Minimax lower bounds for function estimation on graphs

We study minimax lower bounds for function estimation problems on large graph when the target function is smoothly varying over the graph. We derive minimax rates in the context of regression and classification problems on graphs that satisfy an asymptotic shape assumption and with a smoothness condition on the target function, both formulated in terms of the graph Laplacian

Fast Graph Laplacian regularized kernel learning via semidefinite-quadratic-linear programming.

Wu, Xiaoming.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 30-34).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.ivChapter 1 --- Introduction --- p.1Chapter 2 --- Preliminaries --- p.4Chapter 2.1 --- Kernel Learning Theory --- p.4Chapter 2.1.1 --- Positive Semidefinite Kernel --- p.4Chapter 2.1.2 --- The Reproducing Kernel Map --- p.6Chapter 2.1.3 --- Kernel Tricks --- p.7Chapter 2.2 --- Spectral Graph Theory --- p.8Chapter 2.2.1 --- Graph Laplacian --- p.8Chapter 2.2.2 --- Eigenvectors of Graph Laplacian --- p.9Chapter 2.3 --- Convex Optimization --- p.10Chapter 2.3.1 --- From Linear to Conic Programming --- p.11Chapter 2.3.2 --- Second-Order Cone Programming --- p.12Chapter 2.3.3 --- Semidefinite Programming --- p.12Chapter 3 --- Fast Graph Laplacian Regularized Kernel Learning --- p.14Chapter 3.1 --- The Problems --- p.14Chapter 3.1.1 --- MVU --- p.16Chapter 3.1.2 --- PCP --- p.17Chapter 3.1.3 --- Low-Rank Approximation: from SDP to QSDP --- p.18Chapter 3.2 --- Previous Approach: from QSDP to SDP --- p.20Chapter 3.3 --- Our Formulation: from QSDP to SQLP --- p.21Chapter 3.4 --- Experimental Results --- p.23Chapter 3.4.1 --- The Results --- p.25Chapter 4 --- Conclusion --- p.28Bibliography --- p.3

A Generally Semisupervised Dimensionality Reduction Method with Local and Global Regression Regularizations for Recognition

The insufficiency of labeled data is an important problem in image classification such as face recognition. However, unlabeled data are abundant in the real-world application. Therefore, semisupervised learning methods, which corporate a few labeled data and a large number of unlabeled data into learning, have received more and more attention in the field of face recognition. During the past years, graph-based semisupervised learning has been becoming a popular topic in the area of semisupervised learning. In this chapter, we newly present graph-based semisupervised learning method for face recognition. The presented method is based on local and global regression regularization. The local regression regularization has adopted a set of local classification functions to preserve both local discriminative and geometrical information, as well as to reduce the bias of outliers and handle imbalanced data; while the global regression regularization is to preserve the global discriminative information and to calculate the projection matrix for out-of-sample extrapolation. Extensive simulations based on synthetic and real-world datasets verify the effectiveness of the proposed method

A representation learning model based on variational inference and graph autoencoder for predicting lncRNA‑disease associations

Background: Numerous studies have demonstrated that long non-coding RNAs are related to plenty of human diseases. Therefore, it is crucial to predict potential lncRNAdisease associations for disease prognosis, diagnosis and therapy. Dozens of machine learning and deep learning algorithms have been adopted to this problem, yet it is still challenging to learn efficient low-dimensional representations from high-dimensional features of lncRNAs and diseases to predict unknown lncRNA-disease associations accurately. Results: We proposed an end-to-end model, VGAELDA, which integrates variational inference and graph autoencoders for lncRNA-disease associations prediction. VGAELDA contains two kinds of graph autoencoders. Variational graph autoencoders (VGAE) infer representations from features of lncRNAs and diseases respectively, while graph autoencoders propagate labels via known lncRNA-disease associations. These two kinds of autoencoders are trained alternately by adopting variational expectation maximization algorithm. The integration of both the VGAE for graph representation learning, and the alternate training via variational inference, strengthens the capability of VGAELDA to capture efficient low-dimensional representations from high-dimensional features, and hence promotes the robustness and preciseness for predicting unknown lncRNA-disease associations. Further analysis illuminates that the designed co-training framework of lncRNA and disease for VGAELDA solves a geometric matrix completion problem for capturing efficient low-dimensional representations via a deep learning approach. Conclusion: Cross validations and numerical experiments illustrate that VGAELDA outperforms the current state-of-the-art methods in lncRNA-disease association prediction. Case studies indicate that VGAELDA is capable of detecting potential lncRNAdisease associations. The source code and data are available at https:// github. com/ zhang labNKU/ VGAEL DA

Schroedinger Eigenmaps for Manifold Alignment of Multimodal Hyperspectral Images

Multimodal remote sensing is an upcoming field as it allows for many views of the same region of interest. Domain adaption attempts to fuse these multimodal remotely sensed images by utilizing the concept of transfer learning to understand data from different sources to learn a fused outcome. Semisupervised Manifold Alignment (SSMA) maps multiple Hyperspectral images (HSIs) from high dimensional source spaces to a low dimensional latent space where similar elements reside closely together. SSMA preserves the original geometric structure of respective HSIs whilst pulling similar data points together and pushing dissimilar data points apart. The SSMA algorithm is comprised of a geometric component, a similarity component and dissimilarity component. The geometric component of the SSMA method has roots in the original Laplacian Eigenmaps (LE) dimension reduction algorithm and the projection functions have roots in the original Locality Preserving Projections (LPP) dimensionality reduction framework. The similarity and dissimilarity component is a semisupervised component that allows expert labeled information to improve the image fusion process. Spatial-Spectral Schroedinger Eigenmaps (SSSE) was designed as a semisupervised enhancement to the LE algorithm by augmenting the Laplacian matrix with a user-defined potential function. However, the user-defined enhancement has yet to be explored in the LPP framework. The first part of this thesis proposes to use the Spatial-Spectral potential within the LPP algorithm, creating a new algorithm we call the Schroedinger Eigenmap Projections (SEP). Through experiments on publicly available data with expert-labeled ground truth, we perform experiments to compare the performance of the SEP algorithm with respect to the LPP algorithm. The second part of this thesis proposes incorporating the Spatial Spectral potential from SSSE into the SSMA framework. Using two multi-angled HSI’s, we explore the impact of incorporating this potential into SSMA