# Linear Optimal Low-Rank Projection (LOL) 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. ```r lol.project.lol( X, Y, r, second.moment.xfm = FALSE, second.moment.xfm.opts = list(), first.moment = "delta", second.moment = "linear", orthogonalize = FALSE, robust = FALSE, ... ) ``` ## Arguments - `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`: whether to perform PCA on a robust estimate of the covariance matrix or not. Defaults to `FALSE`. - `...`: trailing args. ## Returns 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. ## Details For more details see the help vignette: `vignette("lol", package = "lolR")` ## Examples ```r 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)) ``` ## References Joshua T. Vogelstein, et al. "Supervised Dimensionality Reduction for Big Data" arXiv (2020). ## Author(s) Eric Bridgeford

lol.project.lol function