Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions

Persistent Laplacians are matrix operators that track how the shape and structure of data transform across scales and are popularly adopted in biology, physics, and machine learning. Their eigenvalues are concise descriptors of geometric and topological features in a filtration. Although earlier work established global algebraic stability for these operators, the precise change in a single eigenvalue when one simplex, such as a vertex, edge, or triangle, is added has remained unknown. This is important because downstream tools, including heat-kernel signatures and spectral neural networks, depend directly on these eigenvalues. We close this gap by proving a uniform Lipschitz bound: after inserting one simplex, every up-persistent Laplacian eigenvalue can vary by at most twice the Euclidean norm of that simplex's boundary, independent of filtration scale and complex size. This result delivers the first eigenvalue-level robustness guarantee for spectral topological data analysis. It guarantees that spectral features remain stable under local updates and enables reliable error control in dynamic data settings.

RePCS: Diagnosing Data Memorization in LLM-Powered Retrieval-Augmented Generation

Retrieval-augmented generation (RAG) has become a common strategy for updating large language model (LLM) responses with current, external information. However, models may still rely on memorized training data, bypass the retrieved evidence, and produce contaminated outputs. We introduce Retrieval-Path Contamination Scoring (RePCS), a diagnostic method that detects such behavior without requiring model access or retraining. RePCS compares two inference paths: (i) a parametric path using only the query, and (ii) a retrieval-augmented path using both the query and retrieved context by computing the Kullback-Leibler (KL) divergence between their output distributions. A low divergence suggests that the retrieved context had minimal impact, indicating potential memorization. This procedure is model-agnostic, requires no gradient or internal state access, and adds only a single additional forward pass. We further derive PAC-style guarantees that link the KL threshold to user-defined false positive and false negative rates. On the Prompt-WNQA benchmark, RePCS achieves a ROC-AUC of 0.918. This result outperforms the strongest prior method by 6.5 percentage points while keeping latency overhead below 4.7% on an NVIDIA T4 GPU. RePCS offers a lightweight, black-box safeguard to verify whether a RAG system meaningfully leverages retrieval, making it especially valuable in safety-critical applications.

Spectral Contraction of Boundary-Weighted Filters on delta-Hyperbolic Graphs

Hierarchical graphs often exhibit tree-like branching patterns, a structural property that challenges the design of traditional graph filters. We introduce a boundary-weighted operator that rescales each edge according to how far its endpoints drift toward the graph's Gromov boundary. Using Busemann functions on delta-hyperbolic networks, we prove a closed-form upper bound on the operator's spectral norm: every signal loses a curvature-controlled fraction of its energy at each pass. The result delivers a parameter-free, lightweight filter whose stability follows directly from geometric first principles, offering a new analytic tool for graph signal processing on data with dense or hidden hierarchical structure.

MicroRicci: A Greedy and Local Ricci Flow Solver for Self-Tuning Mesh Smoothing

Real-time mesh smoothing at scale remains a formidable challenge: classical Ricci-flow solvers demand costly global updates, while greedy heuristics suffer from slow convergence or brittle tuning. We present MicroRicci, the first truly self-tuning, local Ricci-flow solver that borrows ideas from coding theory and packs them into just 1K + 200 parameters. Its primary core is a greedy syndrome-decoding step that pinpoints and corrects the largest curvature error in O(E) time, augmented by two tiny neural modules that adaptively choose vertices and step sizes on the fly. On a diverse set of 110 SJTU-TMQA meshes, MicroRicci slashes iteration counts from 950+=140 to 400+=80 (2.4x speedup), tightens curvature spread from 0.19 to 0.185, and achieves a remarkable UV-distortion-to-MOS correlation of r = -0.93. It adds only 0.25 ms per iteration (0.80 to 1.05 ms), yielding an end-to-end 1.8x runtime acceleration over state-of-the-art methods. MicroRicci's combination of linear-time updates, automatic hyperparameter adaptation, and high-quality geometric and perceptual results makes it well suited for real-time, resource-limited applications in graphics, simulation, and related fields.

Topology-Preserving Scaling in Data Augmentation

We propose an algorithmic framework for dataset normalization in data augmentation pipelines that preserves topological stability under non-uniform scaling transformations. Given a finite metric space \( X \subset \mathbb{R}^n \) with Euclidean distance \( d_X \), we consider scaling transformations defined by scaling factors \( s_1, s_2, \ldots, s_n > 0 \). Specifically, we define a scaling function \( S \) that maps each point \( x = (x_1, x_2, \ldots, x_n) \in X \) to \[ S(x) = (s_1 x_1, s_2 x_2, \ldots, s_n x_n). \] Our main result establishes that the bottleneck distance \( d_B(D, D_S) \) between the persistence diagrams \( D \) of \( X \) and \( D_S \) of \( S(X) \) satisfies: \[ d_B(D, D_S) \leq (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), \] where \( s_{\min} = \min_{1 \leq i \leq n} s_i \), \( s_{\max} = \max_{1 \leq i \leq n} s_i \), and \( \operatorname{diam}(X) \) is the diameter of \( X \). Based on this theoretical guarantee, we formulate an optimization problem to minimize the scaling variability \( \Delta_s = s_{\max} - s_{\min} \) under the constraint \( d_B(D, D_S) \leq \epsilon \), where \( \epsilon > 0 \) is a user-defined tolerance. We develop an algorithmic solution to this problem, ensuring that data augmentation via scaling transformations preserves essential topological features. We further extend our analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our contributions provide a rigorous mathematical framework for dataset normalization in data augmentation pipelines, ensuring that essential topological characteristics are maintained despite scaling transformations.

How Analysis Can Teach Us the Optimal Way to Design Neural Operators

This paper presents a mathematics-informed approach to neural operator design, building upon the theoretical framework established in our prior work. By integrating rigorous mathematical analysis with practical design strategies, we aim to enhance the stability, convergence, generalization, and computational efficiency of neural operators. We revisit key theoretical insights, including stability in high dimensions, exponential convergence, and universality of neural operators. Based on these insights, we provide detailed design recommendations, each supported by mathematical proofs and citations. Our contributions offer a systematic methodology for developing next-gen neural operators with improved performance and reliability.

A Mathematical Analysis of Neural Operator Behaviors

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous mathematical framework for analyzing the behaviors of neural operators, with a focus on their stability, convergence, clustering dynamics, universality, and generalization error. By proposing a list of novel theorems, we provide stability bounds in Sobolev spaces and demonstrate clustering in function space via gradient flow interpretation, guiding neural operator design and optimization. Based on these theoretical gurantees, we aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.