Exact heat kernel on a hypersphere and its applications in kernel SVM

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed, demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This paper presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis

Machine learning methods based on diffusion processes

This thesis presents three distinct machine learning algorithms based on the mathematical formalism and physical idea of diffusion processes. First, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed and tested by computer scientists, demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This thesis presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis. The improvement in classification accuracy compared with kernels based on Euclidean geometry may arise from ameliorating the curse of dimensionality on compact manifolds. Second, the effective dissimilarity transformation (EDT) on empirical dissimilarity hyperspheres is proposed and studied using synthetic and gene expression data sets. Iterating the EDT turns a static data distribution into a dynamical process purely driven by the empirical data set geometry and adaptively ameliorates the curse of dimensionality, partly through changing the topology of a Euclidean feature space \mathbb{R}^{n} into a compact hypersphere S^{n}. The EDT often improves the performance of hierarchical clustering via the automatic grouping of information emerging from global interactions of data points. The EDT is not restricted to hierarchical clustering, and other learning methods based on pairwise dissimilarity should also benefit from the many desirable properties of EDT. Finally, quantum time evolution exhibits rich physics, attributable to the interplay between the density and phase of a wave function. However, unlike classical heat diffusion, the wave nature of quantum mechanics has not yet been extensively explored in modern data analysis. We propose that the Laplace transform of quantum transport (QT) can be used to construct an ensemble of maps from a given complex network to a circle S^{1}, such that closely-related nodes on the network are grouped into sharply concentrated clusters on S^{1}. The resulting QT clustering (QTC) algorithm is as powerful as the state-of-the-art spectral clustering in discerning complex geometric patterns and more robust when clusters show strong density variations or heterogeneity in size. The observed phenomenon of QTC can be interpreted as a collective behavior of the microscopic nodes that evolve as macroscopic cluster “orbitals” in an effective tight-binding model recapitulating the network. In summary, the three machine learning methods are based on three distinct diffusion processes. The dynamic diffusion processes serve as a promising foundation for future development in machine learning methods

Exact Heat Kernel on a Hypersphere and Its Applications in Kernel SVM

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed and tested by Lafferty and Lebanon [1], demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This paper presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis

Electric Field Induced Dewetting of Hydrophobic Nanocavities at Ambient Temperature

The understanding of water dewetting in nanoporous materials is of great importance in various fields of science and technology. Herein, we report molecular dynamics simulation results of dewetting of water droplet in hydrophobic nanocavities between graphene walls under the influence of electric field. At ambient temperature, the rate of dewetting induced by electric field is significantly large. Whereas, it is a very low rate of dewetting induced by high temperature (423 K) due to the strong interaction of the hydrogen-bonding networks of water droplets in nanocavities. In addition, the electric filed induced formation of a water column has been found in a vacuum chamber. When the electric field is turned off, the water column will transform into a water droplet. Importantly, the results demonstrate that the rate of electric field-induced dewetting increases with growth of the electric field. Overall, our results suggest that electric field may have a great potential application for nanomaterial dewetting

Serum SELENBP1 and VCL Are Effective Biomarkers for Clinical and Forensic Diagnosis of Coronary Artery Spasm

Coronary artery spasm (CAS) plays an important role in the pathogenesis of many ischemic heart entities; however, there are no established diagnostic biomarkers for CAS in clinical and forensic settings. This present study aimed to identify such serum biomarkers by establishing a rabbit CAS provocation model and integrating quantitative serum proteomics, parallel reaction monitoring/mass spectrometry-based targeted proteomics, and partial least-squares discriminant analysis (PLS-DA). Our results suggested that SELENBP1 and VCL were potential candidate biomarkers for CAS. In independent clinical samples, SELENBP1 and VCL were validated to be significantly lower in serum but not blood cells from CAS patients, with the reasons for this possibly due to the decreased secretion from cardiomyocytes. The areas under the curve of the receiver operating characteristics (ROC) analysis were 0.9384 for SELENBP1 and 0.9180 for VCL when diagnosing CAS. The CAS risk decreased by 32.3% and 53.6% for every 10 unit increases in the serum SELENBP1 and VCL, respectively. In forensic samples, serum SELENBP1 alone diagnosed CAS-induced deaths at a sensitivity of 100.0% and specificity of 72.73%, and its combination with VCL yielded a diagnostic specificity of 100.0%, which was superior to the traditional biomarkers of cTnI and CK-MB. Therefore, serum SELENBP1 and VCL could be effective biomarkers for both the clinical and forensic diagnosis of CAS

A novel adhesive dual-sensitive hydrogel for sustained release of exosomes derived from M2 macrophages promotes repair of bone defects

The repair of bone defects remains a huge clinical challenge. M2 macrophage-derived exosomes (M2-Exos) can act as immunomodulators to promote fracture healing; however, how to retain the sustained release of exosomes to the target area remains a challenge. Here, we report a composite hydrogel loaded with M2-Exos aiming to accelerate bone defect healing. It was verified that the F127/HA-NB hydrogel had a dense network structure, tissue adhesiveness, and dual sensitivity to temperature and light. F127/HA-NB loaded with M2-Exos (M2-Exos@F127/HA-NB) exhibited good biocompatibility and achieved sustained release of exosomes for up to two weeks. The study showed that both M0-Exos and M2-Exos@F127/HA-NB significantly promoted osteogenic differentiation of rat bone marrow mesenchymal stem cells. The mechanism study implied that M2-Exos activates the Wnt/β-catenin signaling pathway to promote osteogenic differentiation of BMSCs. Finally, we evaluated the osteogenetic effects of M2-Exos@F127/HA-NB in a rat cranial defect model, and the results showed that M2-Exos@F127/HA-NB had superior bone regeneration-promoting effects. This study provides a new strategy for cell-free treatment of bone defects