Yueh-Cheng Kuo

Yueh-Cheng Kuo, Wen-Wei Lin, Ching-Sung Liu

In this paper, a homotopy continuation method for the computation of nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair. Additionally, using degree analysis, we show that the number of positive Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented.

Yi-Shan Chu, Yueh-Cheng Kuo

We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry shows that Rectified Linear Unit (ReLU) networks admit decision functions with a combinatorial structure often described as a tropical rational, namely a difference of tropical polynomials. Focusing on planar binary classification, we design purely sigmoidal MLPs that adhere to the finite-sum format of UAT: a finite linear combination of shifted and scaled sigmoids of affine functions. The resulting models yield decision boundaries that already align with prescribed shapes at initialization and can be refined by standard training if desired. This provides a practical bridge between the tropical perspective and smooth MLPs, enabling interpretable, shape-driven initialization without resorting to ReLU architectures. We focus on the construction and empirical demonstrations in two dimensions; theoretical analysis and higher-dimensional extensions are left for future work.

Tsung-Lin Lee, Yueh-Cheng Kuo

The Candecomp/Parafac (CP) decomposition of the tensor whose maximal dimension is greater than its rank is considered. We derive the upper bound of rank under which the generic uniqueness of CP decomposition is guaranteed. The bound only depends on the dimension of the tensor and the proof is constructive. Under these conditions, an algorithm applying homotopy continuation method is developed for computing the CP decomposition of tensors.

Yueh-Cheng Kuo, Wen-Wei Lin, Shih-Feng Shieh

We construct a nonlinear differential equation of matrix pairs $(\mathcal{M}(t),\mathcal{L}(t))$ that is invariant (the \textbf{Structure-Preserving Property}) in the class of symplectic matrix pairs \begin{align*} \mathbb{S}_{\mathcal{S}_1,\mathcal{S}_2}=\left\{\left(\mathcal{M},\mathcal{L}\right)| \ \mathcal{M}=\left[% \begin{array}{cc} X_{12} & 0 X_{22} & I \end{array}% \right]\mathcal{S}_2, \mathcal{L}=\left[% \begin{array}{cc} I & X_{11} 0 & X_{21} \end{array}% \right]\mathcal{S}_1\right.\nonumber \left. \text{ and }X=\left[% \begin{array}{cc} X_{11} & X_{12} X_{21} & X_{22} \end{array}% \right]\text{is Hermitian}\right\} \end{align*} for certain fixed symplectic matrices $\mathcal{S}_1$ and $\mathcal{S}_2$. Its solution also preserves invariant subspaces on the whole orbit (the \textbf{Eigenvector-Preserving Property}). Such a flow is called a \textit{structure-preserving flow} and is governed by a Riccati differential equation (RDE). In addition, Radon's lemma leads to an explicit form. Therefore, blow-ups for the structure-preserving flows may happen at a finite $t$. To continue, we then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole $\mathbb{R}$ subtracting some isolated points.