Manifold Learning


(isomap data k)(isomap data k d c-isomap)
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.


(laplacian data k)(laplacian data k d t)
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.


(lle data k)(lle data k d)
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

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


(tsne data)(tsne data d perplexity eta 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.


(umap data)(umap data distance)(umap data k d iterations learningRate minDist spread negativeSamples repulsionStrength)(umap data distance k d iterations learningRate minDist spread negativeSamples repulsionStrength)
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.