Multiple manifolds

An interesting yet often overlooked area within dimensionality reduction is the case where the data is drawn from multiple manifolds, that is, rather than the data being sampled from a single manifold, X c it is in fact made up of data sampled from more than one manifold, X = (Xi, X2. Xp where Xi c X2 c. A, . ..,Xp c Without extensions, traditional spectral dimensionality reduction methods will often fail to recover the underlying manifolds of such datasets. 

One such algorithm that seeks to overcome the above problems was presented in 

Another similar algorithm is Sparse Manifold Clustering and Embedding (SMCE) 

One boundary case when considering multiple manifold algorithms is when the different manifolds contain different intrinsic dimensionality. This is a problem that is addressed in [27], however the solution given does not directly address spectral dimensionality reduction. It does however contain principles that could be translated into the domain of spectral dimensionality reduction. The method works by associating a probability density with each manifold and then representing the collection of manifolds by a mixture of their associated density models. Although the method 

n = 980, d = 2). As the noise level increases, the error, measured as 1 — /o (where p is the residual variance), also increases indicating an inaccurate low-dimensiontil embedding 

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A number of techniques are available for coextrusion, some of them patented and available only under license. Basically, three types exist feedblock, multiple manifolds, and a combination of these two (Table 9-18). Productions of coextruded products are able to meet product requirements that range from flat to complex profiles. Figure 8-35 (a) shows a typical 3-layer coextrusion die and (b) examples of rather complex profiles that are routinely extruded. 

Fig. 14 A schematic of the photoinduced charge separation and charge recombination processes in 18(w). A simple orbital diagram is provided which captures the essentials of the ET processes. HD, donor HOMO LD, donor LUMO HA, acceptor HOMO LA, acceptor LUMO. Note that all depicted processes are assumed to take place on the singlet multiplicity manifold.

Manual methods using 1-, 3-, and 6-station manifolds are also available (Fig. 11.4). Using the manifold, up to six extractions can be completed simultaneously and multiple manifolds can be managed by a single operator. Extraction time is dependent on the amount of particulate matter in the sample. Typically drinking waters take approximately 30 min per batch for a 1-L sample. For lake and river water, it is necessary to add Filter Aid 400 in order to keep extraction time from taking hours for a 1-L sample. 

Figure 7.11 Principle of multiple manifold split-and-recombination operations...

Sequential injection systems were conceived [89] to simplify flow-based analytical procedures and enhance versatility and robustness for process monitoring. The confluence connections usually present in segmented flow and flow injection analysis are eliminated and different analytical applications can be implemented without the need for multiple manifold re-configuration. 

The first method comes from the idea that the connections among normally hyperbolic invariant manifolds would form a network, which means that one manifold would be connected with multiple manifolds through homoclinic or heteroclinic intersections. Then, a tangency would signify a location in the phase space where their connections change. This idea offers a clue to understand, based on dynamics, those reactions where one transition state is connected with multiple transition states. In these reaction processes, the branching points of the reaction paths and the reaction rates to each of them are important We expect that analysis of the network is the first step toward this direction. 

A number of techniques are available for coextrusion, some of them patented and available only under license. Basically three types exist feed-block, multiple manifold, and a combination of these two (Table 3-7). 

A multiple manifold die involves the combination of melts within the die. Each inlet port leads to a separate manifold for the individual layers involved. The layers are combined at or close to the final land of the die, and they exist as an integral construction through a single lip. Although the multi-manifold die can be more costly than the feedblock type, it has the advantage of more precise control of individual layer thickness. 

There is no doubt as to the importance of the various open problems discussed in this chapter. If the initial manifold modelling stage is inadequate then the final embedding produced as a result of performing spectral dimensionality reduction will itself be inadequate. Therefore, it is important to spend time attempting to understand the nature of the data used and making decisions based on whether the data is expected to be highly noisy, contain loops, or lie on multiple manifolds. Thankfully, there are solutions to help overcome these data problems albeit at an extra computational cost. 

Gong, D.,Zhao, X., Medioni.G. Robust multiple manifolds structure learning. In Proceedings of the 29th International Conference on Machine Learning (2012)... 

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