arXiv subjects

Yaoying Fu

Publications and source records attributed to Yaoying Fu.

3 recordsLinked to original sources

arXiv2025

Persistent Homology for Labeled Datasets: Gromov-Hausdorff Stability and Generalized Landscapes

Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the Gromov-Hausdorff distance and persistent homology, both of which yield isometry-invariant notions of distance between datasets. However, these frameworks do not account for categorical labels, which are intrinsic to many real-world datasets, such as labeled images, pre-clustered data, and semantically segmented shapes. In this paper, we introduce a general framework for labeled metric spaces and develop new notions of Gromov-Hausdorff distance and persistent homology which are adapted to this setting. Our main result shows that our persistent homology construction is stable with respect to our novel notion of Gromov-Hausdorff distance, extending a classic result in topological data analysis. To facilitate computation, we also introduce a labeled version of persistence landscapes and show that the landscape map is Lipschitz.

math.AT↗

arXiv2025

Toric geometry of ReLU neural networks

Given a continuous finitely piecewise linear function $f:\mathbb{R}^{n_0} \to \mathbb{R}$ and a fixed architecture $(n_0,\ldots,n_k;1)$ of feedforward ReLU neural networks, the exact function realization problem is to determine when some network with the given architecture realizes $f$. To develop a systematic way to answer these questions, we establish a connection between toric geometry and ReLU neural networks. This approach enables us to utilize numerous structures and tools from algebraic geometry to study ReLU neural networks. Starting with an unbiased ReLU neural network with rational weights, we define the ReLU fan, the ReLU toric variety, and the ReLU Cartier divisor associated with the network. This work also reveals the connection between the tropical geometry and the toric geometry of ReLU neural networks. As an application of the toric geometry framework, we prove a necessary and sufficient criterion of functions realizable by unbiased shallow ReLU neural networks by computing intersection numbers of the ReLU Cartier divisor and torus-invariant curves.

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arXiv2020

Remarks on the distribution of colors in Gallai colorings

A Gallai coloring of a complete graph $K_n$ is an edge coloring without triangles colored with three different colors. A sequence $e_1\ge \dots \ge e_k$ of positive integers is an $(n,k)$-sequence if $\sum_{i=1}^k e_i=\binom{n}{2}$. An $(n,k)$-sequence is a G-sequence if there is a Gallai coloring of $K_n$ with $k$ colors such that there are $e_i$ edges of color $i$ for all $i,1\le i \le k$. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer $k\ge 3$ there exists an integer $g(k)$ such that every $(n,k)$-sequence is a G-sequence if and only if $n\ge g(k)$. They showed that $g(3)=5, g(4)=8$ and $2k-2\le g(k)\le 8k^2+1$. We show that $g(5)=10$ and give almost matching lower and upper bounds for $g(k)$ by showing that with suitable constants $α,β>0$, $\frac{αk^{1.5}}{\ln k}\le g(k) \le βk^{1.5}$ for all sufficiently large $k$.

math.CO↗