Robust visual tracking using structurally random projection and weighted least squares

Sparse representation based visual tracking approaches have attracted increasing interests in the community in recent years. The main idea is to linearly represent each target candidate using a set of target and trivial templates while imposing a sparsity constraint onto the representation coefficients. After we obtain the coefficients using L1-norm minimization methods, the candidate with the lowest error, when it is reconstructed using only the target templates and the associated coefficients, is considered as the tracking result. In spite of promising system performance widely reported, it is unclear if the performance of these trackers can be maximised. In addition, computational complexity caused by the dimensionality of the feature space limits these algorithms in real-time applications. In this paper, we propose a real-time visual tracking method based on structurally random projection and weighted least squares techniques. In particular, to enhance the discriminative capability of the tracker, we introduce background templates to the linear representation framework. To handle appearance variations over time, we relax the sparsity constraint using a weighed least squares (WLS) method to obtain the representation coefficients. To further reduce the computational complexity, structurally random projection is used to reduce the dimensionality of the feature space while preserving the pairwise distances between the data points in the feature space. Experimental results show that the proposed approach outperforms several state-of-the-art tracking methods.

Two-Stage Orthogonal Least Squares Methods for Neural Network Construction

A number of neural networks can be formulated as the linear-in-the-parameters models. Training such networks can be transformed to a model selection problem where a compact model is selected from all the candidates using subset selection algorithms. Forward selection methods are popular fast subset selection approaches. However, they may only produce suboptimal models and can be trapped into a local minimum. More recently, a two-stage fast recursive algorithm (TSFRA) combining forward selection and backward model refinement has been proposed to improve the compactness and generalization performance of the model. This paper proposes unified two-stage orthogonal least squares methods instead of the fast recursive-based methods. In contrast to the TSFRA, this paper derives a new simplified relationship between the forward and the backward stages to avoid repetitive computations using the inherent orthogonal properties of the least squares methods. Furthermore, a new term exchanging scheme for backward model refinement is introduced to reduce computational demand. Finally, given the error reduction ratio criterion, effective and efficient forward and backward subset selection procedures are proposed. Extensive examples are presented to demonstrate the improved model compactness constructed by the proposed technique in comparison with some popular methods.

The alternating least-squares algorithm for CDPCA

Clustering and Disjoint Principal Component Analysis (CDP CA) is a constrained principal component analysis recently proposed for clustering of objects and partitioning of variables, simultaneously, which we have implemented in R language. In this paper, we deal in detail with the alternating least-squares algorithm for CDPCA and highlight its algebraic features for constructing both interpretable principal components and clusters of objects. Two applications are given to illustrate the capabilities of this new methodology.

An efficient parallel implementation of a least squares problem

Least squares solutions are a very important problem, which appear in a broad range of disciplines (for instance, control systems, statistics, signal processing). Our interest in this kind of problems lies in their use of training neural network controllers.

Face recogniton based on the optimal combination of neural networks, eigenfaces and least squares matching methods

Report of a research project of the Fachhochschule Hannover, University of Applied Sciences and Arts, Department of Information Technologies. Automatic face recognition increases the security standards at public places and border checkpoints. The picture inside the identification documents could widely differ from the face, that is scanned under random lighting conditions and for unknown poses. The paper describes an optimal combination of three key algorithms of object recognition, that are able to perform in real time. The camera scan is processed by a recurrent neural network, by a Eigenfaces (PCA) method and by a least squares matching algorithm. Several examples demonstrate the achieved robustness and high recognition rate.

Detecting Adverse Events in Clinical Trial Free Text

Thesis (Master's)--University of Washington, 2013

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Short-term Wind Speed Forecasting using Support Vector Machines

Wind speed forecasting has been becoming an important field of research to support the electricity industry mainly due to the increasing use of distributed energy sources, largely based on renewable sources. This type of electricity generation is highly dependent on the weather conditions variability, particularly the variability of the wind speed. Therefore, accurate wind power forecasting models are required to the operation and planning of wind plants and power systems. A Support Vector Machines (SVM) model for short-term wind speed is proposed and its performance is evaluated and compared with several artificial neural network (ANN) based approaches. A case study based on a real database regarding 3 years for predicting wind speed at 5 minutes intervals is presented.

On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

Properties of Support Vector Machines

Support Vector Machines (SVMs) perform pattern recognition between two point classes by finding a decision surface determined by certain points of the training set, termed Support Vectors (SV). This surface, which in some feature space of possibly infinite dimension can be regarded as a hyperplane, is obtained from the solution of a problem of quadratic programming that depends on a regularization parameter. In this paper we study some mathematical properties of support vectors and show that the decision surface can be written as the sum of two orthogonal terms, the first depending only on the margin vectors (which are SVs lying on the margin), the second proportional to the regularization parameter. For almost all values of the parameter, this enables us to predict how the decision surface varies for small parameter changes. In the special but important case of feature space of finite dimension m, we also show that there are at most m+1 margin vectors and observe that m+1 SVs are usually sufficient to fully determine the decision surface. For relatively small m this latter result leads to a consistent reduction of the SV number.

Notes on PCA, Regularization, Sparsity and Support Vector Machines

We derive a new representation for a function as a linear combination of local correlation kernels at optimal sparse locations and discuss its relation to PCA, regularization, sparsity principles and Support Vector Machines. We first review previous results for the approximation of a function from discrete data (Girosi, 1998) in the context of Vapnik"s feature space and dual representation (Vapnik, 1995). We apply them to show 1) that a standard regularization functional with a stabilizer defined in terms of the correlation function induces a regression function in the span of the feature space of classical Principal Components and 2) that there exist a dual representations of the regression function in terms of a regularization network with a kernel equal to a generalized correlation function. We then describe the main observation of the paper: the dual representation in terms of the correlation function can be sparsified using the Support Vector Machines (Vapnik, 1982) technique and this operation is equivalent to sparsify a large dictionary of basis functions adapted to the task, using a variation of Basis Pursuit De-Noising (Chen, Donoho and Saunders, 1995; see also related work by Donahue and Geiger, 1994; Olshausen and Field, 1995; Lewicki and Sejnowski, 1998). In addition to extending the close relations between regularization, Support Vector Machines and sparsity, our work also illuminates and formalizes the LFA concept of Penev and Atick (1996). We discuss the relation between our results, which are about regression, and the different problem of pattern classification.

From Regression to Classification in Support Vector Machines

We study the relation between support vector machines (SVMs) for regression (SVMR) and SVM for classification (SVMC). We show that for a given SVMC solution there exists a SVMR solution which is equivalent for a certain choice of the parameters. In particular our result is that for $epsilon$ sufficiently close to one, the optimal hyperplane and threshold for the SVMC problem with regularization parameter C_c are equal to (1-epsilon)^{- 1} times the optimal hyperplane and threshold for SVMR with regularization parameter C_r = (1-epsilon)C_c. A direct consequence of this result is that SVMC can be seen as a special case of SVMR.

A Unified Framework for Regularization Networks and Support Vector Machines

Regularization Networks and Support Vector Machines are techniques for solving certain problems of learning from examples -- in particular the regression problem of approximating a multivariate function from sparse data. We present both formulations in a unified framework, namely in the context of Vapnik's theory of statistical learning which provides a general foundation for the learning problem, combining functional analysis and statistics.

A Note on Support Vector Machines Degeneracy

When training Support Vector Machines (SVMs) over non-separable data sets, one sets the threshold $b$ using any dual cost coefficient that is strictly between the bounds of $0$ and $C$. We show that there exist SVM training problems with dual optimal solutions with all coefficients at bounds, but that all such problems are degenerate in the sense that the "optimal separating hyperplane" is given by ${f w} = {f 0}$, and the resulting (degenerate) SVM will classify all future points identically (to the class that supplies more training data). We also derive necessary and sufficient conditions on the input data for this to occur. Finally, we show that an SVM training problem can always be made degenerate by the addition of a single data point belonging to a certain unboundedspolyhedron, which we characterize in terms of its extreme points and rays.