The Law of Large Numbers in a Metric Space with a Convex Combination Operation

Terán, Pedro; Molchanov, Ilya

doi:10.1007/s10959-006-0043-0

Cited by 33 publications

(56 citation statements)

References 23 publications

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“…This example shows that the definition of expectation does not give natural convex combination as is written in [14]. The property (ii) may not be satisfied if we define the convex combination operation…”

Section: Lemmamentioning

confidence: 98%

“…Example 2 (Example 5, Molchanov, Teran [14]). Define the convex combination operation on a linear normed space E as…”

Section: Examplesmentioning

confidence: 99%

“…We are dealing with definition of expectation given in [14]. The authors presented new and quite general approach basing on properties which usual expectation possess.…”

mentioning

confidence: 99%

“…In the beginning we present the definition of convex combination, convex combination space, the definition of random elements and their expectation and conditional expectation which can be found in [14]. Section 3 presents some problems which can be encountered when one is dealing with definition of expectation in non-linear metric space.…”

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confidence: 99%

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On some definition of expectation of random element in metric space

Bator

Zięba

2009

Annales UMCS, Mathematica

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Abstract. We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results:• conditions under which the metric space is isometric with some real Banach space; • conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.

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Section: Lemmamentioning

confidence: 98%

“…Example 2 (Example 5, Molchanov, Teran [14]). Define the convex combination operation on a linear normed space E as…”

Section: Examplesmentioning

confidence: 99%

“…We are dealing with definition of expectation given in [14]. The authors presented new and quite general approach basing on properties which usual expectation possess.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

On some definition of expectation of random element in metric space

Bator

Zięba

2009

Annales UMCS, Mathematica

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“…The definitions of convex combination operator in a metric space can be found in Terán's work [135].…”

Section: Formulations For Manifold Convex Hullmentioning

confidence: 99%

Fast and accurate image and video analysis on Riemannian manifolds

Zhao¹

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Recent times have seen increasing attention on representing images and videos on Riemannian manifolds. Such representations offer new means of preserving intrinsic data structures, which Euclidean-based representations often fail to capture. Preserving the intrinsic structure can bring superior benefits in terms of richer representations and robustness to variations. However, to intrinsically operate on the manifold incurs expensive computation complexity because of using non-linear operators. Furthermore, it is difficult to extend the existing computer vision techniques which were originally developed in Euclidean space into the manifold. To address these issues, recent research often uses extrinsic approaches such as tangent space projection and kernelised approaches. Unfortunately, these methods are not free of drawbacks in terms of low accuracy and high computational load. To that end, this research studies approaches for analysing manifold features which achieve superior performance from the manifold structures whilst computational complexity can be massively reduced. This thesis illustrates the general steps of manifold approaches for image and video analysis, here called manifold scheme, followed by discussions on the possible ways to reduce the computational complexities. Guided by the manifold scheme, this research further proposes two frameworks to analyse manifold features: random projection on Riemannian manifolds and convex hull on Symmetric Positive Definite manifolds. To solve the problems posed by each framework, we present three solutions. Through experiments on several computer vision tasks, we verified that our proposed methods can massively reduce the computational load, while still achieving competitive performance. In addition, our manifold scheme shows another possible way to reduce the computational load of manifold approaches through the generating manifold model using effective lower-level features with low dimensionality. Thus, this thesis proposes a landmark manifold model generated by effective landmark points for facial emotion recognition, which shows competitive performance with much lower computational complexity compared to state-of-the-art manifold methods. Design of experiments (10%)

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Random Closed Sets and Capacity Functionals

Molchanov

2017

Theory of Random Sets

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The Law of Large Numbers in a Metric Space with a Convex Combination Operation

Cited by 33 publications

References 23 publications

On some definition of expectation of random element in metric space

On some definition of expectation of random element in metric space

Fast and accurate image and video analysis on Riemannian manifolds

Random Closed Sets and Capacity Functionals

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