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Curvilinear Distance Analysis versus Isomap. Presented By: Avadhanula.Laxminarayana kumar. Venishetty Kalyan Kumar. Motivation. Comparing two nonlinear projection methods.
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Curvilinear Distance AnalysisversusIsomap. Presented By: Avadhanula.Laxminarayana kumar. Venishetty Kalyan Kumar.
Motivation Comparing two nonlinear projection methods. 1) Isomap 2) Curvilinear Distance Analysis.
Why we need Dimenstion reduction? Data Coming from the realworld are often difficult to understand because of their high dimensionality. Severeal Dimension reduction techniques to better analyze or visualize Complex datasets.
Types of Techniques Linear Projection Methods Principal Component Analysis(PCA) or the original metric Multidimensional Scaling(MDS). Nonlinear Projection Methods. Locally Linear Embedding. Isomap. Curvilinear Distance Analysis.
These two nonlinear projection methods derived more or less directly from the MDS. Isomap differ from the linear MDS by the innovative metrics used to measure the pairwise distances in the data. Curvilinear Distance Analysis shares the same metrics as Isomap.
What is Isomap? Isometric feature mapping. Basic idea behind Isomap is to overcome the limitations of the traditional metric MDS, which is linear,by replacing the Euclidean distance by another metrics. MDS encounters difficulties when projecting nonlinear structures like the Spiral, Swiss roll, openbox ,Cylinder.
Isomap Actually spiral is embedded in a two-dimensional space, but clearly its intrinsic dimension does not exceed one. Only one parameter is sufficient to describe the spiral. Projection from two dimensional to one dimension is not easy because the spiral needs to be unrolled onto a straight line.
Isomap This unfolding is difficult for MDS because the pairwise Euclidean distances after projection are much larger than in the embedding space. Types of distances: 1) Euclidean Distance 2) Curvilinear or geodesic
Distances Two points in a spiral . Euclidian distance between the two points . Geodesic distance(curvilinear distance)
What is geodesic distance(Curvilinear)? Computing distances along an object. For example, a plane cannot fly from New York to Tokyo by following a straight line (a plane is neither a submarine nor a tunneller!). Instead, it has to follow the curvature of the Earth. This comparison explains curvilinear distances are also known as geodesic distances.
Isomap This geodesic distance is approximated by Isomap in the following way. Neighbourhood of the each point is calculated. Neighbourhood of a point may be the k-nearest points or the set of points closer than a radius. (k and being predetermined constants).
Isomap Once the neighbourhood are known, a graph is build, by linking all neighbouring points. Each arc of the graph is labelled with the Euclidean distance between the correspoinding linked points. The geodesic distance between two points is approximated by the sum of the arc lengths along the shortest path linking both points. (shortest path is computed by Dijkstra`s algorithm).
Isomap Once the neighbourhood are known, a graph Is build, by linking all neighbouring points. Each arc of the graph is labelled with the Euclidean distance between the correspoinding linked points. The geodesic distance between two points is approximated by the sum of the arc lengths along the shortest path linking both points. (shortest path is computed by Dijkstra`s algorithm).
Steps to implement Isomap Randomly select „L“ landmarks points in the n data Vectors. Compute the k neighbourhoods and link neighbouring landmarks. Run Dijkstra algorithm to get the square matrix D, which contains the pairwise geodesic distances. Apply traditional metric MDS on matrix D. Which gives the landmark points in the p-dimensional projection space.
Curvilinear Distance Analysis It also uses the same metrics like Isomap . They differ in the way they use the measured disntace. CDA used neural methods. In CDA we proceed with an error function which is minimized by gradient descent .
CDA The error Function of CDA is writeen as The factor ,weighing each term of the errorfunction decreases as its arguments grows and gives values between 0 and 1. This factor allows the algorithm to focus on the preservation of „small distances“.
CDA Algorithm 1) Apply vector quantitization on the raw data. 2) Compute the k neighbourhoods and link those Neighbouring prototypes. 3) Run Dijkstra`s algorithm to get the square matrix D, that contains pairwise geodesic distances. 4) Optimise Ecda by stochastic gradient descent, in order to get coordinates for the prototypes in the projection space.
Comparision between Isomap and CDA From theoretical point of view, two differences distinguish Isomap from CDA: The way they select the landmark points The way they compute the low-dimensional co-ordinates. The „Vector quantization“ finds landmark points that are more representative of the data distribution than the randomly selected landmark points.
Comparision between Isomap and CDA In the above figure CDA detects the uniform distribution of the manifold, where as Isomap produces strange holes.
Conclusion Both algorithms are useful to explore data sets. Both shares the common innovative ideas like an alternative metrics . Isomap has the advantage of speed and algebraical solidity. CDA relies on more complicated techniques like vector quantization and stochastic gradient.
