the claim that collaborative information is most effective when injected within aligned subbands. Replacing adaptive gating with uniform averaging, disabling intra-subband attention, and removing boundary handling each cause moderate but consistent declines, with boundary handling mattering most for short/sparse sequences.

Sensitivity analysis over the gating regularization coefficient $\ell$2 shows a stable plateau around $\ell$3–$\ell$4, with best performance at $\ell$5 and mild decline at $\ell$6 attributed to over-concentration of gate weights. Performance differences across this range are small (e.g., Beauty HR@10 varies from 0.1187 to 0.1197), indicating the method does not hinge on delicate hyperparameter tuning of this term.

Limitations and open questions

The paper concedes several constraints explicitly. The framework has not been optimized for efficiency in large-scale or online settings; the complexity scales multiplicatively with the number of subbands $\ell$7, and running $\ell$8 separate Chebyshev propagation passes per epoch may be prohibitive at industrial scale. The evaluation is limited to four moderately sized benchmarks (largest: ~25K users), leaving scalability unverified. Additionally, the asymmetric fusion design — learned gating for users but uniform averaging for items — is justified only implicitly, and the paper does not analyze whether learned item-side gates would help. The choice of wavelet family (Coiflets vs. Symlets), depth $\ell$9, and boundary length $B = 2^\ell$0 is resolved by grid search; whether adaptive selection of decomposition depth per user or per sequence would improve over fixed global settings remains an open question the authors flag for future work. Finally, although ablations support each component, the paper does not provide direct evidence (e.g., visualization of learned gate weights or subband interpretability) that the gating mechanism actually suppresses noise-like subbands as intended, leaving the mechanistic claim behind spectral-flatness gating empirically under-examined.

Conclusion

WPGRec unifies stationary wavelet packet decomposition, subband-wise Chebyshev graph propagation, and energy–spectral-flatness-aware gated fusion into a single framework whose defining property is scale alignment between temporal decomposition and collaborative structure injection. Under a rigorous full-ranking protocol with no sampled negatives, it consistently outperforms strong sequential and graph-based baselines across four benchmarks, with the largest gains (+15.4% HR@20 on LastFM) occurring on sparse, behaviorally complex data and the smallest on dense data. The ablation and sensitivity studies substantiate the contribution of each component and the robustness of the gating regularization. The main open issues are computational scalability of multi-subband propagation, adaptive rather than fixed decomposition depth, and empirical validation of the noise-suppression mechanism underlying the fusion design.