Isometric feature mapping.

Isomap is a widely used low-dimensional embedding methods,
where geodesic distances on a weighted graph are incorporated with the
classical multidimensional scaling. Isomap is used for computing a
quasi-isometric, low-dimensional embedding of a set of high-dimensional
data points. Isomap is highly efficient and generally applicable to a broad
range of data sources and dimensionalities.

To be specific, the classical MDS performs low-dimensional embedding based
on the pairwise distance between data points, which is generally measured
using straight-line Euclidean distance. Isomap is distinguished by
its use of the geodesic distance induced by a neighborhood graph
embedded in the classical scaling. This is done to incorporate manifold
structure in the resulting embedding. Isomap defines the geodesic distance
to be the sum of edge weights along the shortest path between two nodes.
The top n eigenvectors of the geodesic distance matrix, represent the
coordinates in the new n-dimensional Euclidean space.

The connectivity of each data point in the neighborhood graph is defined
as its nearest k Euclidean neighbors in the high-dimensional space. This
step is vulnerable to 'short-circuit errors' if k is too large with
respect to the manifold structure or if noise in the data moves the
points slightly off the manifold. Even a single short-circuit error
can alter many entries in the geodesic distance matrix, which in turn
can lead to a drastically different (and incorrect) low-dimensional
embedding. Conversely, if k is too small, the neighborhood graph may
become too sparse to approximate geodesic paths accurately.

This class implements C-Isomap that involves magnifying the regions
of high density and shrink the regions of low density of data points
in the manifold. Edge weights that are maximized in Multi-Dimensional
Scaling(MDS) are modified, with everything else remaining unaffected.

`data` is the data set.
`d` is the dimension of the manifold.
`k` is the number of nearest neighbors.
If `c-isomap` is true, run C-Isomap algorithm. Otherwise standard algorithm.

Kruskal's nonmetric MDS.

In non-metric MDS, only the rank order of entries in the proximity matrix
(not the actual dissimilarities) is assumed to contain the significant
information. Hence, the distances of the final configuration should as
far as possible be in the same rank order as the original data. Note that
a perfect ordinal re-scaling of the data into distances is usually not
possible. The relationship is typically found using isotonic regression.

`proximity` is the non-negative proximity matrix of dissimilarities.
The diagonal should be zero and all other elements should be positive
and symmetric.
`k` is the dimension of the projection.
`tol` is the tolerance for stopping iterations.
`max-iter` is the maximum number of iterations.

Laplacian Eigenmap.

Using the notion of the Laplacian of the nearest neighbor adjacency graph,
Laplacian Eigenmap compute a low dimensional representation of the dataset
that optimally preserves local neighborhood information in a certain sense.
The representation map generated by the algorithm may be viewed as a
discrete approximation to a continuous map that naturally arises from
the geometry of the manifold.

The locality preserving character of the Laplacian Eigenmap algorithm makes
it relatively insensitive to outliers and noise. It is also not prone to
'short circuiting' as only the local distances are used.

`data` is the data set.
`d` is the dimension of the manifold.
`k` is the number of nearest neighbors.
`t` is the smooth/width parameter of heat kernel
e<sup>-||x-y||<sup>2</sup> / t</sup>. Non-positive value means
discrete weights.

Locally Linear Embedding.

LLE has several advantages over Isomap, including faster optimization
when implemented to take advantage of sparse matrix algorithms, and better
results with many problems. LLE also begins by finding a set of the nearest
neighbors of each point. It then computes a set of weights for each point
that best describe the point as a linear combination of its neighbors.
Finally, it uses an eigenvector-based optimization technique to find the
low-dimensional embedding of points, such that each point is still described
with the same linear combination of its neighbors. LLE tends to handle
non-uniform sample densities poorly because there is no fixed unit to
prevent the weights from drifting as various regions differ in sample
densities.

`data` is the data set.
`d` is the dimension of the manifold.
`k` is the number of nearest neighbors.

Classical multidimensional scaling, also known as principal coordinates analysis.

Given a matrix of dissimilarities (e.g. pairwise distances), MDS
finds a set of points in low dimensional space that well-approximates the
dissimilarities in A. We are not restricted to using a Euclidean
distance metric. However, when Euclidean distances are used MDS is
equivalent to PCA.

`proximity` is the non-negative proximity matrix of dissimilarities. The
diagonal should be zero and all other elements should be positive and
symmetric. For pairwise distances matrix, it should be just the plain
distance, not squared.

`k` is the dimension of the projection.

If `positive` is true, estimate an appropriate constant to be added
to all the dissimilarities, apart from the self-dissimilarities, that
makes the learning matrix positive semi-definite. The other formulation of
the additive constant problem is as follows. If the proximity is
measured in an interval scale, where there is no natural origin, then there
is not a sympathy of the dissimilarities to the distances in the Euclidean
space used to represent the objects. In this case, we can estimate a
constant `c` such that proximity + c may be taken as ratio data, and also
possibly to minimize the dimensionality of the Euclidean space required for
representing the objects.

Sammon's mapping.

The Sammon's mapping is an iterative technique for making interpoint
distances in the low-dimensional projection as close as possible to the
interpoint distances in the high-dimensional object. Two points close
together in the high-dimensional space should appear close together in the
projection, while two points far apart in the high dimensional space should
appear far apart in the projection. The Sammon's mapping is a special case
of metric least-square multidimensional scaling.

Ideally when we project from a high dimensional space to a low dimensional
space the image would be geometrically congruent to the original figure.
This is called an isometric projection. Unfortunately it is rarely possible
to isometrically project objects down into lower dimensional spaces. Instead
of trying to achieve equality between corresponding inter-point distances we
can minimize the difference between corresponding inter-point distances.
This is one goal of the Sammon's mapping algorithm. A second goal of the
Sammon's mapping algorithm is to preserve the topology as best as possible
by giving greater emphasize to smaller interpoint distances. The Sammon's
mapping algorithm has the advantage that whenever it is possible to
isometrically project an object into a lower dimensional space it will be
isometrically projected into the lower dimensional space. But whenever an
object cannot be projected down isometrically the Sammon's mapping projects
it down to reduce the distortion in interpoint distances and to limit the
change in the topology of the object.

The projection cannot be solved in a closed form and may be found by an
iterative algorithm such as gradient descent suggested by Sammon. Kohonen
also provides a heuristic that is simple and works reasonably well.

`proximity the non-negative proximity matrix of dissimilarities.
The diagonal should be zero and all other elements should be positive
and symmetric.
`k` is the dimension of the projection.
`lambda` is the initial value of the step size constant in diagonal Newton
method.
`tol` is the  tolerance for stopping iterations.
`step-tol` is the tolerance on step size.
`max-iter` is the maximum number of iterations.

t-distributed stochastic neighbor embedding.

t-SNE is a nonlinear dimensionality reduction technique that is
particularly well suited for embedding high-dimensional data into
a space of two or three dimensions, which can then be visualized
in a scatter plot. Specifically, it models each high-dimensional
object by a two- or three-dimensional point in such a way that
similar objects are modeled by nearby points and dissimilar objects
are modeled by distant points.

`X` is input data. If X is a square matrix, it is assumed to be the
squared distance/dissimilarity matrix.
`d` is the dimension of the manifold.
`perplexity` is the perplexity of the conditional distribution.
`eta` is the learning rate.
`iterations` is the number of iterations.

Unnifold Approximation and Projection.

UMAP is a dimension reduction technique that can be used for visualization
similarly to t-SNE, but also for general non-linear dimension reduction.
The algorithm is founded on three assumptions about the data:

 - The data is uniformly distributed on a Riemannian manifold;
 - The Riemannian metric is locally constant (or can be approximated as such);
 - The manifold is locally connected.

From these assumptions it is possible to model the manifold with a fuzzy
topological structure. The embedding is found by searching for a low
dimensional projection of the data that has the closest possible equivalent
fuzzy topological structure.

`data` is the input data.
`distance` is the distance measure.
`k` is of k-nearest neighbors. Larger values result in more global views
of the manifold, while smaller values result in more local data
being preserved. Generally in the range 2 to 100.
`d` is the target embedding dimensions. defaults to 2 to provide easy
visualization, but can reasonably be set to any integer value
in the range 2 to 100.
`iterations` is the number of iterations to optimize the
low-dimensional representation. Larger values result in more
accurate embedding. Muse be at least 10. Choose wise value
based on the size of the input data, e.g, 200 for large
data (1000+ samples), 500 for small.
`learningRate` is the initial learning rate for the embedding optimization,
default 1.
`minDist` is the desired separation between close points in the embedding
space. Smaller values will result in a more clustered/clumped
embedding where nearby points on the manifold are drawn closer
together, while larger values will result on a more even
disperse of points. The value should be set no-greater than
and relative to the spread value, which determines the scale
at which embedded points will be spread out. default 0.1.
`spread` is the effective scale of embedded points. In combination with
minDist, this determines how clustered/clumped the embedded
points are. default 1.0.
`negativeSamples` is the number of negative samples to select per positive
sample in the optimization process. Increasing this value will result
in greater repulsive force being applied, greater optimization
cost, but slightly more accuracy, default 5.
`repulsionStrength` is the weight applied to negative samples in low
dimensional embedding optimization. Values higher than one will result in
greater weight being given to negative samples, default 1.0.