Fine-Tuning Partition-aware Similarity Refinement

• FPSR is a scalable collaborative filtering framework that partitions large item graphs and refines local similarities with global spectral methods.
• It efficiently reduces computation and storage by decomposing dense similarity matrices into block-diagonals complemented by a low-rank global component.
• FPSR and its variant FPSR+ enhance long-tail recommendations through tailored hub augmentation and balanced partitioning strategies.

Fine-tuning Partition-aware Similarity Refinement (FPSR) is a scalable framework for collaborative filtering (CF) that combines partition-based modeling of item–item similarities with spectral global refinement. FPSR efficiently addresses the quadratic cost of dense similarity learning by decomposing the item graph, selectively fine-tuning subgraphs, and re-incorporating long-range dependencies via spectral components. This approach achieves competitive accuracy, interpretability, and memory efficiency, with demonstrated advantages for long-tail item recommendation in large catalogs (Gioia et al., 18 Dec 2025, Wei et al., 2022).

1. Motivation and Problem Framework

Classic similarity-based CF models, such as SLIM and EASEr, define a user–item interaction matrix $R\in\mathbb{R}^{U\times N}$ and infer a dense item–item similarity matrix $S\in\mathbb{R}^{N\times N}$ to drive recommendations. However, as the number of items $N$ increases, a dense $S$ entails $O(N^2)$ parameters and memory, hampering scalability and efficiency (Gioia et al., 18 Dec 2025, Wei et al., 2022). FPSR circumvents this by recursively partitioning the item graph defined by the co-occurrence matrix $A=R^T R$, resulting in restricted, locally-learned similarities and a global low-rank refinement that captures essential cross-partition information.

2. Model Architecture and Partitioning

FPSR operates through a combination of graph partitioning, local similarity refinement, and global spectral adjustment:

• Item Graph Construction: Define $A=R^T R$; $A_{ij}$ counts shared user interactions for items $i$ and $j$.
• Partitioning: Recursively split the item set using balanced spectral methods (e.g., Fiedler vector bisection), enforcing that no partition exceeds $\tau \cdot N$ items, with $\tau \in (0,1]$. This yields $K$ disjoint parts $P_1,\ldots,P_K$ of sizes $M_k \ll N$.
• Block-diagonal Similarity Matrix: FPSR constructs $$S^{loc} = \mathrm{blockdiag}(S^{(1)},\ldots,S^{(K)})$$, where $S^{(k)} \in \mathbb{R}^{M_k\times M_k}$ are refined within-partition similarity blocks.
• Global Spectral Component: The global low-rank matrix $W$ is extracted using truncated eigendecomposition (top $d$ eigenvectors) of $A$, $W=V_d \Lambda V_d^T$, and addresses inter-partition correlations absent from $S^{loc}$.
• Combined Refined Similarity: The full model combines local and global components: $C = S^{loc} + \lambda W$, with $\lambda\in[0,1]$ regulating the blend.

3. Mathematical Formulation and Optimization Objectives

The FPSR objective consists of local and global terms:

• Local Block-wise Learning: Each partition solves, independently and in parallel,

$$L_{local}^{(k)}(S^{(k)}) = \|A^{(k)} - S^{(k)}\|_F^2 + \alpha \|S^{(k)}\|_F^2,$$

where $A^{(k)} = A_{P_k, P_k}$, a standard blockwise ridge regression.

• Global Spectral Refinement: The global component is obtained through

$$L_{global}(W) = \|A - S^{loc} - \lambda W\|_F^2 + \beta \|W\|_F^2,$$

with $W$ computed via the top spectral components of $A$.

• Alternating Minimization: The total fine-tuning objective is

$$L_{total}(\{S^{(k)}\}, W) = \sum_{k=1}^K L_{local}^{(k)}(S^{(k)}) + \gamma L_{global}(W),$$

optimized by cycling between solving for $S^{(k)}$ with $W$ fixed, and updating $W$ via truncated SVD of $A - S^{loc}$ (Gioia et al., 18 Dec 2025).

4. Algorithmic Variants and Data Handling

FPSR and its variant with hubs, FPSR+, are outlined as follows:

Step	FPSR (Base)	FPSR+ (Hubs)
1	PartitionItems$(A, \tau)$	PartitionItems$(A, \tau)$
2	Local block learning	SelectHubs$(A, P, h, \text{hub\_strategy})$
3	Spectral decomposition for $W$	Augment each $P_k$ with hubs, learn $S^{(k)}$
4	Combine $S^{loc}$ and $\lambda W$ to yield $C$	Spectral decomposition for $W$, combine for final $C$

• Hub Augmentation: FPSR+ introduces a "hub" set of $h$ bridge items selected by degree or Fiedler strategies, augmenting each partition, further connecting the blocks and improving long-tail performance (Gioia et al., 18 Dec 2025).
• User-based Hold-out: Each user's interactions are split into 15% test, 15% validation, and 70% training; evaluation is via Recall@K and nDCG@K, for $K=10,20$.
• Long-tail Analysis: Metrics are computed separately for head (top 10% items) and tail (remaining 90%), revealing the model's behavior across frequency regimes.

5. Empirical Performance, Trade-offs, and Applications

FPSR exhibits the following empirical characteristics (Gioia et al., 18 Dec 2025, Wei et al., 2022):

• Comparative Results: FPSR and FPSR+ are highly competitive—FPSR+ is second only to BISM on Amazon-CDs, and FPSR+ is best on Douban, Gowalla, and Yelp2018 (Recall@20 / nDCG@20). For example, on Douban, FPSR+ₙₑₜ(D) achieved 0.2132 / 0.1928.
• Long-tail Recommendation: FPSR consistently outperforms BISM by 10–20% relative Recall for tail items (e.g., Gowalla-tail: FPSR+ₙₑₜ(F) 0.1149 vs. BISM 0.1081).
• Efficiency: FPSR provides strong speedups (≈10× faster training than leading GCNs) and up to ≈95% parameter storage savings compared to dense similarity approaches (Wei et al., 2022).
• Trade-offs: Lower $\tau$ reduces per-block computation but can degrade signal if blocks are too fine; higher $\lambda$ enhances global coverage but blurs local precision. FPSR+ (with hubs) is robust under popularity skew and often essential for stabilizing tail coverage.

Hyperparameter	Typical Values	Effect
$\tau$	0.1–0.5 (best: 0.3–0.5)	Partition granularity, compute footprint
$\lambda$	0.1–0.5 (best: $\approx$ 0.3)	Global vs. local signal
$\alpha$	$10^{-3}$–$10^{-4}$	Local ridge regularization
$d$	50–100	Rank for $W$
hub_size $h$	0.01$N$ or fixed (e.g., 500 items)	Connectivity in FPSR+
hub_strategy	"degree", "Fiedler"	Head/tail trade-off (FPSR+ₙₑₜ(D) for head, FPSR+ₙₑₜ(F) balanced)

FPSR is appropriate for large catalogs ($N\gg 10^4$) where full dense similarity learning is infeasible, and where long-tail coverage is a priority (e.g., e-commerce, batch recommender deployments).

6. Relation to Prior Work and Evaluation Protocol

FPSR extends and contrasts with several paradigms:

• Graph Convolutional CF: GCN-based CF captures high-order relationships through deep graph structures but suffers from inefficiency and over-smoothing; FPSR achieves comparable or superior accuracy with much faster (10×) training and markedly lower model complexity (Wei et al., 2022).
• Block-aware and Dense Similarity Models: Compared to BISM and classic SLIM/EASEr, FPSR retains block-wise interpretability and manages scalability via heuristic partitioning and block-diagonalization.
• Evaluation Protocol: Recent work emphasizes fair and reproducible evaluation. FPSR's assessment relies on user-based hold-out with transparent reporting of head/tail metrics and rigorous hyperparameter logging (Gioia et al., 18 Dec 2025).

7. Practical Recommendations and Reproducibility Considerations

To employ FPSR effectively:

• Partition Ratio: Begin with $\tau\simeq 0.3$–$0.5$. Finer granularity aids compute, but risks block sparsity.
• Global Weight: Set $\lambda$ to $0.2$–$0.4$ to balance global coverage and local accuracy.
• Hubs: Use FPSR+ with degree-based hubs for optimizing head-item accuracy or Fiedler-based hubs for more balanced gains.
• Reproducibility:
  • Fix RNG seeds for partitioning and data splits.
  • Use the user-based hold-out protocol (15% test, 15% validation).
  • Report separate Recall@K/nDCG@K for head, tail, and overall.
  • Log all hyperparameters: $\tau$, $\lambda$, $\alpha$, $h$, hub strategy, $d$ (Gioia et al., 18 Dec 2025).

A plausible implication is that, by explicitly decomposing the item graph and integrating global spectral structure, FPSR provides a scalable, interpretable framework for large-scale recommendation, achieving robust performance across both frequent and rare items, with practical trade-offs governing accuracy and computational cost.

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References

• (Gioia et al., 18 Dec 2025) A Reproducible and Fair Evaluation of Partition-aware Collaborative Filtering
• (Wei et al., 2022) Fine-tuning Partition-aware Item Similarities for Efficient and Scalable Recommendation