Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights

Gabriel, K. Ruben; Zamir, Shmuel

doi:10.1080/00401706.1979.10489819

Cited by 220 publications

(110 citation statements)

References 17 publications

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“…Wiberg [23] has proposed a method based on the weighted least squares technique, which was later extended by Shum et al [20]. Gabriel and Zamir [7] proposed a method for subspace learning with any choice of weights, where each data point can have a different weight determined on the basis of reliability.…”

Section: Related Workmentioning

confidence: 99%

A Robust PCA Algorithm for Building Representations from Panoramic Images

Skočaj

Bischof

Leonardis

2002

Computer Vision — ECCV 2002

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Abstract. Appearance-based modeling of objects and scenes using PCA has been successfully applied in many recognition tasks. Robust methods which have made the recognition stage less susceptible to outliers, occlusions, and varying illumination have further enlarged the domain of applicability. However, much less research has been done in achieving robustness in the learning stage. In this paper, we propose a novel robust PCA method for obtaining a consistent subspace representation in the presence of outlying pixels in the training images. The method is based on the EM algorithm for estimation of principal subspaces in the presence of missing data. By treating the outlying points as missing pixels, we arrive at a robust PCA representation. We demonstrate experimentally that the proposed method is efficient. In addition, we apply the method to a set of panoramic images to build a representation that enables surveillance and view-based mobile robot localization.

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Section: Related Workmentioning

confidence: 99%

A Robust PCA Algorithm for Building Representations from Panoramic Images

Skočaj

Bischof

Leonardis

2002

Computer Vision — ECCV 2002

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“…The environmental indexes adjusted in this way are called L 2 environmental indexes, because the L 2 norm was used. The described zigzag algorithm is a version of the iterative algorithms existing in the literature; see for example, Digby (1979); Gabriel and Zamir (1979) and Ng and Williams (2001). Pereira and Mexia (2010) proved the convergence of the zigzag algorithm and that the adjusted parameters could be seen as maximum likelihood estimators.…”

Section: The Zigzag Algorithmmentioning

confidence: 99%

Analyzing genotype-by-environment interaction using curvilinear regression

Pereira

Rodrigues

Mejza

et al. 2012

Sci. agric. (Piracicaba, Braz.)

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In the context of multi-environment trials, where a series of experiments is conducted across different environmental conditions, the analysis of the structure of genotype-byenvironment interaction is an important topic. This paper presents a generalization of the joint regression analysis for the cases where the response (e.g. yield) is not linear across environments and can be written as a second (or higher) order polynomial or another non-linear function. After identifying the common form regression function for all genotypes, we propose a selection procedure based on the adaptation of two tests: (i) a test for parallelism of regression curves; and (ii) a test of coincidence for those regressions. When the hypothesis of parallelism is rejected, subgroups of genotypes where the responses are parallel (or coincident) should be identified.The use of the Scheffé multiple comparison method for regression coefficients in second-order polynomials allows to group the genotypes in two types of groups: one with upward-facing concavity (i.e. potential yield growth), and the other with downward-facing concavity (i.e. the yield approaches saturation). Theoretical results for genotype comparison and genotype selection are illustrated with an example of yield from a non-orthogonal series of experiments with winter rye (Secalecereale L.). We have deleted 10 % of that data at random to show that our meteorology is fully applicable to incomplete data sets, often observed in multi-environment trials.

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“…Mixing matrices and coefficients are learned in a supervised manner using a criss-cross regression algorithm [18] [15]. As the number of required parameters to represent shape and appearance within a given accuracy can greatly differ, two reduced-rank matrices R 1 and R 2 are also learned during the model construction process, coupling the two spaces.…”

Section: Learning the Bilinear Modelmentioning

confidence: 99%

Bilinear Active Appearance Models

González–Mora

Torre

Murthi

et al. 2007

2007 IEEE 11th International Conference on Computer Vision

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Appearance Models have been applied to model the space of human faces over the last two decades. In particular, Active Appearance Models (AAMs) have been successfully used for face tracking, synthesis and recognition, and they are one of the state-of-the-art approaches due to its efficiency and representational power. Although widely employed, AAMs suffer from a few drawbacks, such as the inability to isolate pose, identity and expression changes. This paper proposes Bilinear Active Appearance Models (BAAMs), an extension of AAMs, that effectively decouple changes due to pose and expression/identity. We derive a gradient-descent algorithm to efficiently fit BAAMs to new images. Experimental results show how BAAMs improve generalization and convergence with respect to the linear model. In addition, we illustrate decoupling benefits of BAAMs in face recognition across pose. We show how the pose normalization provided by BAAMs increase the recognition performance of commercial systems.

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Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights

Abstract: Reduced rank approximation of matrices has hitherto been possible only by unweighted least squares. This paper presents iterative techniques for obtaining such approximations when weights are introduced. The techniques involve criss-cross regressions with careful initialization.

Cited by 220 publications

References 17 publications

A Robust PCA Algorithm for Building Representations from Panoramic Images

A Robust PCA Algorithm for Building Representations from Panoramic Images

Analyzing genotype-by-environment interaction using curvilinear regression

Bilinear Active Appearance Models

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