Topology-aware Positive Sample Construction

• Topology-aware positive sample set construction is a framework that uses data topology such as co-occurrence, community structure, and persistent regions to redefine positive supervision.
• The approach spans recommendation and active learning, employing strategies like auxiliary positives in graph contrastive learning, community-based relabeling, and region-level label propagation.
• Empirical results indicate performance improvements in various tasks, though the methods demand extra computation for graph construction, community detection, and parameter tuning.

Searching arXiv for the specified papers and closely related work to ground the article. Topology-aware positive sample set construction denotes a family of learning strategies in which positive supervision is expanded or reorganized using structural information derived from data topology rather than relying solely on observed labels, uniform negative sampling, or the current model representation. In the literature represented by recommendation and active learning, the term refers to distinct but related operations: converting semantically similar false negatives into auxiliary positives in graph contrastive learning, re-labeling topologically plausible missing interactions as positives in implicit collaborative filtering, and constructing labeled sets from topological regions in pool-based active learning (Han et al., 2023, Wu et al., 20 Feb 2026, Hadjadj et al., 2023). Across these settings, the unifying premise is that proximity induced by interaction topology, community structure, or persistent topological regions can reveal latent positive relations that conventional sampling procedures neglect.

1. Conceptual scope and problem formulation

In recommendation, topology-aware positive sample set construction arises from the observation that many unobserved user–item pairs or randomly sampled node pairs are not genuine negatives. In graph-based collaborative filtering with graph contrastive learning, random negative sampling can introduce false negatives, defined as negatives that are similar to the anchor user or item, while simultaneously ignoring potential positive samples (Han et al., 2023). In implicit collaborative filtering, unobserved entries in the interaction matrix $R$ contain both true negatives and false negatives, so sampling from $S'_u = \{ i \in I \mid r_{ui}=0 \}$ can train the model to suppress items that fit the user’s latent preference (Wu et al., 20 Feb 2026).

The same expression has a broader interpretation in pool-based active learning. There, topology-aware positive sample set construction refers to building a labeled training subset from Proper Topological Regions, so that acquired examples are organized into connected, density-coherent regions rather than isolated queried points (Hadjadj et al., 2023). The “positive set” in this setting is the set of examples promoted from unlabeled status to labeled training data.

A common misconception is that topology-aware construction is equivalent to hard-negative mining in embedding space. The surveyed methods do not support that equivalence. TDSGL measures semantic similarity directly from interaction topology rather than from latent embeddings, and TPSC-FO identifies candidate false negatives through community structure and ALS-based personalized filtration rather than through the target recommender’s current scores (Han et al., 2023, Wu et al., 20 Feb 2026). This distinction matters because both papers explicitly criticize representation-dependent criteria as unstable or biased under sparsity.

2. Topological signals used to define latent positives

The topological signal depends on the task and the graph object being analyzed. In TDSGL, the underlying structure is a user–item bipartite graph with interaction matrix $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$. Semantic similarity is computed from co-occurrence matrices

$$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$

where $\mathbf{P}_{user}(u,v)$ counts the number of items co-interacted by users $u$ and $v$, and $\mathbf{P}_{item}(i,j)$ analogously counts co-occurring users for items $i$ and $j$ (Han et al., 2023). These counts are interpreted as semantic similarity in terms of purchasing intent or item characteristics, so the positive-set construction is directly anchored in the original interaction topology.

TPSC-FO uses a different topological primitive: community structure in the user–item bipartite graph $S'_u = \{ i \in I \mid r_{ui}=0 \}$0. Its central intuition is that topological communities reflect groups of users and items with similar preferences, and that a missing edge $S'_u = \{ i \in I \mid r_{ui}=0 \}$1 within the same community has a strong chance of being a false negative (Wu et al., 20 Feb 2026). The method operationalizes this through a differential community detection strategy, combining Leiden and Infomap and keeping only user–item pairs for which both methods assign the same community.

PTR-based active learning relies on topological data analysis rather than collaborative filtering graphs. It constructs Rips or $S'_u = \{ i \in I \mid r_{ui}=0 \}$2-Rips graphs over a point cloud $S'_u = \{ i \in I \mid r_{ui}=0 \}$3, uses upper-star filtrations of a density estimator $S'_u = \{ i \in I \mid r_{ui}=0 \}$4, and applies ToMATo clustering to obtain Proper Topological Regions (Hadjadj et al., 2023). These regions are topologically meaningful components associated with prominent density peaks and their basins of attraction. In this setting, topology-aware positive set construction means selecting representative regions and propagating labels within each region.

These three variants show that “topology-aware” is not a single algorithmic recipe. It can mean co-occurrence topology, community topology, or persistent topological regions. What remains invariant is that positive candidates are identified from structure external to the target model’s current prediction state.

3. Contrastive positive construction in graph recommendation

TDSGL provides a direct formulation of topology-aware positive sample set construction in graph contrastive learning for recommendation. The method first discriminates semantically similar and dissimilar nodes with a threshold $S'_u = \{ i \in I \mid r_{ui}=0 \}$5, defining masks

$S'_u = \{ i \in I \mid r_{ui}=0 \}$6

When $S'_u = \{ i \in I \mid r_{ui}=0 \}$7, user $S'_u = \{ i \in I \mid r_{ui}=0 \}$8 is semantically dissimilar and can be treated as a negative for anchor $S'_u = \{ i \in I \mid r_{ui}=0 \}$9; when $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$0, the pair is a potential positive or false negative and should not enter the negative set (Han et al., 2023).

The positive-set construction does not stop at masking. TDSGL explicitly converts semantically similar nodes into auxiliary positives through a one-layer linear GCN-based feature extraction module. On the user side,

$\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$1

and analogously for items. The resulting row $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$2 is an aggregated semantic feature from all users that are potential positives for $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$3, and the paper states that these features are treated as positive samples (Han et al., 2023).

The resulting positive sample set for each anchor node contains two elements. The primary positive is the node’s own representation in the second augmented view, and the auxiliary positive is the aggregated feature from semantically similar nodes. This leads to a multi-positive contrastive objective in which the numerator includes both $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$4 and $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$5, while the denominator retains only topologically dissimilar negatives. The construction is thus topology-aware in two senses: topology prunes the negative pool, and topology also generates an auxiliary positive representation.

Empirically, the paper attributes performance gains to this combination of debiased contrasting and exploitation of potential positives. It reports improvements over SGL variants on Yelp2018, Movielens-1M, and LastFM, and the ablation in which the feature extraction module is removed underperforms the full model (Han et al., 2023). This suggests that positive enrichment contributes beyond simple false-negative masking.

4. Community-based positive relabeling in implicit collaborative filtering

TPSC-FO generalizes the idea from auxiliary positives in contrastive learning to explicit positive-set augmentation for BPR-based implicit collaborative filtering. Its topology-aware positive sample set is

$\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$6

where $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$7 is the observed positive set and $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$8 is the final false negative set identified by topology and personalized filtration (Wu et al., 20 Feb 2026). The corresponding negative pool is reduced to $\mathbf{R} \in \mathbb{R}^{N_U \times N_I}$9.

The construction proceeds in stages. ComFNI first builds a bipartite graph from $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$0, applies a community detection algorithm, and for each community takes the Cartesian product of user nodes and item nodes, retaining only non-interacted pairs. This yields candidate false negatives inside communities. TPSC then strengthens this by running ComFNI with Leiden and Infomap, and defining the candidate set

$$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$1

The intersection acts as a consensus filter over two distinct views of community structure (Wu et al., 20 Feb 2026).

Because topological closeness alone may still be noisy, the method adds personalized noise filtration in ALS embedding space. For each user $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$2, it computes a threshold $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$3 as a quantile of cosine similarities between $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$4 and the items in $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$5, then retains only those items in $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$6 whose similarity exceeds $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$7. This produces $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$8, the final set of false negatives re-labeled as positives (Wu et al., 20 Feb 2026). The filtration is personalized because the threshold is derived from the user’s own positive similarity distribution, not from a global cutoff.

The practical significance of this design is that the recommendation objective remains standard BPR in form, but the supervision changes. Positives are sampled from $$\mathbf{P}_{user} = \mathbf{R}\mathbf{R}^T,\qquad \mathbf{P}_{item} = \mathbf{R}^T\mathbf{R},$$9, and negatives are sampled from $\mathbf{P}_{user}(u,v)$0. The paper characterizes this as a model-agnostic, topology-based way to identify false negatives and convert them into useful positive samples, rather than merely avoiding them during negative sampling (Wu et al., 20 Feb 2026).

5. Feature optimization and denoising of constructed positives

In TPSC-FO, positive construction and positive refinement are separate modules. Even after topology-aware augmentation, the positive set may contain noise: original positives can include accidental clicks, and added false negatives may still contain misidentified items. The neighborhood-guided Feature Optimization module is introduced to denoise positive sample embeddings without dropping or down-weighting positives (Wu et al., 20 Feb 2026).

For each positive interaction pair $\mathbf{P}_{user}(u,v)$1 with $\mathbf{P}_{user}(u,v)$2, the method randomly selects $\mathbf{P}_{user}(u,v)$3 other positive items from $\mathbf{P}_{user}(u,v)$4 to form a neighborhood set $\mathbf{P}_{user}(u,v)$5. The base recommender produces embeddings for $\mathbf{P}_{user}(u,v)$6, $\mathbf{P}_{user}(u,v)$7, and $\mathbf{P}_{user}(u,v)$8, after which the neighborhood features are aggregated by mean pooling: $\mathbf{P}_{user}(u,v)$9 The optimized positive embedding is then obtained via mixup,

$u$0

with $u$1 (Wu et al., 20 Feb 2026).

The paper provides a variance-reduction analysis under the assumption $u$2, with Gaussian noise. Under the stated assumptions, the expectation of the optimized embedding is approximately preserved, while the noise variance satisfies

$u$3

Accordingly, FO is framed as preserving the expected true preference component while reducing noise variance (Wu et al., 20 Feb 2026).

A methodological distinction emerges here. TDSGL compresses a set of false-negative neighbors into a single auxiliary positive vector via a one-layer GCN, whereas TPSC-FO keeps the augmented positive set explicit and then denoises each sampled positive through neighborhood mixup. Both are topology-aware, but they act at different levels: one constructs an auxiliary positive representation for contrastive learning, and the other constructs an expanded positive supervision set for ranking loss and then refines the associated features.

6. Region-level positive construction in active learning

The PTR framework uses the same phrase in a different domain. Proper Topological Regions are obtained by applying ToMATo to a density-adaptive $u$4-Rips graph, with parameters chosen by maximizing a Silhouette-Size objective over $u$5 (Hadjadj et al., 2023). Each region corresponds to a topologically coherent, density-informed cluster.

In the cold-start setting, the method selects the $u$6 largest PTR regions, queries the oracle only on the peak of each region, and propagates that label to all points in the region. This yields an initial labeled set

$u$7

where $u$8 is the union of the chosen regions (Hadjadj et al., 2023). The paper explicitly describes this as a topology-aware construction of an initial positive sample set.

In subsequent active learning rounds, standard strategies such as uncertainty, entropy, or margin sampling still operate at the point level, but their queried points are lifted to region-level acquisitions. If the queried points intersect certain PTR regions, the full regions are added to the labeled set; if fewer than $u$9 distinct regions are activated, the leftover budget is allocated to the largest unlabeled PTR regions (Hadjadj et al., 2023). The effect is a topology-based non-redundancy mechanism: acquisition favors topologically new areas rather than repeatedly querying within the same region.

This usage broadens the meaning of topology-aware positive sample set construction. It need not involve false negatives or recommendation at all. Instead, it can mean selecting positive training examples so that they cover prominent modes of the data distribution and remain aligned with topological components recovered from persistent homology.

7. Assumptions, limitations, and methodological implications

The surveyed methods share a reliance on the quality of the topological proxy. TDSGL assumes that co-occurrence count is a reliable similarity measure; the authors note that on denser Movielens-1M the topology-aware method has more difficulty in recognizing false negatives, and that performance varies non-monotonically with the threshold $v$0 (Han et al., 2023). TPSC-FO assumes that community structure is informative for preference inference and that the interaction graph exhibits homophily; the paper notes that very sparse or noisy graphs may yield weak communities, that modularity-based methods can merge small but meaningful communities, and that communities and thresholds are computed once and remain static during training (Wu et al., 20 Feb 2026). PTR assumes that labels are sufficiently coherent within topological regions; on the highly imbalanced nursery dataset, label propagation over PTR tends to amplify imbalance and may degrade active learning performance (Hadjadj et al., 2023).

Another important limitation concerns scale and preprocessing cost. TDSGL requires offline computation of $v$1 and $v$2, while TPSC-FO requires community detection, ALS embeddings, and personalized noise filtration before recommender training (Han et al., 2023, Wu et al., 20 Feb 2026). PTR similarly incurs repeated graph construction and hyperparameter optimization over TPE trials (Hadjadj et al., 2023). None of these methods presents topology-aware positive construction as cost-free.

A broader implication is that topology-aware positive set construction is best viewed as a structural prior over supervision. In recommendation, the prior states that missing or contrastively excluded relations inside semantically coherent or community-coherent neighborhoods should sometimes be promoted to positives. In active learning, the prior states that labeled data should be acquired in topologically meaningful blocks rather than as isolated points. This suggests a common research direction, though only as an inference from the surveyed works: topology is being used not merely to regularize representations, but to redefine what counts as positive evidence before or during learning.

Taken together, the literature establishes topology-aware positive sample set construction as a distinct methodological theme rather than a single algorithm. TDSGL formulates it through co-occurrence-based auxiliary positives in debiased graph contrastive learning, TPSC-FO formulates it through community-consensus false negative relabeling and subsequent feature optimization, and PTR formulates it through region-level labeled set construction based on persistent topological structure (Han et al., 2023, Wu et al., 20 Feb 2026, Hadjadj et al., 2023). The shared objective is to exploit topological structure to recover latent positive supervision that standard sampling, labeling, or contrastive schemes leave unused.