Tema- Calidad de pontecia.

...

Reemplazando (28) en (25) se tiene,

[pic 45]

Sustituyendo en la ecuación (29) se obtiene,[pic 46]

) (32)[pic 47]

Para encontrar el argumento que minimiza la expresión (32) se calcula,

(33)[pic 48]

Resolviendo la derivada de la expresión (33) e igualándola a 0 se tiene,

(34)[pic 49]

Si entonces reemplazando en (34) se tiene,[pic 50]

(35)[pic 51]

De la ecuación (35) se despeja y se obtiene,[pic 52]

(36)[pic 53]

Finalmente reemplazando (30) en (36) se tiene,

[pic 54]

(37)[pic 55]

Para desarrollar el algoritmo del gradiente conjugado se llevan a cabo las siguientes etapas:

- Se define un punto inicial .[pic 56]

- Se obtiene el gradiente evaluado en de mediante la expresión . Si , entonces , donde es la dirección inicial.[pic 57][pic 58][pic 59][pic 60][pic 61][pic 62]

- Se calcula mediante la ecuación;[pic 63]

[pic 64]

- Se determina,

[pic 65]

- Se establece,

[pic 66]

Si , entonces se detiene el algoritmo por que se ha encontrado al mínimo de la función. Si entonces se calcula,[pic 67][pic 68]

[pic 69]

Donde es el coeficiente que se calcula para que sea Q-conjugada. [pic 70][pic 71]

[pic 72][pic 73]

Dimensionality reduction (DR) is a key stage for designing pattern recognition and data mining systems when dealing with high-dimensional data sets [1]. The aim of DR methods is to extract lower dimensional, relevant information (called embedded data) from high-dimensional input data, so that both the performance of a pattern recognition system can be improved and data representation becomes more intelligible [2]. Since DR methods are often developed under determined design parameters and pre-established optimization criteria, they still lack the properties of user interaction and controllability, which are characteristic of information visualization procedures [3]. The field of information visualization (Info Vis) is aimed at developing graphical ways of representing data so that information can be more usable and intelligible for the user. Then, one can intuit that DR can be improved by importing some properties of Info Vis methods. This is in fact the premise on which this research is based [4].

This paper presents an attempt to link the field of dimensionality reduction with that of information visualization, in order to harness the special properties of the latter within DR frameworks. In particular, the properties of controllability and interactivity are of interest, which should make the DR outcomes significantly more understandable and tractable for the (no-necessarily-expert) user [5]. These two properties allow the user to have freedom to select the best way for representing data. Specifically, we propose a geometrical strategy to set the weighting factors for linearly combining DR methods. This is done from kernel approximations [6, 7] of conventional methods (Classical Multidimensional Scaling - CMDS [3], Laplacian Eigenmaps – LE, and Locally Linear Embedding - LLE), which are combined to reach a mixture of kernels. To involve the user in the selection of a method, we use a polygonal approach so the points inside the polygon surface defines the degree or level that a kernel is used, that is, the set of weighting factors. Such polygon has as many edges as the number of considered kernels. This approach allows to evaluating visually the behavior of the embedding data regarding the kernel mixture.

For experiments, we use publicly available databases from the UCI Machine Learning Repository [8] as well as a subset of images from Columbia University Image Library [9]. To assess the performance of the kernel mixture, we consider conventional methods of spectral dimensionality reduction such as multidimensional scaling, locally linear embedding and laplacian eigenmaps [10]. The quality of obtained embedded data is quantified by a scaled version of the average agreement rate between K-ary neighborhoods [19]. Provided mixture represents every single dimensionality reduction approach as well as it helps users to find a suitable representation of embedded data within a visual and intuitive framework.

The remaining of the paper is organized as follows: In section 2, data visualization using DR is outlined. Section 3 introduces a novel mathematical geometric model based on a polygonal approach aimed at performing customized DR tasks. Experimental setup and results are shown in Sections 4 and 5, respectively. Finally, Section 6 gathers some final remarks as conclusions and future work.

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Metodos de optimización

An intuitive way of visualizing numerical data is via a 2D or 3D scatter plot, which is a natural and intelligible visualization fashion for human beings. Therefore, it entails that the initial data should be represented into a lower-dimensional space. In this sense, dimensionality reduction takes places, being an important stage within both the pattern recognition and data visualization systems. Correspondingly, DR is aiming at reaching a low-dimensional data representation, upon which both the classification task performance is improved in terms of accuracy, as well as the intrinsic nature of data is properly represented [1]. So, a more realistic and intelligible visualization for the user is obtained [2]. In other words, the goal of dimensionality reduction is to embed a high dimensional data matrix, such that into a low-dimensional, latent data matrix, being , where d D. Figure 1 depicts an instance where a manifold, so-called Swiss roll, is embedded into a 2D representation, which resembles to an unfolded version of the original manifold.[pic 74][pic 75][pic 76][pic 77]

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