lol.project.lol: Linear Optimal Low-Rank Projection (LOL)
In neurodata/lol: Linear Optimal Low-Rank Projection

Description Usage Arguments Value Details Author(s) References Examples

A function for implementing the Linear Optimal Low-Rank Projection (LOL) Algorithm. This algorithm allows users to find an optimal projection from 'd' to 'r' dimensions, where 'r << d', by combining information from the first and second moments in thet data.

lol.project.lol(
  X,
  Y,
  r,
  second.moment.xfm = FALSE,
  second.moment.xfm.opts = list(),
  first.moment = "delta",
  second.moment = "linear",
  orthogonalize = FALSE,
  robust.first = TRUE,
  robust.second = FALSE,
  ...
)

`X`	`[n, d]` the data with `n` samples in `d` dimensions.
`Y`	`[n]` the labels of the samples with `K` unique labels.
`r`	the rank of the projection. Note that `r >= K`, and `r < d`.
`second.moment.xfm`	whether to use extraneous options in estimation of the second moment component. The transforms specified should be a numbered list of transforms you wish to apply, and will be applied in accordance with `second.moment`.
`second.moment.xfm.opts`	optional arguments to pass to the `second.moment.xfm` option specified. Should be a numbered list of lists, where `second.moment.xfm.opts[[i]]` corresponds to the optional arguments for `second.moment.xfm[[i]]`. Defaults to the default options for each transform scheme.
`first.moment`	the function to capture the first moment. Defaults to `'delta'`. `'delta'` capture the first moment with the hyperplane separating the per-class means. `FALSE` do not capture the first moment.
`second.moment`	the function to capture the second moment. Defaults to `'linear'`. `'linear'` performs PCA on the class-conditional data to capture the second moment, retaining the vectors with the top singular values. Transform options for `second.moment.xfm` and arguments in `second.moment.opts` should be in accordance with the trailing arguments for lol.project.lrlda. `'quadratic'` performs PCA on the data for each class separately to capture the second moment, retaining the vectors with the top singular values from each class's PCA. Transform options for `second.moment.xfm` and arguments in `second.moment.opts` should be in accordance with the trailing arguments for lol.project.pca. `'pls'` performs PLS on the data to capture the second moment, retaining the vectors that maximize the correlation between the different classes. Transform options for `second.moment.xfm` and arguments in `second.moment.opts` should be in accordance with the trailing arguments for lol.project.pls. `FALSE` do not capture the second moment.
`orthogonalize`	whether to orthogonalize the projection matrix. Defaults to `FALSE`.
`robust.first`	whether to perform PCA on a robust estimate of the first moment component or not. A robust estimate corresponds to usage of medians. Defaults to `TRUE`.
`robust.second`	whether to perform PCA on a robust estimate of the second moment component or not. A robust estimate corresponds to usage of a robust covariance matrix, which requires `d < n`. Defaults to `FALSE`.
`...`	trailing args.

A list containing the following:

`A`	`[d, r]` the projection matrix from `d` to `r` dimensions.
`ylabs`	`[K]` vector containing the `K` unique, ordered class labels.
`centroids`	`[K, d]` centroid matrix of the `K` unique, ordered classes in native `d` dimensions.
`priors`	`[K]` vector containing the `K` prior probabilities for the unique, ordered classes.
`Xr`	`[n, r]` the `n` data points in reduced dimensionality `r`.
`cr`	`[K, r]` the `K` centroids in reduced dimensionality `r`.
`second.moment`	the method used to estimate the second moment.
`first.moment`	the method used to estimate the first moment.

For more details see the help vignette: vignette("lol", package = "lolR")

Eric Bridgeford

Joshua T. Vogelstein, et al. "Supervised Dimensionality Reduction for Big Data" arXiv (2020).

library(lolR)
data <- lol.sims.rtrunk(n=200, d=30)  # 200 examples of 30 dimensions
X <- data$X; Y <- data$Y
model <- lol.project.lol(X=X, Y=Y, r=5)  # use lol to project into 5 dimensions

# use lol to project into 5 dimensions, and produce an orthogonal basis for the projection matrix
model <- lol.project.lol(X=X, Y=Y, r=5, orthogonalize=TRUE)

# use LRQDA to estimate the second moment by performing PCA on each class
model <- lol.project.lol(X=X, Y=Y, r=5, second.moment='quadratic')

# use PLS to estimate the second moment
model <- lol.project.lol(X=X, Y=Y, r=5, second.moment='pls')

# use LRLDA to estimate the second moment, and apply a unit transformation
# (according to scale function) with no centering
model <- lol.project.lol(X=X, Y=Y, r=5, second.moment='linear', second.moment.xfm='unit',
                         second.moment.xfm.opts=list(center=FALSE))