Reduction dimensionality

As the previous section illustrates, descriptor selection is an important concept. The problem is amplified due to inherently high-dimensional representations (dne to large numbers of descriptors), which have a significant impact on the speed of computation 

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Svozil also suggests a third possibility, whereby a discretized field theory is strictly local in a higher dimensional space d > 4 but appears to be nonlocal in d = 4. While the physical reasons for a such a dimensional reduction remain unclear, such a dimensional shadowing clearly circumvents the no-go theorem by postulating a local dynamics in a higher dimension (see figure 12.9). 

Pal NR, Eluri VK. Two efficient connectionist schemes for structure preserving dimensionality reduction. IEEE Trans Neural Networks 1998 9 1142-54. 

The most serious problem with input analysis methods such as PCA that are designed for dimension reduction is the fact that they focus only on pattern representation rather than on discrimination. Good generalization from a pattern recognition standpoint requires the ability to identify characteristics that both define and discriminate between pattern classes. Methods that do one or the other are insufficient. Consequently, methods such as PLS that simultaneously attempt to reduce the input and output dimensionality while finding the best input-output model may perform better than methods such as PCA that ignore the input-output relationship, or OLS that does not emphasize input dimensionality reduction. 

The proceeding of common methods of data analysis can be traced back to a few fundamental principles the most essential of which are dimensionality reduction, transformation of coordinates, and eigenanalysis. 

Figure 13-4 Illustration of two-dimensional reduction to one dimension by an x-directional rotation of 71.57°.

Solution Processing of Chalcogenide Semiconductors via Dimensional Reduction... 

Particularly desirable among film deposition processes are solution-based techniques, because of the relative simplicity and potential economy of these approaches. However, the covalent character of the metal chalcogenides, which provides the benefit of the desired electronic properties (e.g., high electrical mobility), represents an important barrier for solution processing. Several methods have been developed to overcome the solubility problem, including spray deposition, bath-based techniques, and electrochemical routes, each of which will be discussed in later chapters. In this chapter, a very simple dimensional reduction approach will be considered as a means of achieving a convenient solution-based route to film deposition. 

Figure 3.1. Schematic representation of dimensional reduction for a framework of corner-sharing MX6 octahedra. The M and X atoms are represented by black and white spheres, respectively. In a) though d), reaction with AbX incorporates additional X atoms into the M—X framework, progressively reducing the connectedness and effective dimensionality of the M—X framework. In d), after incorporating n units of AbX (n > 2), the structure is reduced to isolated oligomeric or monomeric components. For clarity, the A atoms are not shown in the figure. [Adapted with permission from [Ref. 16]. Copyright 2001 American Chemical Society.]...

The process for film formation using a dimensional reduction approach requires three conceptually simple steps (Fig. 3.2). The first step involves... 

Figure 3.2. Film formation using a dimensional reduction approach involves three steps 1) breaking up the insoluble extended inorganic framework (a) into more soluble-isolated anionic species, which are separated by some small and volatile cationic species (b). 2) Solution-processing thin films of the precursor (b). 3) Heating the precursor films such that the cationic species and corresponding chalcogen anions are dissociated, leaving behind the targeted inorganic semiconductor (c).

Tulsky, E. G Long, J. R. 2001. Dimensional reduction A practical formalism for manipulating solid structures. Chem. Mater. 13 1149-1166. 

The high dimensional nature of LIBS signals can lead to several computational issues when used in conjunction with many machine learning techniques. Dimensionality reduction is the process by which the high dimensional signals are mapped into a lower dimensional space. The resulting lower dimensional space can enable more robust performance when used in conjunction with pattern recognition techniques. 

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

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