Deli Zhao

Linear Laplacian Discrimination for Feature Extraction

Discriminant feature extraction plays a fundamental role in pattern recognition. In this paper, we propose the linear Laplacian discrimination (LLD) algorithm/or discriminant feature extraction. LLD is an extension of linear discriminant analysis (LDA). Our motivation is to address the issue that LDA cannot work well in cases where sample spaces are non-Euclidean. Specifically, we define the within-class scatter and the between-class scatter using similarities which are based on pairwise distances in sample spaces. Thus the structural information of classes is contained in the within-class and the between-class Laplacian matrices which are free from metrics of sample spaces. The optimal discriminant subspace can be derived by controlling the structural evolution of Laplacian matrices. Experiments are performed on the facial database for FRGC version 2. Experimental results show that LLD is effective in extracting discriminant features.

Laplacian PCA and Its Applications

Dimensionality reduction plays a fundamental role in data processing, for which principal component analysis (PCA) is widely used. In this paper, we develop the Laplacian PCA (LPCA) algorithm which is the extension of PCA to a more general form by locally optimizing the weighted scatter. In addition to the simplicity of PCA, the benefits brought by LPCA are twofold: the strong robustness against noise and the weak metric-dependence on sample spaces. The LPCA algorithm is based on the global alignment of locally Gaussian or linear subspaces via an alignment technique borrowed from manifold learning. Based on the coding length of local samples, the weights can be determined to capture the local principal structure of data. We also give the exemplary application of LPCA to manifold learning. Manifold unfolding (non-linear dimensionality reduction) can be performed by the alignment of tangential maps which are linear transformations of tangent coordinates approximated by LPCA. The superiority of LPCA to PCA and kernel PCA is verified by the experiments on face recognition (FRGC version 2 face database) and manifold (Scherk surface) unfolding.

Distant Supervision for Relation Extraction with Matrix Completion

The essence of distantly supervised relation extraction is that it is an incomplete multi-label classification problem with sparse and noisy features. To tackle the sparsity and noise challenges, we propose solving the classification problem using matrix completion on factorized matrix of minimized rank. We formulate relation classification as completing the unknown labels of testing items (entity pairs) in a sparse matrix that concatenates training and testing textual features with training labels. Our algorithmic framework is based on the assumption that the rank of item-byfeature and item-by-label joint matrix is low. We apply two optimization models to recover the underlying low-rank matrix leveraging the sparsity of feature-label matrix. The matrix completion problem is then solved by the fixed point continuation (FPC) algorithm, which can find the global optimum. Experiments on two widely used datasets with different dimensions of textual features demonstrate that our low-rank matrix completion approach significantly outperforms the baseline and the state-of-the-art methods.

Contextual Distance for Data Perception

Structural perception of data plays a fundamental role in pattern analysis and machine learning. In this paper, we develop a new structural perception of data based on local contexts. We first identify the contextual set of a point by finding its nearest neighbors. Then the contextual distance between the point and one of its neighbors is defined by the difference between their contribution to the integrity of the geometric structure of the contextual set, which is depicted by a structural descriptor. The centroid and the coding length are introduced as the examples of descriptors of the contextual set. Furthermore, a directed graph (digraph) is built to model the asymmetry of perception. The edges of the digraph are weighted based on the contextual distances. Thus direction is brought to the undirected data. And the structural perception of data can be performed by mining the properties of the digraph. We also present the method for deriving the global digraph Laplacian from the alignment of the local digraph Laplacians. Experimental results on clustering and ranking of toy problems and real data show the superiority of asymmetric perception.

Classification via semi-Riemannian spaces

In this paper, we develop a geometric framework for linear or nonlinear discriminant subspace learning and classification. In our framework, the structures of classes are conceptualized as a semi-Riemannian manifold which is considered as a submanifold embedded in an ambient semi-Riemannian space. The class structures of original samples can be characterized and deformed by local metrics of the semi-Riemannian space. Semi-Riemannian metrics are uniquely determined by the smoothing of discrete functions and the nullity of the semi-Riemannian space. Based on the geometrization of class structures, optimizing class structures in the feature space is equivalent to maximizing the quadratic quantities of metric tensors in the semi-Riemannian space. Thus supervised discriminant subspace learning reduces to unsupervised semi-Riemannian manifold learning. Based on the proposed framework, a novel algorithm, dubbed as semi-Riemannian discriminant analysis (SRDA), is presented for subspace-based classification. The performance of SRDA is tested on face recognition (singular case) and handwritten capital letter classification (nonsingular case) against existing algorithms. The experimental results show that SRDA works well on recognition and classification, implying that semi-Riemannian geometry is a promising new tool for pattern recognition and machine learning.

Face Recognition via Archetype Hull Ranking

The archetype hull model is playing an important role in large-scale data analytics and mining, but rarely applied to vision problems. In this paper, we migrate such a geometric model to address face recognition and verification together through proposing a unified archetype hull ranking framework. Upon a scalable graph characterized by a compact set of archetype exemplars whose convex hull encompasses most of the training images, the proposed framework explicitly captures the relevance between any query and the stored archetypes, yielding a rank vector over the archetype hull. The archetype hull ranking is then executed for every block of face images to generate a block wise similarity measure that is achieved by comparing two different rank vectors with respect to the same archetype hull. After integrating block wise similarity measurements with learned importance weights, we accomplish a sensible face similarity measure which can support robust and effective face recognition and verification. We evaluate the face similarity measure in terms of experiments performed on three benchmark face databases Multi-PIE, Pubfig83, and LFW, demonstrating its performance superior to the state-of-the-arts.

Sparse Coding and Dictionary Learning with Linear Dynamical Systems

Linear Dynamical Systems (LDSs) are the fundamental tools for encoding spatio-temporal data in various disciplines. To enhance the performance of LDSs, in this paper, we address the challenging issue of performing sparse coding on the space of LDSs, where both data and dictionary atoms are LDSs. Rather than approximate the extended observability with a finite-order matrix, we represent the space of LDSs by an infinite Grassmannian consisting of the orthonormalized extended observability subspaces. Via a homeomorphic mapping, such Grassmannian is embedded into the space of symmetric matrices, where a tractable objective function can be derived for sparse coding. Then, we propose an efficient method to learn the system parameters of the dictionary atoms explicitly, by imposing the symmetric constraint to the transition matrices of the data and dictionary systems. Moreover, we combine the state covariance into the algorithm formulation, thus further promoting the performance of the models with symmetric transition matrices. Comparative experimental evaluations reveal the superior performance of proposed methods on various tasks including video classification and tactile recognition.

Face Recognition Using Face Patch Networks

When face images are taken in the wild, the large variations in facial pose, illumination, and expression make face recognition challenging. The most fundamental problem for face recognition is to measure the similarity between faces. The traditional measurements such as various mathematical norms, Hausdorff distance, and approximate geodesic distance cannot accurately capture the structural information between faces in such complex circumstances. To address this issue, we develop a novel face patch network, based on which we define a new similarity measure called the random path (RP) measure. The RP measure is derived from the collective similarity of paths by performing random walks in the network. It can globally characterize the contextual and curved structures of the face space. To apply the RP measure, we construct two kinds of networks: the in-face network and the out-face network. The in-face network is drawn from any two face images and captures the local structural information. The out-face network is constructed from all the training face patches, thereby modeling the global structures of face space. The two face networks are structurally complementary and can be combined together to improve the recognition performance. Experiments on the Multi-PIE and LFW benchmarks show that the RP measure outperforms most of the state-of-art algorithms for face recognition.

A new retraction for accelerating the Riemannian three-factor low-rank matrix completion algorithm

The Riemannian three-factor matrix completion (R3MC) algorithm is one of the state-of-the-art geometric optimization methods for the low-rank matrix completion problem. It is a nonlinear conjugate-gradient method optimizing on a quotient Riemannian manifold. In the line search step, R3MC approximates the minimum point on the searching curve by minimizing on the line tangent to the curve. However, finding the exact minimum point by iteration is too expensive. We address this issue by proposing a new retraction with a minimizing property. This special property provides the exact minimization for the line search by establishing correspondences between points on the searching curve and points on the tangent line. Accelerated R3MC, which is R3MC equipped with this new retraction, outperforms the original algorithm and other geometric algorithms for matrix completion in our empirical study.

Determining step sizes in geometric optimization algorithms

Optimization on Riemannian manifolds is an intuitive generalization of the traditional optimization algorithms in Euclidean spaces. In these algorithms, minimizing along a search direction becomes minimizing along a search curve lying on a manifold. Computing such a curve to be subsequently searched upon is itself computational intensive. We propose a new minimization scheme aiming to find a better step size utilizing the first order information of the search curve. We prove that this scheme can provide further reduction for the cost function when the retraction and the vector transport are collinear. Then we adapt this scheme to propose a heuristic strategy for line search. In numerical experiments, we apply this heuristic strategy to one of the geometric algorithms for matrix completion and show its feasibility and the potential in accelerating computation.