Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Self-weighted Multiple Kernel Learning for Graph-based Clustering and Semi-supervised Classification

Multiple kernel learning (MKL) method is generally believed to perform better than single kernel method. However, some empirical studies show that this is not always true: the combination of multiple kernels may even yield an even worse performance than using a single kernel. There are two possible reasons for the failure: (i) most existing MKL methods assume that the optimal kernel is a linear combination of base kernels, which may not hold true; and (ii) some kernel weights are inappropriately assigned due to noises and carelessly designed algorithms. In this paper, we propose a novel MKL framework by following two intuitive assumptions: (i) each kernel is a perturbation of the consensus kernel; and (ii) the kernel that is close to the consensus kernel should be assigned a large weight. Impressively, the proposed method can automatically assign an appropriate weight to each kernel without introducing additional parameters, as existing methods do. The proposed framework is integrated into a unified framework for graph-based clustering and semi-supervised classification. We have conducted experiments on multiple benchmark datasets and our empirical results verify the superiority of the proposed framework.Comment: Accepted by IJCAI 2018, Code is availabl

A Convex Formulation for Spectral Shrunk Clustering

Spectral clustering is a fundamental technique in the field of data mining and information processing. Most existing spectral clustering algorithms integrate dimensionality reduction into the clustering process assisted by manifold learning in the original space. However, the manifold in reduced-dimensional subspace is likely to exhibit altered properties in contrast with the original space. Thus, applying manifold information obtained from the original space to the clustering process in a low-dimensional subspace is prone to inferior performance. Aiming to address this issue, we propose a novel convex algorithm that mines the manifold structure in the low-dimensional subspace. In addition, our unified learning process makes the manifold learning particularly tailored for the clustering. Compared with other related methods, the proposed algorithm results in more structured clustering result. To validate the efficacy of the proposed algorithm, we perform extensive experiments on several benchmark datasets in comparison with some state-of-the-art clustering approaches. The experimental results demonstrate that the proposed algorithm has quite promising clustering performance.Comment: AAAI201

SADIH: Semantic-Aware DIscrete Hashing

Due to its low storage cost and fast query speed, hashing has been recognized to accomplish similarity search in large-scale multimedia retrieval applications. Particularly supervised hashing has recently received considerable research attention by leveraging the label information to preserve the pairwise similarities of data points in the Hamming space. However, there still remain two crucial bottlenecks: 1) the learning process of the full pairwise similarity preservation is computationally unaffordable and unscalable to deal with big data; 2) the available category information of data are not well-explored to learn discriminative hash functions. To overcome these challenges, we propose a unified Semantic-Aware DIscrete Hashing (SADIH) framework, which aims to directly embed the transformed semantic information into the asymmetric similarity approximation and discriminative hashing function learning. Specifically, a semantic-aware latent embedding is introduced to asymmetrically preserve the full pairwise similarities while skillfully handle the cumbersome n times n pairwise similarity matrix. Meanwhile, a semantic-aware autoencoder is developed to jointly preserve the data structures in the discriminative latent semantic space and perform data reconstruction. Moreover, an efficient alternating optimization algorithm is proposed to solve the resulting discrete optimization problem. Extensive experimental results on multiple large-scale datasets demonstrate that our SADIH can clearly outperform the state-of-the-art baselines with the additional benefit of lower computational costs.Comment: Accepted by The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19

Multi-task CNN Model for Attribute Prediction

This paper proposes a joint multi-task learning algorithm to better predict attributes in images using deep convolutional neural networks (CNN). We consider learning binary semantic attributes through a multi-task CNN model, where each CNN will predict one binary attribute. The multi-task learning allows CNN models to simultaneously share visual knowledge among different attribute categories. Each CNN will generate attribute-specific feature representations, and then we apply multi-task learning on the features to predict their attributes. In our multi-task framework, we propose a method to decompose the overall model's parameters into a latent task matrix and combination matrix. Furthermore, under-sampled classifiers can leverage shared statistics from other classifiers to improve their performance. Natural grouping of attributes is applied such that attributes in the same group are encouraged to share more knowledge. Meanwhile, attributes in different groups will generally compete with each other, and consequently share less knowledge. We show the effectiveness of our method on two popular attribute datasets.Comment: 11 pages, 3 figures, ieee transaction pape

Near-Optimal Coresets of Kernel Density Estimates

We construct near-optimal coresets for kernel density estimate for points in R^d when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size O(sqrt{d log (1/epsilon)}/epsilon), and we show a near-matching lower bound of size Omega(sqrt{d}/epsilon). The upper bound is a polynomial in 1/epsilon improvement when d in [3,1/epsilon^2) (for all kernels except the Gaussian kernel which had a previous upper bound of O((1/epsilon) log^d (1/epsilon))) and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative