4.5. Updating a Linear Projection

AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

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Eigenfaces vs. Fisherfaces: recognition using class specific linear projection

IEEE Transactions on Pattern Analysis and Machine Intelligence ◽

10.1109/34.598228 ◽

1997 ◽

Vol 19 (7) ◽

pp. 711-720 ◽

Cited By ~ 7738

Author(s):

P.N. Belhumeur ◽

J.P. Hespanha ◽

D.J. Kriegman

Keyword(s):

Linear Projection

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Over-clocking of Linear Projection Designs through Device Specific Optimisations

2014 IEEE International Parallel & Distributed Processing Symposium Workshops ◽

10.1109/ipdpsw.2014.25 ◽

2014 ◽

Cited By ~ 3

Author(s):

Rui Policarpo Duarte ◽

Christos-Savvas Bouganis

Keyword(s):

Linear Projection

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Review of Linear Projection Methods

Non-Standard Parameter Adaptation for Exploratory Data Analysis - Studies in Computational Intelligence ◽

10.1007/978-3-642-04005-4_3 ◽

2009 ◽

pp. 29-48

Author(s):

Wesam Ashour Barbakh ◽

Ying Wu ◽

Colin Fyfe

Keyword(s):

Projection Methods ◽

Linear Projection

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Discriminant Criteria for Pattern Classification

Advances in Information and Communication Technology Education - Advanced Pattern Recognition Technologies with Applications to Biometrics ◽

10.4018/978-1-60566-200-8.ch003 ◽

2011 ◽

pp. 30-57

Author(s):

David Zhang ◽

Fengxi Song ◽

Yong Xu ◽

Zhizhen Liang

Keyword(s):

Feature Extraction ◽

Pattern Classification ◽

The Other ◽

Linear Projection ◽

Linear Discriminant ◽

Large Margin ◽

The Third ◽

Fisher Discriminant ◽

Squared Error ◽

Fisher Discriminant Criterion

As mentioned in Chapter II, there are two kinds of LDA approaches: classification- oriented LDA and feature extraction-oriented LDA. In most chapters of this session of the book, we focus our attention on the feature extraction aspect of LDA for SSS problems. On the other hand,, with this chapter we present our studies on the pattern classification aspect of LDA for SSS problems. In this chapter, we present three novel classification-oriented linear discriminant criteria. The first one is large margin linear projection (LMLP) which makes full use of the characteristic of the SSS problems. The second one is the minimum norm minimum squared-error criterion which is a modification of the minimum squared-error discriminant criterion. The third one is the maximum scatter difference which is a modification of the Fisher discriminant criterion.

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A characterization of M-Summands

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100048817 ◽

1974 ◽

Vol 76 (1) ◽

pp. 157-159 ◽

Cited By ~ 1

Author(s):

Richard Evans

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Closed Subspace ◽

Structure Theory ◽

Linear Projection ◽

Image Position

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection ifand an L-projection if, insteadA closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

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Bringing Chaos to Order: Historical Memory and the Manipulation of History

Religion and Theology ◽

10.1163/157430108x376546 ◽

2008 ◽

Vol 15 (3-4) ◽

pp. 280-303 ◽

Cited By ~ 2

Author(s):

Richard Ascough

Keyword(s):

Historical Memory ◽

Modern Science ◽

Scientific Models ◽

Social Scientists ◽

Linear Projection ◽

Classical Studies ◽

Non Linear ◽

Historical Investigation ◽

Roman Antiquity

AbstractIn modern approaches to biblical and classical studies enlightenment scientific models have dominated historical investigation. As such, the notion of memories and traditions, even when they are assumed to be invented, are presented as reflecting a linear projection of history, with singular causes of change. Modern science, however, has moved beyond the Newtonian view of mechanics that undergirds such a view and is working with models of chaos and complexity. Social scientists and humanists are lagging behind and are only now beginning to understand the implications for their disciplines. This paper adds another voice to the small but growing cadre of promoters of a non-linear notion of historiography by noting its implications for a project of redescribing Graeco-Roman antiquity.

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A multi-resolution approximation via linear projection for large spatial datasets

Japanese Journal of Statistics and Data Science ◽

10.1007/s42081-020-00092-x ◽

2020 ◽

Author(s):

Toshihiro Hirano

Keyword(s):

Linear Projection ◽

Spatial Datasets

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On Some Classes of Primary Banach Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-088-7 ◽

1977 ◽

Vol 29 (4) ◽

pp. 856-873 ◽

Cited By ~ 6

Author(s):

P. G. Casazza ◽

C. A. Kottman ◽

Bor-Luh Lin

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Bounded Linear ◽

Linear Projection

A Banach spaceXis calledprimary(respectively,prime)if for every (bounded linear) projectionPonXeitherPXor (I—P)X(respectively,PXwith dimPX =∞ ) is isomorphic toX.It is well-known that C0andlp, 1 ≦p≦ ∞ [8;14] are prime. However, it is unknown whether there are other prime Banach spaces. For a discussion on prime and primary Banach spaces, we refer the reader to [9].

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