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Graph-based collaborative filtering learns user and item representations by propagating embeddings over a user--item interaction graph. We investigate a modular, sheaf-inspired scoring extension placed on top of a three-layer LightGCN backbone. The motivation comes from cellular sheaves, where local vector spaces are related by linear restriction maps; here, this idea motivates structured, pair-dependent transformations of user and item embeddings rather than a change to graph propagation. We compare the identity inner-product scorer, diagonal, rank-one low-rank, and full matrix residual scorers, and a parameter-matched residual MLP baseline (\texttt{mlp\_full}) on Amazon-Book, Gowalla, MovieLens-1M, and Yelp2018. We report ranking quality and catalog coverage at cutoffs 10 and 20. The selected structured variants show modest, dataset-dependent changes in ranking quality and sometimes substantial changes in coverage. Our method is inspired by sheaf geometry but does not construct a cellular sheaf or implement sheaf-Laplacian diffusion; the experiments therefore do not establish a causal benefit from sheaf geometry itself.

Dismantling reduces a graph to one vertex by successive deletions. At level zero, a vertex is eligible if one of its neighbours is adjacent to all its other neighbours. At level $k\geq1$, its neighbours must induce a graph dismantlable at level $k-1$. A graph has strict level $k$ if it admits a reduction at level $k$ and none at level $k-1$. For every $k\geq1$, we construct a graph of strict level $k$ with $3k+8$ vertices. Each step adds three vertices: the neighbours of one induce the preceding graph, and the other two provide the adjacencies used to rule out other first deletions at the lower level. A second construction gives a graph of order $4k+4$ at strict level $k$. Together the two families give the upper bound $\min\{4k+4,3k+8\}$ on the least order at strict level $k$.

Multi-criteria decision making over large databases is fundamental to many real-world applications. Interactive regret minimization has emerged recently as a promising approach that combines the controllable output size of top-$k$ queries with the flexibility of skyline queries, without requiring users to explicitly specify a utility function. Instead, the system iteratively presents a small set of options, learns the user's implicit preferences from their selections, and progressively refines its recommendations. However, modern applications such as housing search and financial product selection often involve hundreds of attributes. Existing interactive methods either fail to scale to such high-dimensional data or require excessive user interaction, frequently exceeding $1000$ rounds. We address this limitation with FHDR (Fast High-Dimensional Reduction), a framework that exploits sparse linear preferences to discover key attributes through lightweight interactions. We also propose AttributeSubset, an efficient single-round algorithm that returns a compact recommendation set without further feedback. In our experiments on real datasets, FHDR completes in under $0.03$s and requires fewer than $30$ interaction rounds under the default setting. Extensive experiments on both synthetic and real-world datasets show that FHDR outperforms the best-known baselines by at least one order of magnitude in execution time and by up to several orders of magnitude in interaction cost. These results establish FHDR as a new state of the art for scalable interactive regret minimization.

We introduce Multi-scale Closing Index (MCI), a new dissimilarity measure for evaluating the alignment of two 2D point sets. MCI is bounded between 0 and 1 and is based on multiscale mathematical morphology operations, with smaller values indicating greater structural agreement. The measure was motivated by the need to assess the quality of registration between point sets representing cell populations in biological images. To evaluate its behaviour under controlled conditions, we generated synthetic point clouds from Perlin-noise-based spatial structures, following established approaches for simulating biologically plausible spatial organization. We assessed MCI between those point clouds and different transformed versions, obtained by local perturbation, point removal, translation, rotation and addition of impulsive noise. Across these experiments, the dissimilarity MCI increased gradually and consistently with perturbation severity, while showing a clear discontinuity between clouds originating from the same underlying spatial structure and point clouds generated from different structures. We further compared MCI with commonly used point-set distances, including Hausdorff-based measures and Sliced Wasserstein distance. These experiments show the ability of MCI to capture structural agreement and quantify deviations between point sets, highlighting its potential as a complementary metric for evaluating two-dimensional point-set registration.