Xiaoyin Pan | Searcharxiv

arXiv2026

Existence-Field Diffusion Model for Spatial Point Processes with Variable Cardinality

We study generative modeling of spatial point processes (SPP), where both the number of points and their spatial configuration are governed by a joint distribution. While diffusion models have achieved strong performance in modeling complex distributions, extending them to variable-cardinality SPP remains challenging. Existing approaches either decouple the modeling of cardinality and spatial structure, or rely on discrete trans-dimensional operations to modify the number of points, resulting in inflexible and asymmetric generative dynamics. We propose the existence-field diffusion model (EFDM) for spatial point processes modeling, where each potential point is associated with an existence variable representing its degree of presence. This enables a unified diffusion process that jointly models both spatial locations and cardinality without requiring explicit discrete transitions. We demonstrate that our approach provides a flexible and general framework for generative modeling of spatial point processes, achieving improved modeling capability on datasets with varying cardinality.

cs.LG↗

arXiv2016

Randomized approximate nearest neighbor search with limited adaptivity

We study the fundamental problem of approximate nearest neighbor search in $d$-dimensional Hamming space $\{0,1\}^d$. We study the complexity of the problem in the famous cell-probe model, a classic model for data structures. We consider algorithms in the cell-probe model with limited adaptivity, where the algorithm makes $k$ rounds of parallel accesses to the data structure for a given $k$. For any $k\ge 1$, we give a simple randomized algorithm solving the approximate nearest neighbor search using $k$ rounds of parallel memory accesses, with $O(k(\log d)^{1/k})$ accesses in total. We also give a more sophisticated randomized algorithm using $O(k+(\frac{1}{k}\log d)^{O(1/k)})$ memory accesses in $k$ rounds for large enough $k$. Both algorithms use data structures of size polynomial in $n$, the number of points in the database. For the lower bound, we prove an $Ω(\frac{1}{k}(\log d)^{1/k})$ lower bound for the total number of memory accesses required by any randomized algorithm solving the approximate nearest neighbor search within $k\le\frac{\log\log d}{2\log\log\log d}$ rounds of parallel memory accesses on any data structures of polynomial size. This lower bound shows that our first algorithm is asymptotically optimal for any constant round $k$. And our second algorithm approaches the asymptotically optimal tradeoff between rounds and memory accesses, in a sense that the lower bound of memory accesses for any $k_1$ rounds can be matched by the algorithm within $k_2=O(k_1)$ rounds. In the extreme, for some large enough $k=Θ\left(\frac{\log\log d}{\log\log\log d}\right)$, our second algorithm matches the $Θ\left(\frac{\log\log d}{\log\log\log d}\right)$ tight bound for fully adaptive algorithms for approximate nearest neighbor search due to Chakrabarti and Regev.

cs.DS↗