Face recognition using Elasticfaces

Though principle component analysis (PCA) and locality preserving projections (LPPs) are two of the most popular linear methods for face recognition, PCA can only see the Euclidean structure of the training set and LPP preserves the nonlinear submanifold structure hidden in the training set. In this paper, we propose the elastic preserving projections (EPPs) which by incorporating the merits of the local geometry and the global information of the training set. EPP outputs a sample subspace which simultaneously preserves the local geometrical structure and exploits the global information of the training set. Different from some other linear dimensionality reduction methods, EPP can be deemed as learning both the coordinates and the affinities between sample points. Furthermore, the effectiveness of our proposed algorithm is analyzed theoretically and confirmed by some experiments on several well-known face databases. The obtained results indicate that EPP significantly outperforms its other rival algorithms.     

Robust structure and motion from outlines of smooth curved surfaces

This paper addresses the problem of estimating the motion of a camera as it observes the outline (or apparent contour) of a solid bounded by a smooth surface in successive image frames. In this context, the surface points that project onto the outline of an object depend on the viewpoint and the only true correspondences between two outlines of the same object are the projections of frontier points where the viewing rays intersect in the tangent plane of the surface. In turn, the epipolar geometry is easily estimated once these correspondences have been identified. Given the apparent contours detected in an image sequence, a robust procedure based on RANSAC and a voting strategy is proposed to simultaneously estimate the camera configurations and a consistent set of frontier point projections by enforcing the redundancy of multiview epipolar geometry. The proposed approach is, in principle, applicable to orthographic, weak-perspective, and affine projection models. Experiments with nine real image sequences are presented for the orthographic projection case, including a quantitative comparison with the ground-truth data for the six data sets for which the latter information is available. Sample visual hulls have been computed from all image sequences for qualitative evaluation.     

On projection methods,convergence and robust formulations in topology optimization

Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.     

Finding Essential Attributes from Binary Data

We consider data sets that consist of n-dimensional binary vectors representing positive and negative examples for some (possibly unknown) phenomenon. A subset S of the attributes (or variables) of such a data set is called a support set if the positive and negative examples can be distinguished by using only the attributes in S. In this paper we study the problem of finding small support sets, a frequently arising task in various fields, including knowledge discovery, data mining, learning theory, logical analysis of data, etc. We study the distribution of support sets in randomly generated data, and discuss why finding small support sets is important. We propose several measures of separation (real valued set functions over the subsets of attributes), formulate optimization models for finding the smallest subsets maximizing these measures, and devise efficient heuristic algorithms to solve these (typically NP-hard) optimization problems. We prove that several of the proposed heuristics have a guaranteed constant approximation ratio, and we report on computational experience comparing these heuristics with some others from the literature both on randomly generated and on real world data sets.     

Density-preserving projections for large-scale local anomaly detection

Outlier or anomaly detection is a fundamental data mining task with the aim to identify data points, events, transactions which deviate from the norm. The identification of outliers in data can provide insights about the underlying data generating process. In general, outliers can be of two kinds: global and local. Global outliers are distinct with respect to the whole data set, while local outliers are distinct with respect to data points in their local neighbourhood. While several approaches have been proposed to scale up the process of global outlier discovery in large databases, this has not been the case for local outliers. We tackle this problem by optimising the use of local outlier factor (LOF) for large and high-dimensional data. We propose projection-indexed nearest-neighbours (PINN), a novel technique that exploits extended nearest-neighbour sets in a reduced-dimensional space to create an accurate approximation for k-nearest-neighbour distances, which is used as the core density measurement within LOF. The reduced dimensionality allows for efficient sub-quadratic indexing in the number of items in the data set, where previously only quadratic performance was possible. A detailed theoretical analysis of random projection (RP) and PINN shows that we are able to preserve the density of the intrinsic manifold of the data set after projection. Experimental results show that PINN outperforms the standard projection methods RP and PCA when measuring LOF for many high-dimensional real-world data sets of up to 300,000 elements and 102,600 dimensions. A further investigation into the use of high-dimensionality-specific indexing such as spatial approximate sample hierarchy (SASH) shows that our novel technique holds benefits over even these types of highly efficient indexing. We cement the practical applications of our novel technique with insights into what it means to find local outliers in real data including image and text data, and include potential applications for this knowledge.     

Document analysis and visualization with zero-inflated poisson

Data visualization is aimed at obtaining a graphic representation of high dimensional information. A data projection over a lower dimensional space is pursued, looking for some structure on the projections. Among the several data projection based methods available, the Generative Topographic Mapping (GTM) has become an important probabilistic framework to model data. The application to document data requires a change in the original (Gaussian) model in order to consider binary or multinomial variables. There have been several modifications on GTM to consider this kind of data, but the resulting latent projections are all scattered on the visualization plane. A document visualization method is proposed in this paper, based on a generative probabilistic model consisting of a mixture of Zero-inflated Poisson distributions. The performance of the method is evaluated in terms of cluster forming for the latent projections with an index based on Fisher’s classifier, and the topology preservation capability is measured with the Sammon’s stress error. A comparison with the GTM implementation with Gaussian, multinomial and Poisson distributions and with a Latent Dirichlet model is presented, observing a greater performance for the proposed method. A graphic presentation of the projections is also provided, showing the advantage of the developed method in terms of visualization and class separation. A detailed analysis of some documents projected on the latent representation showed that most of the documents appearing away from the corresponding cluster could be identified as outliers.     

Two hybrid algorithms for solving split equilibrium problems

The paper considers split equilibrium problems (EPs) in Hilbert spaces and proposes two hybrid algorithms for finding their solution approximations. Three methods including the diagonal subgradient method, the projection method and the proximal method have been used to design the algorithms. Using the diagonal subgradient method for EPs has allowed us to reduce complex computations on bifunctions and feasible sets. The first algorithm is designed with two projections on feasible set and with the prior knowledge of operator norm while the second algorithm is simpler in computations where only one projection on feasible set needs to be implemented and the information of operator norm is not necessary to construct solution approximations. The strongly convergent theorems are established under suitable assumptions imposed on equilibrium bifunctions. The computational performance of the proposed algorithms over existing methods is also illustrated by several preliminary numerical experiments.     

Efficient Euclidean projections via Piecewise Root Finding and its application in gradient projection

Recently, the gradient (subgradient) projection method, especially by incorporating the idea of Nesterov's method, has aroused more and more attention and achieved great successes on constrained optimization problems arising in the field of machine learning, data mining and signal processing. In the gradient projection method, a critical step is how to efficiently project a vector onto a constraint set. In this paper, we propose a unified method called Piecewise Root Finding (PRF) to efficiently calculate Euclidean projections onto three typical constraint sets: ?₁-ball, Elastic Net (EN) and the Intersection of a Hyperplane and a Halfspace (IHH). In our PRF method, we first formulate a Euclidean projection problem as a root finding problem. Then, a Piecewise Root Finding algorithm is applied to find the root and global convergence is guaranteed. Finally, the Euclidean projection result is obtained as a function of the found root in a closed form. Moreover, the sparsity of the projected vector is considered, leading to reduced computational cost for projection onto the ?₁-ball and EN. Empirical studies demonstrate that our PRF algorithm is efficient by comparing it with several state of the art algorithms for Euclidean projections onto the three typical constraint sets mentioned above. Besides, we apply our efficient Euclidean projection algorithm (PRF) to the Gradient Projection with Nesterov's Method (GPNM), which efficiently solves the popular logistic regression problem with the ?₁-ball/EN/IHH constraint. Experimental results on real-world data sets indicate that GPNM has a fast convergence speed.     

Probabilistic principal component subspaces: a hierarchical finitemixture model for data visualization

Visual exploration has proven to be a powerful tool for multivariate data mining and knowledge discovery. Most visualization algorithms aim to find a projection from the data space down to a visually perceivable rendering space. To reveal all of the interesting aspects of multimodal data sets living in a high-dimensional space, a hierarchical visualization algorithm is introduced which allows the complete data set to be visualized at the top level, with clusters and subclusters of data points visualized at deeper levels. The methods involve hierarchical use of standard finite normal mixtures and probabilistic principal component projections, whose parameters are estimated using the expectation-maximization and principal component neural networks under the information theoretic criteria. We demonstrate the principle of the approach on several multimodal numerical data sets, and we then apply the method to the visual explanation in computer-aided diagnosis for breast cancer detection from digital mammograms.     

Video synchronization from human motion using rank constraints

This paper presents a method of synchronizing video sequences that exploits the non-rigidity of sets of 3D point features (e.g., anatomical joint locations) within the scene. The theory is developed for homography, perspective and affine projection models within a unified rank constraint framework that is computationally cheap. An efficient method is then presented that recovers potential frame correspondences, estimates possible synchronization parameters via the Hough transform and refines these parameters using non-linear optimization methods in order to recover synchronization to sub-frame accuracy, even for sequences of unknown and different frame rates. The method is evaluated quantitatively using synthetic data and demonstrated qualitatively on several real sequences.