# Expand an input matrix X using raptObj. Expand an input matrix X using a Random Affine Projection Transformation (RAPT) object. Such objects use random affine projection transformation to the resulting matrix. This allows RAPT objects serve as a basis for a large number of kinds of expansions. If p are the number of features of X, and q are number of expanded features, the applications fall into two broad categories: * p \> q using the Johnson-Lindenstrauss theorem: * Compressed sensing. * Manifold learning. * Dimension reduction. * Graph embedding. * ... * p \< q using Bochner's theorem: * Approximate kernel projection. * Fast approximate SVD. * Estimation of dependence. * ... ```r rapt(X, raptObj) ``` ## Arguments - `X`: Input data matrix - `raptObj`: raptObj generated by raptMake ## Returns A matrix of randomly (but repeatable) features. ## Details Computes $$ X W + b $$ where W = raptObj$W b = raptObj$b ## Examples ```r # Toy problem set.seed(1) nObs <- 100 # Number of observations X <- matrix(seq(-4,+4,length.out = nObs),ncol=1) Ytrue <- sin(5*X) + 2*cos(2*X) # True value Ytrue = g(X) Y <- Ytrue + rnorm(nObs) # Noisy measurement Y # Standardize X Xstd <- scale(X) attributes(Xstd) <- attributes(X) # Bochner (random Fourier) projection object nDim <- NCOL(X) h <- 10 # Estimated by goodness of fit Adj R^2. # Normally this would be fit by cross validation. raptObj <- raptMake(nDim,nDim*200,WdistOpt = list(sd=h), bDistOpt=list(min=-pi,max=+pi)) # Apply raptObj to Xstd to generate features, # keeping unaltered features Xstd as well. Xrapt <- cbind( Xstd, cos( rapt(Xstd,raptObj) ) ) # Standardize results XraptStd <- scale(Xrapt) attributes(XraptStd) <- attributes(Xrapt) # A linear fitting of Y to the features XraptStd lmObj <- lm(Y ~ XraptStd) summary(lmObj) # Plot measurements (Y), predictions (Yhat), # Kernel smoothing with Gaussian kernel and same bandwidth, # true Y without noise. Yhat <- predict(lmObj) plot (X,Y ,main="Linear Fitting", ylim=c(-6,10)) lines(X,Yhat,col="red",lty=1,lwd=2) grid(col="darkgray") kFit <- ksmooth(X,Y,kernel="normal",bandwidth=1/h) lines(kFit$x,kFit$y,lty=1,col="green",lwd=2) lines(X,Ytrue,lty=1,col="blue",lwd=2) legend("topleft", legend=c("Noisy measurements", "Estimated Y from RAPT", "Estimated Y from Kernel Smooth", "True Y"), col=1:4, pch=c( 1,NA,NA,NA), lty=c(NA, 1, 1, 1), lwd=2, bty="n") # Fit sparse model w/LASSO and # lambda criteria = 1 standard deviation. # This avoids overgeneralization errors usually # associated with fitting large numbers of features # to relatively few data points. It also improves # the end effects, which are of paramount importance # in high dimensional problems (since by the curse # of dimensionality, almost all points are close an edge # in high dimensional problems). lassoObj <- easyLASSO(XraptStd,Y) Yhat <- predict(lassoObj, newx = XraptStd) # Use linear fit of prediction Yhat as goodness of fit. summary(lm(Y ~ Yhat)) # Plot results of LASSO fitting # These show LASSO does a better job fitting edges. plot(X,Y,main="LASSO Fitting",ylim=c(-6,10)) lines(X,Yhat,col="red",lty=1,lwd=2) grid(col="darkgray") kFit <- ksmooth(X,Y,kernel="normal",bandwidth=1/h) lines(kFit$x,kFit$y,lty=1,col="green",lwd=2) lines(X,Ytrue,lty=1,col="blue",lwd=2) legend("topleft", legend=c("Noisy measurements", "Estimated Y from RAPT", "Estimated Y from Kernel Smooth", "True Y"), col=1:4, pch=c( 1,NA,NA,NA), lty=c(NA, 1, 1, 1), lwd=2, bty="n") ``` ## References [https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma), [https://en.wikipedia.org/wiki/Bochner%27s_theorem](https://en.wikipedia.org/wiki/Bochner%27s_theorem) ## See Also Details of how the rapt object is built are in `raptMake`.

rapt function