CURec: Recommender & Krylov Solver

• The paper introduces CURec as a framework that leverages LLMs to bridge the semantic-collaborative gap, generating user interest patterns and recommendation reasons with improved Recall and NDCG metrics.
• CURec integrates recommendation alignment pretraining, reward modeling, RL fine-tuning, and chronological correction to produce comprehensible and adaptive recommendations.
• In numerical linear algebra, CURec applies CUR decomposition and Krylov iteration to efficiently compute low-rank solutions for large-scale matrix equations without forming the full dense problem.

CURec is an overloaded research name used in two distinct 2025 arXiv works. In recommender systems, CURec denotes Content Understanding from a Collaborative Perspective, an LLM-centered framework for generating recommendation-perspective content features—specifically user interest patterns and item recommendation reasons—so that recommendation is both accurate and comprehensible. In numerical linear algebra, CURec denotes a CUR-based Krylov solver for large-scale linear matrix equations, combining interpolatory CUR decomposition, DEIM-based sampling, iterative decoupling, and matrix-form Krylov methods to compute low-rank solutions without forming the full dense problem (Luo et al., 11 Aug 2025, Akbari et al., 18 Nov 2025).

1. Nomenclature and domain disambiguation

The acronym CURec is attached to two unrelated methods in different research areas. One arises in LLM-enhanced recommendation, and the other in low-rank numerical algorithms for matrix equations. The shared label is therefore a matter of naming coincidence rather than methodological continuity.

Usage of CURec	Domain	Core task
Content Understanding from a Collaborative Perspective	Recommender systems	Generate collaborative-aligned reasons and patterns for comprehensible recommendation
CUR-based Krylov solver	Numerical linear algebra	Solve large-scale multi-term linear matrix equations on low-rank matrix manifolds

The recommendation paper, "Towards Comprehensible Recommendation with LLM Fine-tuning" (Luo et al., 11 Aug 2025), uses CURec to address the semantic-collaborative gap in recommendation. The numerical linear algebra paper, "A CUR Krylov Solver for Large-Scale Linear Matrix Equations" (Akbari et al., 18 Nov 2025), uses CURec to denote a solver based on CUR decomposition and Krylov/GMRES iteration. In bibliographic or technical discussion, the meaning of CURec must therefore be recovered from context.

2. CURec in recommender systems: objective and problem formulation

In the recommendation literature, CURec is introduced as a framework that uses LLMs not merely to summarize item content, but to generate recommendation-perspective content features aligned with collaborative filtering signals. Its central objective is comprehensible recommendation: the system should recommend items accurately while also exposing why a particular user would like a particular item (Luo et al., 11 Aug 2025).

The motivating diagnosis is a divide between two established families of methods. ID-based collaborative filtering models are strong at exploiting interaction histories but are characterized as memorization-based and therefore weak at recovering the semantic reasons behind preferences. Content-based or LLM-assisted methods encode item semantics, but they typically describe what an item is rather than why it matches a user. CURec frames this mismatch as the semantic-collaborative gap. The same item may be preferred for different reasons by different users, and those reasons are not directly recoverable from item text alone.

The target setting uses user interaction histories and item descriptions. For each training sample $$((u,\mathcal{S}_u),(i,\mathcal{S}_i))\in\mathcal{D}_{\text{train}}$$, the framework constructs a Recommendation Alignment Prompt $prompt_{\text{rec}}$ that asks the LLM to inspect the user’s historical item descriptions, infer the user’s interest pattern, and choose the correct target item from a candidate list $\mathcal{N}_{ui}$ containing the ground-truth item and negative items. The intended output is a structured reasoning trace with tags such as > and <answer>. This makes the model’s latent reasoning operationally relevant to downstream recommendation rather than merely explanatory after the fact.

3. Recommendation architecture: alignment, reward modeling, and chronological correction

The recommendation CURec pipeline has four coupled components: recommendation alignment pretraining, a recommendation-inspired reward model, RL fine-tuning, and chronological correction (Luo et al., 11 Aug 2025).

The first stage aligns the LLM to recommendation objectives. The paper defines the model output as

$$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$

and optimizes it with GRPO using a rule-based reward $r_{\text{pre}}$. That reward has three parts: Format Reward, which checks whether reasoning is inside <think> ... and the answer inside <answer> ... </answer>; Legal Reward, which checks whether the predicted item is in the candidate list; and Correctness Reward, which checks whether the predicted item equals the ground-truth item. The GRPO objective samples a group of responses and normalizes their rewards through a groupwise advantage,

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$

This stage produces $LLM_{\text{pre}}$, described as capable of instruction-following, recommendation-oriented chain-of-thought reasoning, user-interest inference, and recommendation-reason generation.

The second stage constructs a reward model that evaluates whether generated reasons are good from a collaborative recommendation perspective. Using $LLM_{\text{pre}}$, the framework generates a user interest pattern

$$p_u = LLM_{\text{pre}}(prompt_{\text{pattern}}, \mathcal{S}_u)$$

and a recommendation reason

$$q_{ui} = LLM_{\text{pre}}(prompt_{\text{reason}}, \mathcal{S}_u, i).$$

For each item, all generated reasons are collected into

$prompt_{\text{rec}}$0

The reward model encodes $prompt_{\text{rec}}$1 and the reason list $prompt_{\text{rec}}$2, then computes a user-item matching representation with multi-head attention,

$prompt_{\text{rec}}$3

where the user pattern is the query and the item reasons are keys and values. It integrates this with a sequential recommender,

$prompt_{\text{rec}}$4

and predicts

$prompt_{\text{rec}}$5

The third stage uses the recommender score itself as an RL reward. For a newly generated reason $prompt_{\text{rec}}$6, the paper defines

$prompt_{\text{rec}}$7

GRPO is then used again so that reasons that improve collaborative matching receive positive advantage, while a KL penalty keeps the updated policy close to a reference model.

The fourth stage is the framework’s chronological correction mechanism. After fine-tuning, the updated LLM revises both the user pattern and the item reason list through a Pattern Update Prompt $prompt_{\text{rec}}$8, producing

$prompt_{\text{rec}}$9

The reported purpose is twofold: suppress hallucinated or inaccurate reasons over time and keep user patterns and item reason lists current as interactions accumulate. The corrected features are then used directly by the downstream recommender.

4. Empirical profile of the recommendation framework

The recommendation CURec is evaluated on three public timestamped datasets: MovieLens-1M, Video Games from Amazon-2023, and Movies and TV from Amazon-2023. The preprocessing retains ratings above 3 as positive interactions, users with at least 30 interactions, and items with at least 10 interactions. The reported dataset statistics are 6,041 users, 3,952 items, 575,292 interactions, and density 2.41% for MovieLens; 15,459 users, 12,540 items, 227,575 interactions, and density 1.17‰ for Video Games; and 12,722 users, 20,551 items, 405,189 interactions, and density 1.55‰ for Movies and TV (Luo et al., 11 Aug 2025).

The experimental protocol uses leave-one-out evaluation, with the most recent interaction held out for test and full ranking over all items, not negative sampling. Metrics are Recall@K and NDCG@K for $\mathcal{N}_{ui}$0. The comparison includes ten baselines spanning traditional collaborative filtering (NCF, SASRec, Bert4Rec, DuoRec, MAERec), content-based recommendation (UniSRec, RecFormer), and LLM-enhanced recommendation (LLMRec, KAR, LEARN). Implementation details specify Qwen2.5 as the base LLM, AdamW with learning rate 5e-7 and cosine scheduling for LLM optimization, 2 epochs for LLM pretraining/correction, Adam with reward-model learning rate 1e-3, batch size 128, and embedding dimension 64.

The paper reports that CURec achieves the best overall performance on all datasets and metrics. On MovieLens, reported values include Recall@1 = 0.0447, Recall@5 = 0.1457, Recall@10 = 0.2195, Recall@20 = 0.3096, NDCG@5 = 0.0952, NDCG@10 = 0.1192, and NDCG@20 = 0.1419. On Video Games, the corresponding values include Recall@1 = 0.0212, Recall@5 = 0.0587, Recall@10 = 0.0896, Recall@20 = 0.1330, NDCG@5 = 0.0403, NDCG@10 = 0.0502, and NDCG@20 = 0.0610. On Movies and TV, reported values include Recall@1 = 0.0492, Recall@5 = 0.0721, Recall@10 = 0.0906, Recall@20 = 0.1153, NDCG@5 = 0.0602, NDCG@10 = 0.0661, and NDCG@20 = 0.0723. The reported relative gains reach +21.24% on Recall@20 and +19.70% on NDCG@20 for Movies and TV.

The ablation studies attribute the gains to the framework’s collaborative reasons rather than raw text, to the chronological correction mechanism, to the feature updating process, and to the attention-based reason integration. The paper further reports that models under 3B parameters perform much worse, which it interprets as insufficient reasoning capacity below that scale. Efficiency measurements distinguish CURec from direct LLM recommendation: Qwen2.5 API inference is reported as 7.421 s on MovieLens and 7.228 s on Video Games, whereas CURec is reported as 2.941e-3 s and 2.942e-3 s, respectively, nearly matching classical recommenders such as SASRec and MAERec. The case study on Heartbreak Ridge is used to illustrate how different user histories induce different inferred patterns and different recommendation reasons.

5. CURec in numerical linear algebra: target equations and low-rank formulation

In numerical linear algebra, CURec is a solver for large-scale multi-term linear matrix equations (LMEs) designed to compute low-rank matrix solutions without ever forming or solving the full dense problem (Akbari et al., 18 Nov 2025).

The motivating difficulty is that even when coefficient matrices are sparse, the solution matrix $\mathcal{N}_{ui}$1 may be dense and so large that storage and factorization become infeasible. The paper emphasizes this for generalized Sylvester and Lyapunov problems and for matrix differential equations arising from PDE discretizations, including representative settings with up to $\mathcal{N}_{ui}$2 unknown scalar entries. The time-dependent equation under study is

$\mathcal{N}_{ui}$3

together with its implicit Euler discretization. For exposition, the paper centers the three-term case with $\mathcal{N}_{ui}$4 and $\mathcal{N}_{ui}$5: $\mathcal{N}_{ui}$6

The solver is intended for several equation classes. These include generalized Sylvester equations

$\mathcal{N}_{ui}$7

with the classical Sylvester equation $\mathcal{N}_{ui}$8 as a special case; generalized Lyapunov equations

$\mathcal{N}_{ui}$9

reducing to $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$0 when $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$1; and non-Sylvester equations with Hadamard terms

$$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$2

The time-dependent formulation is solved on the low-rank manifold

$$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$3

The central modeling assumption is that the solution is well approximated by low rank. This is expressed through an interpolatory CUR decomposition of the unknown matrix: $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$4 with selected columns $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$5, selected rows $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$6, and coupling matrix $$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$7. The paper uses the specific interpolatory form

$$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$8

so that

$$o_{\text{rec}} = LLM(prompt_{\text{rec}}, \mathcal{S}_u, \mathcal{N}_{ui}),$$9

This exact matching on sampled rows and columns distinguishes the method from SVD-based global factorization.

6. Solver workflow: DEIM sampling, decoupled subproblems, Krylov iteration, and demonstrated scope

The numerical CURec method combines adaptive sampling with a decoupling strategy that converts the original large LME into independent reduced subproblems for selected columns and selected rows (Akbari et al., 18 Nov 2025).

The row and column indices are selected by DEIM from dominant singular vectors of the current low-rank approximation $r_{\text{pre}}$0: $r_{\text{pre}}$1 The conditioning of the sampled submatrices is quantified through

$r_{\text{pre}}$2

and the CUR-DEIM approximation error is stated as

$r_{\text{pre}}$3

with $r_{\text{pre}}$4 depending on conditioning. Because current singular vectors are not always available a priori in the time-dependent setting, the paper uses the previous iterate’s singular vectors to update $r_{\text{pre}}$5 and $r_{\text{pre}}$6.

A central contribution is the fixed-point decoupling of the selected subproblems. A naive extracted selected-column equation still depends on all of $r_{\text{pre}}$7. CURec removes this dependence via local interpolation relations,

$r_{\text{pre}}$8

where

$r_{\text{pre}}$9

This yields a closed reduced selected-column system

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$0

with

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$1

and an analogous reduced selected-row system

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$2

with

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$3

The selected row and column problems are therefore solved independently, and the full matrix is never formed.

The reduced systems are solved by matrix-form GMRES/Krylov rather than by explicit vectorization. For the selected columns, the operator is

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$4

and the right-hand side is

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$5

Although vectorization would produce a Kronecker matrix,

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$6

the solver avoids constructing it explicitly and instead works directly with matrix blocks under the Frobenius inner product

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$7

The method is also rank-adaptive. After each outer CUR iteration, it monitors the residual

$$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$8

If $$A_n = \frac{r_{\text{pre}_n} - \mathrm{mean}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}{\mathrm{std}(\{r_{\text{pre}_1}, r_{\text{pre}_2}, \cdots, r_{\text{pre}_G}\})}.$$9 is too large, the rank $LLM_{\text{pre}}$0 is increased by $LLM_{\text{pre}}$1. The stated computational advantages are memory savings of $LLM_{\text{pre}}$2 rather than $LLM_{\text{pre}}$3, scalability without an explicit full Kronecker matrix, independence of the selected subproblems, adaptivity of the rank and sample sets, and applicability beyond Sylvester structure.

The paper demonstrates the method on implicit time integration of the 3D heat equation on a low-rank manifold, steady-state generalized Lyapunov equations including a case with around $LLM_{\text{pre}}$4 scalar algebraic equations, and non-Sylvester LMEs with Hadamard terms arising from nonlinear PDEs. The solver is explicitly presented as applicable to generalized Sylvester equations, generalized Lyapunov equations, and non-Sylvester linear matrix equations with Hadamard product terms, and as scaling effectively to representative settings with up to about $LLM_{\text{pre}}$5 unknown entries. Its stated assumptions are that the solution is well approximated by low rank, the interpolation submatrices are invertible or well conditioned, the coefficient matrices are sparse or structured enough that matrix-block products are cheap, and the residual is compressible enough to monitor efficiently.