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# We compute a solution path of the sparse fused lasso dual problem:
#
# \hat{u}(\lambda) =
# \argmin_u \|y - (X^+)^T D^T u\|_2^2 \rm{s.t.} \|\u\|_\infty \leq \lambda
#
# where D is (a multiple of) incidence matrix of a given graph, row-
# binded with (a multiple of) the identity matrix, and X is a full column
# rank predictor matrix, X^+ being its pseudoinverse.
#
# Fortuitously, we never have to fully invert X (i.e. compute its pseudo-
# inverse).
#
# Note: the df estimates at each lambda_k can be thought of as the df
# for all solutions corresponding to lambda in (lambda_k,lambda_{k-1}),
# the open interval to the *right* of the current lambda_k.
dualpathFusedL1X <- function(y, X, D, D0, gamma, approx=FALSE, maxsteps=2000,
minlam=0, rtol=1e-7, btol=1e-7, eps=1e-4,
verbose=FALSE, object=NULL) {
# If we are starting a new path
if (is.null(object)) {
m = nrow(D)
p = ncol(D)
n = length(y)
numedges = m-p
numnodes = p
# Modify y,X,n in the case of a ridge penalty, but
# keep the originals
y0 = y
X0 = X
if (eps>0) {
y = c(y,rep(0,p))
X = rbind(X,diag(sqrt(eps),p))
n = n+p
}
# Find the minimum 2-norm solution, using some linear algebra
# tricks and a little bit of graph theory
L = abs(crossprod(D0))
diag(L) = 0
gr = graph.adjacency(L,mode="upper") # Underlying graph
cl = clusters(gr)
q = cl$no # Number of clusters
i = cl$membership # Cluster membership
# First we project y onto the row space of D*X^+
xy = t(X)%*%y
g = xy
# Here we perform our usual fused lasso solve but
# with g in place of y
x = numeric(p)
# For efficiency, don't loop over singletons
tab = tabulate(i)
oo = which(tab[i]==1)
if (length(oo)>0) {
x[oo] = g[oo]/(diag(L)[oo])
}
# Now all groups with at least two elements
oi = order(i)
cs = cumsum(tab)
grps = which(tab>1)
for (j in grps) {
oo = oi[Seq(cs[j]-tab[j]+1,cs[j])]
Lj = crossprod(Matrix(D[,oo],sparse=TRUE))
x[oo] = as.numeric(solve(Lj,g[oo]))
}
uhat = as.numeric(D%*%x) # Dual solution
betahat = numeric(p) # Primal solution
ihit = which.max(abs(uhat)) # Hitting coordinate
hit = abs(uhat[ihit]) # Critical lambda
s = sign(uhat[ihit]) # Sign
if (verbose) {
cat(sprintf("1. lambda=%.3f, adding coordinate %i, |B|=%i...",
hit,ihit,1))
}
# Now iteratively find the new dual solution, and
# the next critical point
# Things to keep track of, and return at the end
buf = min(maxsteps,1000)
lams = numeric(buf) # Critical lambdas
h = logical(buf) # Hit or leave?
df = numeric(buf) # Degrees of freedom
lams[1] = hit
h[1] = TRUE
df[1] = 0
u = matrix(0,m,buf) # Dual solutions
beta = matrix(0,p,buf) # Primal solutions
u[,1] = uhat
beta[,1] = betahat
# Special interior set over nodes
I0 = rep(TRUE,numnodes)
# Update the graph if we need to, otherwise
# update the special interior set
if (ihit <= numedges) {
ed = which(D[ihit,]!=0)
gr[ed[1],ed[2]] = 0 # Delete edge
newcl = subcomponent(gr,ed[1]) # New cluster
oldcl = which(i==i[ed[1]]) # Old cluster
# If these two clusters aren't the same, update
# the memberships
if (length(newcl)!=length(oldcl) || any(sort(newcl)!=sort(oldcl))) {
i[newcl] = q+1
q = q+1
}
}
else {
I0[ihit-numedges] = FALSE
}
# Other things to keep track of, but not return
r = 1 # Size of boundary set
B = ihit # Boundary set
I = Seq(1,m)[-ihit] # Interior set
D1 = D[-ihit,,drop=FALSE] # Matrix D[I,]
D2 = D[ihit,,drop=FALSE] # Matrix D[B,]
k = 2 # What step are we at?
}
# If iterating already started path
else {
# Grab variables from outer object
lambda = NULL
for (j in 1:length(object)) {
if (names(object)[j] != "pathobjs") {
assign(names(object)[j], object[[j]])
}
}
# Trick: save y,X from outer object
y0 = y
X0 = X
# Grab variables from inner object
for (j in 1:length(object$pathobjs)) {
assign(names(object$pathobjs)[j], object$pathobjs[[j]])
}
lams = lambda
# In the case of a ridge penalty, modify X
if (eps>0) X = rbind(X,diag(sqrt(eps),p))
}
tryCatch({
while (k<=maxsteps && lams[k-1]>=minlam) {
##########
# Check if we've reached the end of the buffer
if (k > length(lams)) {
buf = length(lams)
lams = c(lams,numeric(buf))
h = c(h,logical(buf))
df = c(df,numeric(buf))
u = cbind(u,matrix(0,m,buf))
beta = cbind(beta,matrix(0,p,buf))
}
##########
Ds = as.numeric(t(D2)%*%s)
# Precomputation for the hitting times: first we project
# y and Ds onto the row space of D1*X^+
A = matrix(0,n,q)
z = matrix(0,q,2)
nz = rep(FALSE,q)
# For efficiency, don't loop over singletons
tab = tabulate(i)
oo = which(tab[i]==1)
oo2 = oo[!I0[oo]]
if (length(oo2)>0) {
j = i[oo2]
A[,j] = X[,oo2,drop=FALSE]
z[j,1] = xy[oo2]
z[j,2] = Ds[oo2]
nz[j] = TRUE
}
# Now consider all groups with at least two elements
grps = which(tab>1)
for (j in grps) {
oo = which(i==j)
if (all(!I0[oo])) {
A[,j] = rowMeans(X[,oo,drop=FALSE])
z[j,1] = mean(xy[oo])
z[j,2] = mean(Ds[oo])
nz[j] = TRUE
}
}
nzq = sum(nz)
e = matrix(0,q,2)
if (nzq>0) {
R = qr.R(qr(A[,nz]))
e[nz,] = backsolve(R,forwardsolve(R,z[nz,,drop=FALSE],upper.tri=TRUE,transpose=TRUE))
# Note: using a QR here is preferable than simply calling
# e[nz,] = solve(crossprod(A[,nz]),z[nz,,drop=FALSE]), for
# numerical stablity. Plus, it's not really any slower
}
ea = e[,1]
eb = e[,2]
ga = xy-t(X)%*%(A%*%ea)
gb = Ds-t(X)%*%(A%*%eb)
# If the interior is empty, then nothing will hit
if (r==m) {
fa = ea[i]
fb = eb[i]
hit = 0
}
# Otherwise, find the next hitting time
else {
# Here we perform our usual fused lasso solve but
# with ga in place of y and gb in place of Ds
xa = xb = numeric(p)
fa = fb = numeric(p)
# For efficiency, don't loop over singletons
oo = which(tab[i]==1)
fa[oo] = ea[i][oo]
fb[oo] = eb[i][oo]
oo1 = oo[I0[oo]]
if (length(oo1)>0) {
Ldiag = diag(crossprod(Matrix(D1[,oo1],sparse=TRUE)))
xa[oo1] = ga[oo1]/Ldiag
xb[oo1] = gb[oo1]/Ldiag
}
# Now all groups with at least two elements
oi = order(i)
cs = cumsum(tab)
grps = which(tab>1)
for (j in grps) {
oo = oi[Seq(cs[j]-tab[j]+1,cs[j])]
fa[oo] = ea[j]/length(oo)
fb[oo] = eb[j]/length(oo)
gaj = ga[oo]
gbj = gb[oo]
if (any(I0[oo])) {
Lj = crossprod(Matrix(D1[,oo],sparse=TRUE))
xa[oo] = as.numeric(solve(Lj,gaj))
xb[oo] = as.numeric(solve(Lj,gbj))
}
else {
Lj = crossprod(Matrix(D1[,oo[-1]],sparse=TRUE))
xa[oo][-1] = as.numeric(solve(Lj,(gaj-mean(gaj))[-1]))
xb[oo][-1] = as.numeric(solve(Lj,(gbj-mean(gbj))[-1]))
}
}
a = as.numeric(D1%*%xa)
b = as.numeric(D1%*%xb)
shits = Sign(a)
hits = a/(b+shits)
# Make sure none of the hitting times are larger
# than the current lambda (precision issue)
hits[hits>lams[k-1]+btol] = 0
hits[hits>lams[k-1]] = lams[k-1]
ihit = which.max(hits)
hit = hits[ihit]
shit = shits[ihit]
}
##########
# If nothing is on the boundary, then nothing will leave
# Also, skip this if we are in "approx" mode
if (r==0 || approx) {
leave = 0
}
# Otherwise, find the next leaving time
else {
c = as.numeric(s*(D2%*%fa))
d = as.numeric(s*(D2%*%fb))
leaves = c/d
# c must be negative
leaves[c>=0] = 0
# Make sure none of the leaving times are larger
# than the current lambda (precision issue)
leaves[leaves>lams[k-1]+btol] = 0
leaves[leaves>lams[k-1]] = lams[k-1]
ileave = which.max(leaves)
leave = leaves[ileave]
}
##########
# Stop if the next critical point is negative
if (hit<=0 && leave<=0) break
# If a hitting time comes next
if (hit > leave) {
# Record the critical lambda and properties
lams[k] = hit
h[k] = TRUE
df[k] = nzq
uhat = numeric(m)
uhat[B] = hit*s
uhat[I] = a-hit*b
betahat = fa-hit*fb
# Update our graph if we need to, otherwise
# update the special interior set
if (I[ihit] <= numedges) {
ed = which(D1[ihit,]!=0)
gr[ed[1],ed[2]] = 0 # Delete edge
newcl = subcomponent(gr,ed[1]) # New cluster
oldcl = which(i==i[ed[1]]) # Old cluster
# If these two clusters aren't the same, update
# the memberships
if (length(newcl)!=length(oldcl) || any(sort(newcl)!=sort(oldcl))) {
i[newcl] = q+1
q = q+1
}
}
else {
I0[I[ihit]-numedges] = FALSE
}
# Update all other variables
r = r+1
B = c(B,I[ihit])
I = I[-ihit]
s = c(s,shit)
D2 = rbind(D2,D1[ihit,])
D1 = D1[-ihit,,drop=FALSE]
if (verbose) {
cat(sprintf("\n%i. lambda=%.3f, adding coordinate %i, |B|=%i...",
k,hit,B[r],r))
}
}
# Otherwise a leaving time comes next
else {
# Record the critical lambda and properties
lams[k] = leave
h[k] = FALSE
df[k] = nzq
uhat = numeric(m)
uhat[B] = leave*s
uhat[I] = a-leave*b
betahat = fa-leave*fb
# Update our graph if we need to, otherwise
# update the special interior set
if (B[ileave] <= numedges) {
ed = which(D2[ileave,]!=0)
gr[ed[1],ed[2]] = 1 # Add edge
newcl = subcomponent(gr,ed[1]) # New cluster
oldcl = which(i==i[ed[1]]) # Old cluster
# If these two clusters aren't the same, update
# the memberships
if (length(newcl)!=length(oldcl) || !all(sort(newcl)==sort(oldcl))) {
newno = i[ed[2]]
oldno = i[ed[1]]
i[oldcl] = newno
i[i>oldno] = i[i>oldno]-1
q = q-1
}
}
else {
I0[B[ileave]-numedges] = TRUE
}
# Update all other variables
r = r-1
I = c(I,B[ileave])
B = B[-ileave]
s = s[-ileave]
D1 = rbind(D1,D2[ileave,])
D2 = D2[-ileave,,drop=FALSE]
if (verbose) {
cat(sprintf("\n%i. lambda=%.3f, deleting coordinate %i, |B|=%i...",
k,leave,I[m-r],r))
}
}
u[,k] = uhat
beta[,k] = betahat
# Step counter
k = k+1
}
}, error = function(err) {
err$message = paste(err$message,"\n(Path computation has been terminated;",
" partial path is being returned.)",sep="")
warning(err)})
# Trim
lams = lams[Seq(1,k-1)]
h = h[Seq(1,k-1)]
df = df[Seq(1,k-1)]
u = u[,Seq(1,k-1),drop=FALSE]
beta = beta[,Seq(1,k-1),drop=FALSE]
# If we reached the maximum number of steps
if (k>maxsteps) {
if (verbose) {
cat(sprintf("\nReached the max number of steps (%i),",maxsteps))
cat(" skipping the rest of the path.")
}
completepath = FALSE
}
# If we reached the minimum lambda
else if (lams[k-1]<minlam) {
if (verbose) {
cat(sprintf("\nReached the min lambda (%.3f),",minlam))
cat(" skipping the rest of the path.")
}
completepath = FALSE
}
# Otherwise, note that we completed the path
else completepath = TRUE
# The least squares solution (lambda=0)
bls = NULL
if (completepath) bls = fa
if (verbose) cat("\n")
# Save needed elements for continuing the path
pathobjs = list(type="fused.l1.x" ,r=r, B=B, I=I, Q1=NA, approx=approx,
Q2=NA, k=k, df=df, D1=D1, D2=D2, Ds=Ds, ihit=ihit, m=m, n=n, p=p, q=q,
h=h, q0=NA, rtol=rtol, btol=btol, eps=eps, s=s, y=y, gr=gr, i=i,
numedges=numedges, I0=I0, xy=xy)
colnames(u) = as.character(round(lams,3))
colnames(beta) = as.character(round(lams,3))
return(list(lambda=lams,beta=beta,fit=X0%*%beta,u=u,hit=h,df=df,y=y0,X=X0,
completepath=completepath,bls=bls,gamma=gamma,pathobjs=pathobjs))
}
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