# GrassmannOptim: Grassmann Manifold Optimization In GrassmannOptim: Grassmann Manifold Optimization

## Description

Maximizing a function F(U), where U is a semi-orthogonal matrix and the function is invariant under an orthogonal transformation of U. An explicit expression of the gradient is not required and the hessian is not used. It includes a global search option using simulated annealing.

 1 2 3 4 GrassmannOptim(objfun, W, sim_anneal = FALSE, temp_init = 20, cooling_rate = 2, max_iter_sa = 100, eps_conv = 1e-05, max_iter = 100, eps_grad = 1e-05, eps_f = .Machine$double.eps, verbose = FALSE)  ## Arguments  objfun a required R function that evaluates value and possibly gradient of the function to be maximized. It returns a list of components in which the component value is required whereas gradient is optional. When gradient is not provided, an approximation is used by default. The parameter of objfun is W that is a list of components described next. W a list object of arguments to be passed to objfun. It contains all arguments required to compute the objective function and eventually the gradient. It has a required component that is the dimension of the matrix U as dim=c(d, p) where d is the number of columns and p is the number of rows d ## Details The algorithm was adapted from Liu, Srivastava and Gallivan (2004) who discussed the geometry of Grassmann manifolds. See also Edelman, Arias and Smith (1998) for more expositions. This is a non-linear optimization program. We describe a basic gradient algorithm for Grassmann manifolds. Let G_{p,d} be the set of all d-dimensional subspaces of \mathrm{R}^n. It is a compact, connected manifold of dimension d(p-d). An element of this manifold is a subspace. It can be represented by a basis or by a projection matrix. Here, the computations are carried in terms of the bases. Let U such that \texttt{Span}(U) \in G_{p,d}. We consider an objective function F(U) to be optimized. Let D(U) be the gradient of the objective function F at the point U. The algorithm starts with an initial value U_i of U. For step size δ in \mathrm{R}^1, a single step of the gradient algorithm is Q_{t+1} = \exp(-δ A) Q_t where Q_t=[U_t,V_t] and V_t is the orthogonal completion of U_t so that Q_t is orthogonal. The matrix A is computed using the directional derivatives of F. The new value of the objective function is F(U_t). The matrix A is skewed-symmetric and \exp(-δ \bold{A}) is orthogonal. The algorithm works by rotating the starting orthonormal basis Q_t to a new basis by left multiplication by an orthogonal matrix. The iterations continues until a stopping criterion is met. Ideally, convergence is met when the norm of the gradient is sufficiently small. But stopping can be set at a fixed number of iterations. An explicit expression of the gradient may not be provided; finite difference approximations are used instead. However, deriving the gradient expression may pay off in terms of the efficiency and reliability of the algorithm. But a differentiable function F that maps G_{p,d} to \mathrm{R}^1 is necessary. The choice of the initial starting value U_i of U is important. We recommend not to use random start for the optimization to avoid a local maximum. Liu et al. (2004) suggested a simulated annealing method to attain a global optimum. ## Value A list containing the following components  Qt optimal orthogonal matrix such that Qt[,1:d] maximizes the objective function. norm_grads  a vector of successive norms of the directional derivative throughout all iterations. The last scalar is the norm of the gradient at the optimal Qt. fvalues  a vector of successive values of the objective function throughout all iterations. The last scalar is the value of the objective function at the optimal Qt. converged if TRUE, the final iterate was considered optimal by the specified termination criteria. ## Warning This program may search for a global maximizer using a simulated annealing stochastic gradient. The choice of the initial temperature, cooling rate and also of the maximum allowable number of iterations within the simulated annealing process affect the success of reaching that global maximum. ## Note This program uses the objective function objfun provided by the user. An expression of the objective function needs to follow the format illustrated in the example. ## Author(s) Kofi Placid Adragni <[email protected]> and Seongho Wu ## References Liu, X.; Srivastava, A,; Gallivan, K. (2004) Optimal linear representations of images for object recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol 26, No. 5, pp 662-666 Edelman, A.; Arias, T. A.; Smith, S. T. (1998) The Geometry of Algorithms with Orthogonality Constraints. SIAM J. Matrix Anal. Appl. Vol. 20, No. 2, pp 303-353 ## See Also nlm, nlminb, optim, optimize, constrOptim for other optimization functions. ## Examples   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 objfun <- function(W){value <- f(W); gradient <- Grad(W); return(list(value=value, gradient=gradient))} f <- function(W){d <- W$dim[1]; Y<-matrix(W$Qt[,1:d], ncol=d); return(0.5*sum(diag(t(Y)%*%W$A%*%Y)))} Grad <- function(W){ Qt <- W$Qt; d <- W$dim[1]; p <- nrow(Qt); grad <- matrix (0, p, p); Y <- matrix(Qt[,1:d], ncol=d); Y0 <- matrix(Qt[,(d+1):p], ncol=(p-d)); return(t(Y) %*% W$A %*% Y0)} p=5; d=2; set.seed(234); a <- matrix(rnorm(p**2), ncol=p); A <- t(a)%*%a; # Exact Solution W <- list(Qt=eigen(A)$vectors[,1:p], dim=c(d,p), A=A); ans <- GrassmannOptim(objfun, W, eps_conv=1e-5, verbose=TRUE); ans$converged # Random starting matrix m<-matrix(rnorm(p**2), ncol=p); m<-t(m)%*%m; W <- list(Qt=eigen(m)$vectors, dim=c(d,p), A=A); ans <- GrassmannOptim(objfun, W, eps_conv=1e-5, verbose=TRUE); plot(ans$fvalues) # Simulated Annealing W <- list(dim=c(d,p), A=A); ans <- GrassmannOptim(objfun, W, sim_anneal=TRUE, max_iter_sa=35, verbose=TRUE); ######## set.seed(13); p=8; nobs=200; d=3; sigma=1.5; sigma0=2; rmvnorm <- function (n, mean = rep(0, nrow(sigma)), sigma = diag(length(mean))) { # This function generates random numbers from the multivariate normal - # see library "mvtnorm" ev <- eigen(sigma, symmetric = TRUE) retval <- ev$vectors %*% diag(sqrt(ev$values), length(ev$values)) %*% t(ev$vectors) retval <- matrix(rnorm(n * ncol(sigma)), nrow = n) %*% retval; retval <- sweep(retval, 2, mean, "+"); colnames(retval) <- names(mean); retval } objfun <- function(W){return(list(value=f(W), gradient=Gradient(W)))} f <- function(W){ Qt <- W$Qt; d <- W$dim[1]; p <- ncol(Qt); Sigmas <- W$sigmas; U <- matrix(Qt[,1:d], ncol=d); V <- matrix(Qt[,(d+1):p], ncol=(p-d)); return(-log(det(t(V)%*%Sigmas$S%*%V))-log(det(t(U)%*%Sigmas$S_res%*%U)))} Gradient <- function(W) {Qt <- W$Qt; d <- W$dim[1]; p <- ncol(Qt); Sigmas <- W$sigmas; U <- matrix(Qt[,1:d], ncol=d); V <- matrix(Qt[,(d+1):p], ncol=(p-d)); terme1 <- solve(t(U)%*%Sigmas$S_res%*%U)%*% t(U)%*%Sigmas$S_res%*%V; terme2 <- t(U)%*%Sigmas$S%*%V%*%solve(t(V)%*%Sigmas$S%*%V); return(2*(terme1 - terme2))} y<-array(runif(n=nobs, min=-2, max=2), c(nobs, 1)); fy<-scale(cbind(y, y^2, y^3),TRUE,FALSE); #Structured error PFC model; Gamma<-diag(p)[,c(1:3)]; Gamma0<-diag(p)[,-c(1:3)]; Omega <-sigma^2*matrix(0.5, ncol=3, nrow=3); diag(Omega)<-sigma^2; Delta<- Gamma%*%Omega%*%t(Gamma) + sigma0^2*Gamma0%*%t(Gamma0); Err <- t(rmvnorm(n=nobs, mean = rep(0, p), sigma = Delta)) beta <- diag(3*c(1, 0.4, 0.4)); X <- t(Gamma%*%beta%*%t(fy) + Err); Xc <- scale(X, TRUE, FALSE); P_F <- fy%*%solve(t(fy)%*%fy)%*%t(fy); S <- t(Xc)%*%Xc/nobs; S_fit <- t(Xc)%*%P_F%*%Xc/nobs; S_res <- S-S_fit; sigmas <- list(S=S, S_fit=S_fit, S_res=S_res, p=p, nobs=nobs); # Random starting matrix; Qt <- svd(matrix(rnorm(p^2), ncol=p))$u; W <- list(Qt=Qt, dim=c(d, p), sigmas=sigmas) ans <- GrassmannOptim(objfun, W, eps_conv=1e-4); ans$converged; ans$fvalues; ans$Qt[,1:3]; # Good starting matrix; Qt <- svd(S_fit)$u; W <- list(Qt=Qt, dim=c(d, p), sigmas=sigmas) ans <- GrassmannOptim(objfun, W, eps_conv=1e-4, verbose=TRUE); ans\$converged; `

GrassmannOptim documentation built on May 30, 2017, 7:41 a.m.