While neural rendering methods such as 3D Gaussian Splatting achieve remarkable visual fidelity, traditional polygonal meshes remain the backbone of established graphics pipelines. Triangle splatting bridges this gap by optimizing triangle primitives as differentiable splats, producing representations that are closer to mesh-based workflows. Central to these methods is the kernel function that softens triangle boundaries to propagate gradients to vertex positions. Existing triangle splatting methods make inconsistent choices of kernel functions, and analysis of these kernels' optimization behavior has been limited to unstructured triangle soups for novel-view synthesis. In this work, we consider triangle splatting as a generic tool for photometric optimization, comparing kernel properties through two complementary tasks: mesh optimization for shape reconstruction and triangle soup optimization for novel-view synthesis. Along with the analysis, we introduce an elastic kernel function that features bilateral gradient support across the boundary and an adaptive boundary value, which are shown to be essential for robust optimization. Under isolated comparison, our elastic kernel outperforms existing kernels on shape reconstruction and in the majority of novel-view synthesis benchmarks, demonstrating the importance of kernel design in the effectiveness and versatility of triangle splatting.
The quadratic cost of self-attention makes long-context inference prohibitively expensive, and proxy-based block-sparse attention has become a practical remedy. Existing methods typically rely on a proxy to predict a binary sparse mask and a kernel to consume this mask and perform sparse attention computation. Such an approach is effective under moderate budgets. However, as the budget tightens, the estimated proxy inevitably drops some salient blocks, while the kernel can only apply the sparse mask mechanically, leading to an evident drop in model accuracy. We propose CoSA, a two-stage training-free Sparse Attention under proxy-kernel CO-design, which couples a Kernel-Aware Proxy (KAP) with an Ordered-Skipping Kernel (OSK). In the first stage, the KAP selects blocks under a moderate budget and produces an ordered mask that prescribes the order in which KV pages are visited in the kernel inner loop. In the second stage, the OSK applies this mask and skips more blocks under a tightened budget given online-softmax statistics. Across mainstream LLM backbones and long-context benchmarks, CoSA attains higher accuracy at lower budgets. Impressively, CoSA achieves a 4.93$\times$ attention speedup and reduces end-to-end Time-to-First-Token by 2.53$\times$ under a context length of 128K with negligible performance degradation.
Bayesian Optimization is an iterative method, tailored to optimizing expensive black box objective functions. Surrogate models like Gaussian Processes, which are the gold standard in Bayesian Optimization, can be inefficient for inputs with permutation symmetries, as the most common kernels employed are better suited for vector inputs rather than unordered sets of items. Motivated by this issue, we turn to permutation invariant Bayesian Optimization for well placement in Carbon Capture and Storage projects. The high fidelity black box simulator is instructed to operate wells under group control, giving rise to permutation symmetries within injector and producer groups that cannot be exploited with standard GP kernels. In this work, our main contribution is a novel Gaussian Process kernel (GP-Perm) that encodes permutation invariance by comparing sets through a stable divergence between their induced empirical representations, and can be combined with standard kernels for additional vector-valued inputs. As a learned invariant baseline, we also consider a Deep Kernel Learning model (DKL-DS) using the Deep Sets architecture to learn a permutation-invariant embedding. We evaluate the proposed methodology across 8 use cases, comprising seven synthetic benchmarks and one realistic CCS case study (Johansen formation)