Daily Papers

The widespread application of graph data inevitably brings significant privacy risks, as its unprotected use can lead to the leakage of sensitive information. These risks are particularly acute in the setting with an untrusted curator, where the data remains decentralized and each user only holds the connections to their neighbors. To mitigate such privacy risks, we adopt local differential privacy (LDP) to collect users' private information and generate a synthetic graph. However, existing methods suffer from either excessive noise injection by perturbing the local adjacency lists or significant structural information loss due to the simplistic graph encoding process. To address these issues, we propose LDPGraph, an effective graph synthesis algorithm that takes one step further by exploiting neighborhood structures under LDP. To obtain neighborhood statistics beyond degrees, LDPGraph aggregates a noisy global view from perturbed adjacency lists and combines it with projected local connections to estimate node-level triangle counts. To correct the structural inconsistency caused by separately perturbing degree and triangle count, LDPGraph jointly refines them into feasible degree-triangle targets. To reconstruct a global graph from these estimated targets, LDPGraph adopts a triangle-first strategy that first preserves local clustering structures and then fulfills remaining degree requirements. Extensive experiments on four real-world datasets and multiple commonly used graph metrics validate the superiority of LDPGraph. Source code is available at https://github.com/ZJU-TrustAID/LDPGraph.

For a vertex v of a graph G, the triangle-degree td_G(v) is the number of triangles containing v. A graph is triangle-distinct, or K_3-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erdős, and Oellermann asked whether a regular K_3-irregular graph exists. We prove that such graphs exist for every regularity rge 9. More precisely, for every integer kge 15 we construct a 2k-regular triangle-distinct graph on 4k+2 vertices; complementation gives a (2k+1)-regular example of the same order. The construction uses two antiregular threshold blocks joined by a zero-one matrix with prescribed margins, followed by matrix 2-switches that preserve those margins. After a uniform reference perturbation, exactly three triangle-degree collisions remain. Switches chosen according to parity remove two of them, and the last possible collision is controlled by a short quadratic discriminant argument. Odd kge 17 and even kge 62 are handled symbolically, while the remaining 24 values are settled by exact finite verification. Together with the known examples for 9le rle 29, this closes the positive existence problem for every rge 9.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

Reconstructing 3D scenes and synthesizing novel views has seen rapid progress in recent years. Neural Radiance Fields demonstrated that continuous volumetric radiance fields can achieve high-quality image synthesis, but their long training and rendering times limit practicality. 3D Gaussian Splatting (3DGS) addressed these issues by representing scenes with millions of Gaussians, enabling real-time rendering and fast optimization. However, Gaussian primitives are not natively compatible with the mesh-based pipelines used in VR headsets, and real-time graphics applications. Existing solutions attempt to convert Gaussians into meshes through post-processing or two-stage pipelines, which increases complexity and degrades visual quality. In this work, we introduce Triangle Splatting+, which directly optimizes triangles, the fundamental primitive of computer graphics, within a differentiable splatting framework. We formulate triangle parametrization to enable connectivity through shared vertices, and we design a training strategy that enforces opaque triangles. The final output is immediately usable in standard graphics engines without post-processing. Experiments on the Mip-NeRF360 and Tanks & Temples datasets show that Triangle Splatting+achieves state-of-the-art performance in mesh-based novel view synthesis. Our method surpasses prior splatting approaches in visual fidelity while remaining efficient and fast to training. Moreover, the resulting semi-connected meshes support downstream applications such as physics-based simulation or interactive walkthroughs. The project page is https://trianglesplatting2.github.io/trianglesplatting2/.

We determine the 4-point correlation function and amplitude in planar, maximally supersymmetric Yang-Mills theory to 12 loops. We find that the recently-introduced 'double-triangle' rule in fact implies the previously described square and pentagon rules; and when applied to 12 loops, it fully determines the 11-loop correlator and fixes all but 3 of the (22,024,902) 12-loop coefficients; these remaining coefficients can be subsequently fixed using the '(single-)triangle' rule. Not only do we confirm the Catalan conjecture for anti-prism graphs, but we discover evidence for a greatly generalized Catalan conjecture for the coefficients of all polygon-framed fishnet graphs. We provide all contributions through 12 loops as ancillary files to this work.

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in O(mbarδ) time and O(m+n) space, where barδle O(m) is the average degeneracy introduced by Burkhardt, Faber, and Harris (2020). In practical applications, barδ is much smaller than m so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes O(log n) time using O(mm) CREW PRAM processors. In MapReduce, we show it takes four rounds using O(mm) communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in O(mbarδ) time on sparse graphs.

The field of computer graphics was revolutionized by models such as Neural Radiance Fields and 3D Gaussian Splatting, displacing triangles as the dominant representation for photogrammetry. In this paper, we argue for a triangle comeback. We develop a differentiable renderer that directly optimizes triangles via end-to-end gradients. We achieve this by rendering each triangle as differentiable splats, combining the efficiency of triangles with the adaptive density of representations based on independent primitives. Compared to popular 2D and 3D Gaussian Splatting methods, our approach achieves higher visual fidelity, faster convergence, and increased rendering throughput. On the Mip-NeRF360 dataset, our method outperforms concurrent non-volumetric primitives in visual fidelity and achieves higher perceptual quality than the state-of-the-art Zip-NeRF on indoor scenes. Triangles are simple, compatible with standard graphics stacks and GPU hardware, and highly efficient: for the Garden scene, we achieve over 2,400 FPS at 1280x720 resolution using an off-the-shelf mesh renderer. These results highlight the efficiency and effectiveness of triangle-based representations for high-quality novel view synthesis. Triangles bring us closer to mesh-based optimization by combining classical computer graphics with modern differentiable rendering frameworks. The project page is https://trianglesplatting.github.io/

Graph Neural Networks (GNNs) update node representations through message passing, which is primarily based on the homophily principle, assuming that adjacent nodes share similar features. However, in real-world graphs with long-tailed degree distributions, high-degree nodes dominate message passing, causing a degree bias where low-degree nodes remain under-represented due to inadequate messages. The main challenge in addressing degree bias is how to discover non-adjacent nodes to provide additional messages to low-degree nodes while reducing excessive messages for high-degree nodes. Nevertheless, exploiting non-adjacent nodes to provide valuable messages is challenging, as it could generate noisy information and disrupt the original graph structures. To solve it, we propose a novel Degree Fairness Graph Transformer, named DegFairGT, to mitigate degree bias by discovering structural similarities between non-adjacent nodes through learnable structural augmentation and structural self-attention. Our key idea is to exploit non-adjacent nodes with similar roles in the same community to generate informative edges under our augmentation, which could provide informative messages between nodes with similar roles while ensuring that the homophily principle is maintained within the community. To enable DegFairGT to learn such structural similarities, we then propose a structural self-attention to capture the similarities between node pairs. To preserve global graph structures and prevent graph augmentation from hindering graph structure, we propose a Self-Supervised Learning task to preserve p-step transition probability and regularize graph augmentation. Extensive experiments on six datasets showed that DegFairGT outperformed state-of-the-art baselines in degree fairness analysis, node classification, and node clustering tasks.

A classical problem in convex and discrete geometry asks for the convex polyhedron of greatest volume whose vertices are chosen from the unit sphere S^2. For a prescribed number N of vertices, the problem is known only in a small number of cases. In this paper we resolve the next outstanding case, N=9. We prove that every convex polyhedron with at most nine vertices on S^2 has volume at most 32sqrt{3-3}, with equality, up to rotation, precisely for a triaugmented triangular prism of an explicitly determined shape. The proof combines combinatorial and geometric reductions with sharp volume estimates. By a theorem of Berman and Hanes (Mathematische Annalen, 1970), a volume maximizer must be simplicial, reducing the 2,606 combinatorial types of 9-vertex polyhedra to 50. We prove that a maximizer cannot have a trivalent vertex, leaving only five combinatorial types, which are treated using geometric and combinatorial arguments. In particular, we determine the exact maximizer within the triaugmented triangular prism class, and characterize the equality case.

Evaluation metrics are central to binary target segmentation because they determine how progress is measured, compared, and interpreted. In this paper, target denotes the task-defined positive region to be segmented rather than a generic foreground object. It may be salient, camouflaged, transparent, glass-like, mirror-like, shadow-like, lesion-like, or defined by other application-specific semantics. We treat existing metrics as compositions of modular design choices rather than isolated formulas. The proposed framework decomposes each metric into five stages covering prediction representation, target extraction, target matching, score computation, and metric reporting. We use this framework to analyze representative metrics and show how newer metrics address specific limits in earlier protocols. The stage choices keep each metric's assumptions visible. We then discuss the design space opened by the framework and its implications for task-aware evaluation protocols. Reference code is available at https://github.com/lartpang/PySODMetrics.

Approximate nearest neighbor (ANN) graph indices such as HNSW and Vamana construct their edge topology in full-precision or high-fidelity quantized metric spaces, relegating binary quantization (BQ) to a post-hoc distance estimator during search. This paper asks a different question: Can binary quantization define the graph topology itself -- and if so, under what conditions? We study this question through QuIVer (Quantized Index for Vector Retrieval), a training-free ANN graph index that performs Vamana edge selection, diversity pruning, and beam-search navigation entirely within a 2-bit Sign-Magnitude BQ metric space, accessing float32 vectors only for final reranking. Systematic evaluation on twelve million-scale datasets reveals a sharp applicability boundary: BQ-native topology is highly effective on cosine-native contrastive-learning embeddings (>=88% Recall@10 at ef=64 across five datasets, 384--3072 dimensions), moderately effective on multimodal CLIP data (71--78%), and empirically unsuitable for Euclidean-native or structureless distributions (<15%). Our results suggest an empirical "impossible triangle" between aggressive compression, high throughput, and universal data compatibility. The central contribution is not merely the system, but the boundary it reveals: falsifiable criteria for when industrial vector search systems can safely trade metric fidelity for compact BQ-native navigation. On compatible workloads, the system benefits are substantial: QuIVer's BQ-native hot path (<1.3 GB for 1M vectors) yields 2.5--5.5x higher multi-threaded throughput than DiskANN Rust and HNSW variants at matched recall, with 4.7x less hot memory and no codebook or rotation training (unlike PQ/OPQ/RaBitQ).

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

We introduce an inference-time scene optimization algorithm utilizing triangle soup, a collection of disconnected translucent triangle primitives, as the representation for the geometry and appearance of a scene. Unlike full-rank Gaussian kernels, triangles are a natural, locally-flat proxy for surfaces that can be connected to achieve highly complex geometry. When coupled with per-vertex Spherical Harmonics (SH), triangles provide a rich visual representation without incurring an expensive increase in primitives. We leverage our new representation to incorporate optimization objectives and enforce spatial regularization directly on the underlying primitives. The main differentiator of our approach is the definition and enforcement of soft connectivity forces between triangles during optimization, encouraging explicit, but soft, surface continuity in 3D. Experiments on representative 3D reconstruction and novel view synthesis datasets show improvements in geometric accuracy compared to current state-of-the-art algorithms without sacrificing visual fidelity.

Let G be a graph with adjacency matrix A(G) and degree matrix D(G), and let L_μ(G):=A(G)-μD(G). Two graphs G_1 and G_2 are called degree-similar if there exists an invertible matrix M such that M^{-1} A(G_1) M =A(G_2) and M^{-1} D(G_1) M =D(G_2). In this paper, we address three problems concerning degree-similar graphs proposed by Godsil and Sun. First, we present a new characterization of degree-similar graphs using degree partition, from which we derive methods and examples for constructing cospectral graphs and degree-similar graphs. Second, we construct infinite pairs of non-degree-similar trees G_1 and G_2 such that tI- L_μ(G_1) and tI-L_μ(G_2) have the same Smith normal form over Q(μ)[t], which provides a negative answer to a problem posed by Godsil and Sun. Third, we establish several invariants of degree-similar graphs and obtain results on unicyclic graphs that are degree-similar determined. Lastly we prove that for a strongly regular graph G and any two edges e and f of G, G backslash e and G backslash f have identical μ-polynomial, i.e., det(tI-L_μ(G backslash e))=det(tI-L_μ(G backslash f)), which enables the construction of pairs of non-isomorphic graphs with same μ-polynomial, where G backslash e denotes the graph obtained from G by deleting the edge e.

Aiming at better representing multivariate relationships, this paper investigates a motif dimensional framework for higher-order graph learning. The graph learning effectiveness can be improved through OFFER. The proposed framework mainly aims at accelerating and improving higher-order graph learning results. We apply the acceleration procedure from the dimensional of network motifs. Specifically, the refined degree for nodes and edges are conducted in two stages: (1) employ motif degree of nodes to refine the adjacency matrix of the network; and (2) employ motif degree of edges to refine the transition probability matrix in the learning process. In order to assess the efficiency of the proposed framework, four popular network representation algorithms are modified and examined. By evaluating the performance of OFFER, both link prediction results and clustering results demonstrate that the graph representation learning algorithms enhanced with OFFER consistently outperform the original algorithms with higher efficiency.

Cancer is the second leading cause of death, with chemotherapy as one of the primary forms of treatment. As a result, researchers are turning to drug combination therapy to decrease drug resistance and increase efficacy. Current methods of drug combination screening, such as in vivo and in vitro, are inefficient due to stark time and monetary costs. In silico methods have become increasingly important for screening drugs, but current methods are inaccurate and generalize poorly to unseen anticancer drugs. In this paper, I employ a geometric deep-learning model utilizing a graph attention network that is equivariant to 3D rotations, translations, and reflections with structural motifs. Additionally, the gene expression of cancer cell lines is utilized to classify synergistic drug combinations specific to each cell line. I compared the proposed geometric deep learning framework to current state-of-the-art (SOTA) methods, and the proposed model architecture achieved greater performance on all 12 benchmark tasks performed on the DrugComb dataset. Specifically, the proposed framework outperformed other SOTA methods by an accuracy difference greater than 28%. Based on these results, I believe that the equivariant graph attention network's capability of learning geometric data accounts for the large performance improvements. The model's ability to generalize to foreign drugs is thought to be due to the structural motifs providing a better representation of the molecule. Overall, I believe that the proposed equivariant geometric deep learning framework serves as an effective tool for virtually screening anticancer drug combinations for further validation in a wet lab environment. The code for this work is made available online at: https://github.com/WeToTheMoon/EGAT_DrugSynergy.

Generating explorable 3D scenes from a single image requires strong generative priors and accurate geometric representations suitable for downstream use. Current video diffusion models offer high-quality generation and implicitly encode multi-view geometric structure in latent space. However, existing feedforward latent scene decoders typically output volumetric 3D Gaussians that lack a well-defined surface, limiting their use in simulation or standard graphics pipelines. This motivates decoding surface-aligned primitives that are not only renderable but also closer to explicit geometric assets. We ask whether compressed video diffusion latents can be mapped directly to explicit surface primitives in a single pass. To this end, we introduce FLAT and, for the first time, show that triangle splats can be decoded directly from video diffusion latents. Compared with decoding 3D Gaussians, predicting flat primitives is notoriously more challenging due to high sensitivity to primitive orientations, oftentimes leading to poor gradient flow. FLAT solves with two key ingredients: a ray-centered rotation parameterization for triangle regression and a novel product window function that improves gradient flow during differentiable triangle rendering. On standard benchmarks, FLAT achieves significantly better geometric accuracy while maintaining competitive visual quality compared to state-of-the-art feedforward baselines. We further show that a lightweight test-time refinement step converts the predicted triangle soup into a fully opaque, game-engine-ready representation that supports real-time rendering. By evaluating 3DGS, 2DGS, and triangle splatting variants under an identical training setup, we provide the first systematic analysis of representation tradeoffs in feedforward scene generation. The project page is available at https://flat-splat.github.io