Jae-Hun Jung

Debananda Chakraborty, Jae-Hun Jung

We consider the Klein-Gordon and sine-Gordon type equations with a point-like potential, which describes the wave phenomenon in disordered media with a defect. The singular potential term yields a critical phenomenon--that is, the solution behavior around the critical parameter value bifurcates into two extreme cases. Pinpointing the critical value with arbitrary accuracy is even more challenging. In this work, we adopt the generalized polynomial chaos (gPC) method to determine the critical values and the mean solutions around such values. First, we consider the critical value associated with the strength of the singular potential for the Klein-Gordon equation. We expand the solution in the random variable associated with the parameter. The obtained partial differential equations are solved using the Chebyshev collocation method. Due to the existence of the singularity, the Gibbs phenomenon appears in the solution, yielding a slow convergence of the numerically computed critical value. To deal with the singularity, we adopt the consistent spectral collocation method. The gPC method, along with the consistent Chebyshev method, determines the critical value and the mean solution highly efficiently. We then consider the sine-Gordon equation, for which the critical value is associated with the initial velocity of the kink soliton solution. The critical behavior in this case is that the solution passes through (particle-pass), is trapped by (particle-capture), or is reflected by (particle-reflection) the singular potential if the initial velocity of the soliton solution is greater than, equal to, or less than the critical value, respectively. We use the gPC mean value rather than reconstructing the solution to find the critical parameter. Numerical results show that the critical value can be determined efficiently and accurately by using the proposed method.

Keunsu Kim, Hanbaek Lyu, Jinsu Kim et al.

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.

Junwon You, Eunwoo Heo, Jae-Hun Jung

Link prediction (LP), inferring the connectivity between nodes, is a significant research area in graph data, where a link represents essential information on relationships between nodes. Although graph neural network (GNN)-based models have achieved high performance in LP, understanding why they perform well is challenging because most comprise complex neural networks. We employ persistent homology (PH), a topological data analysis method that helps analyze the topological information of graphs, to interpret the features used for prediction. We propose a novel method that employs PH for LP (PHLP) focusing on how the presence or absence of target links influences the overall topology. The PHLP utilizes the angle hop subgraph and new node labeling called degree double radius node labeling (Degree DRNL), distinguishing the information of graphs better than DRNL. Using only a classifier, PHLP performs similarly to state-of-the-art (SOTA) models on most benchmark datasets. Incorporating the outputs calculated using PHLP into the existing GNN-based SOTA models improves performance across all benchmark datasets. To the best of our knowledge, PHLP is the first method of applying PH to LP without GNNs. The proposed approach, employing PH while not relying on neural networks, enables the identification of crucial factors for improving performance.

Megan Johnson, Jae-Hun Jung

Persistent Homology (PH) is a useful tool to study the underlying structure of a data set. Persistence Diagrams (PDs), which are 2D multisets of points, are a concise summary of the information found by studying the PH of a data set. However, PDs are difficult to incorporate into a typical machine learning workflow. To that end, two main methods for representing PDs have been developed: kernel methods and vectorization methods. In this paper we propose a new finite-dimensional vector, called the interconnectivity vector, representation of a PD adapted from Bag-of-Words (BoW). This new representation is constructed to demonstrate the connections between the homological features of a data set. This initial definition of the interconnectivity vector proves to be unstable, but we introduce a stabilized version of the vector and prove its stability with respect to small perturbations in the inputs. We evaluate both versions of the presented vectorization on several data sets and show their high discriminative power.