trait Operators extends AnyRef
High level manifold learning operators.
 Alphabetic
 By Inheritance
 Operators
 AnyRef
 Any
 by any2stringadd
 by StringFormat
 by Ensuring
 by ArrowAssoc
 Hide All
 Show All
 Public
 All
Value Members

final
def
!=(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

final
def
##(): Int
 Definition Classes
 AnyRef → Any
 def +(other: String): String
 def >[B](y: B): (Operators, B)

final
def
==(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

final
def
asInstanceOf[T0]: T0
 Definition Classes
 Any

def
clone(): AnyRef
 Attributes
 protected[java.lang]
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )
 def ensuring(cond: (Operators) ⇒ Boolean, msg: ⇒ Any): Operators
 def ensuring(cond: (Operators) ⇒ Boolean): Operators
 def ensuring(cond: Boolean, msg: ⇒ Any): Operators
 def ensuring(cond: Boolean): Operators

final
def
eq(arg0: AnyRef): Boolean
 Definition Classes
 AnyRef

def
equals(arg0: Any): Boolean
 Definition Classes
 AnyRef → Any

def
finalize(): Unit
 Attributes
 protected[java.lang]
 Definition Classes
 AnyRef
 Annotations
 @throws( classOf[java.lang.Throwable] )
 def formatted(fmtstr: String): String

final
def
getClass(): Class[_]
 Definition Classes
 AnyRef → Any

def
hashCode(): Int
 Definition Classes
 AnyRef → Any

final
def
isInstanceOf[T0]: Boolean
 Definition Classes
 Any

def
isomap(data: Array[Array[Double]], d: Int, k: Int, CIsomap: Boolean = true): IsoMap
Isometric feature mapping.
Isometric feature mapping. Isomap is a widely used lowdimensional embedding methods, where geodesic distances on a weighted graph are incorporated with the classical multidimensional scaling. Isomap is used for computing a quasiisometric, lowdimensional embedding of a set of highdimensional 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 lowdimensional embedding based on the pairwise distance between data points, which is generally measured using straightline 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 ndimensional Euclidean space.
The connectivity of each data point in the neighborhood graph is defined as its nearest k Euclidean neighbors in the highdimensional space. This step is vulnerable to "shortcircuit 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 shortcircuit error can alter many entries in the geodesic distance matrix, which in turn can lead to a drastically different (and incorrect) lowdimensional embedding. Conversely, if k is too small, the neighborhood graph may become too sparse to approximate geodesic paths accurately.
This class implements CIsomap 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 MultiDimensional Scaling(MDS) are modified, with everything else remaining unaffected.
References:
 J. B. Tenenbaum, V. de Silva and J. C. Langford A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290(5500):23192323, 2000.
 data
the data set.
 d
the dimension of the manifold.
 k
knearest neighbor.
 CIsomap
CIsomap algorithm if true, otherwise standard algorithm.

def
laplacian(data: Array[Array[Double]], d: Int, k: Int, t: Double = 1): LaplacianEigenmap
Laplacian Eigenmap.
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.
References:
 Mikhail Belkin and Partha Niyogi. Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS, 2001.
 data
the data set.
 d
the dimension of the manifold.
 k
knearest neighbor.
 t
the smooth/width parameter of heat kernel e^{xy2 / t}. Nonpositive value means discrete weights.

def
lle(data: Array[Array[Double]], d: Int, k: Int): LLE
Locally Linear Embedding.
Locally Linear Embedding. It 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 eigenvectorbased optimization technique to find the lowdimensional embedding of points, such that each point is still described with the same linear combination of its neighbors. LLE tends to handle nonuniform sample densities poorly because there is no fixed unit to prevent the weights from drifting as various regions differ in sample densities.
References:
 Sam T. Roweis and Lawrence K. Saul. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science 290(5500):23232326, 2000.
 data
the data set.
 d
the dimension of the manifold.
 k
knearest neighbor.

final
def
ne(arg0: AnyRef): Boolean
 Definition Classes
 AnyRef

final
def
notify(): Unit
 Definition Classes
 AnyRef

final
def
notifyAll(): Unit
 Definition Classes
 AnyRef

final
def
synchronized[T0](arg0: ⇒ T0): T0
 Definition Classes
 AnyRef

def
toString(): String
 Definition Classes
 AnyRef → Any

def
tsne(X: Array[Array[Double]], d: Int = 2, perplexity: Double = 20.0, eta: Double = 200.0, iterations: Int = 1000): TSNE
tdistributed stochastic neighbor embedding.
tdistributed stochastic neighbor embedding. tSNE is a nonlinear dimensionality reduction technique that is particularly well suited for embedding highdimensional data into a space of two or three dimensions, which can then be visualized in a scatter plot. Specifically, it models each highdimensional object by a two or threedimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points.
References:
 L.J.P. van der Maaten. Accelerating tSNE using TreeBased Algorithms. Journal of Machine Learning Research 15(Oct):32213245, 2014.
 L.J.P. van der Maaten and G.E. Hinton. Visualizing NonMetric Similarities in Multiple Maps. Machine Learning 87(1):3355, 2012.
 L.J.P. van der Maaten. Learning a Parametric Embedding by Preserving Local Structure. In Proceedings of the Twelfth International Conference on Artificial Intelligence & Statistics (AISTATS), JMLR W&CP 5:384391, 2009.
 L.J.P. van der Maaten and G.E. Hinton. Visualizing HighDimensional Data Using tSNE. Journal of Machine Learning Research 9(Nov):25792605, 2008.
 X
input data. If X is a square matrix, it is assumed to be the squared distance/dissimilarity matrix.
 d
the dimension of the manifold.
 perplexity
the perplexity of the conditional distribution.
 eta
the learning rate.
 iterations
the number of iterations.

final
def
wait(): Unit
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )

final
def
wait(arg0: Long, arg1: Int): Unit
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )

final
def
wait(arg0: Long): Unit
 Definition Classes
 AnyRef
 Annotations
 @throws( ... )
 def →[B](y: B): (Operators, B)
High level Smile operators in Scala.