def classifier2(Y, soft=False):
M = Y.size[0]
# K = Y*X' / sigma
K = matrix(theta, (width, M))
blas.gemm(X,
Y,
K,
transB='T',
alpha=1.0 / sigma,
beta=-1.0,
m=width)
K = exp(K)
K = div(K - K**-1, K + K**-1)
# complete K
lapack.potrs(L11, K)
K = matrix([K, matrix(0., (N - width, M))], (N, M))
chompack.solve(Lc, K, mode=0)
chompack.solve(Lc, K, mode=1)
x = matrix(b, (M, 1))
blas.gemv(K, z, x, trans='T', beta=1.0)
if soft: return x
else: return matrix([2 * (xk > 0.0) - 1 for xk in x])
def classifier2(Y, soft=False):
M = Y.size[0]
# K = Y*X' / sigma
K = matrix(0.0, (width, M))
blas.gemm(X, Y, K, transB='T', alpha=1.0 / sigma, m=width)
# c[i] = ||Yi||^2 / sigma
ones = matrix(1.0, (max(width, n, M), 1))
c = Y**2 * ones[:n]
blas.scal(1.0 / sigma, c)
# Kij := Kij - 0.5 * (ci + aj)
# = || yi - xj ||^2 / (2*sigma)
blas.ger(ones[:width], c, K, alpha=-0.5)
blas.ger(a[:width], ones[:M], K, alpha=-0.5)
# Kij = exp(Kij)
K = exp(K)
# complete K
lapack.potrs(L11, K)
K = matrix([K, matrix(0., (N - width, M))], (N, M))
chompack.solve(Lc, K, mode=0)
chompack.solve(Lc, K, mode=1)
x = matrix(b, (M, 1))
blas.gemv(K, z, x, trans='T', beta=1.0)
if soft: return x
else: return matrix([2 * (xk > 0.0) - 1 for xk in x])
def classifier2(Y, soft=False):
M = Y.size[0]
W = matrix(0., (width, M))
blas.gemm(X, Y, W, transB='T', alpha=1.0 / sigma, m=width)
lapack.potrs(L11, W)
W = matrix([W, matrix(0., (N - width, M))])
chompack.solve(Lc, W, mode=0)
chompack.solve(Lc, W, mode=1)
x = matrix(b, (M, 1))
blas.gemv(W, z, x, trans='T', beta=1.0)
if soft: return x
else: return matrix([2 * (xk > 0.0) - 1 for xk in x])
def g(x, y, z):
# Solve
#
# [ K d3 ] [ ux_y ]
# [ ] [ ] =
# [ d3' 1'*d3 ] [ ux_b ]
#
# [ bx_y ] [ D ]
# [ ] - [ ] * D3 * (D2 * bx_v + bx_z - bx_w).
# [ bx_b ] [ d' ]
x[:N] -= mul(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:]))
x[N] -= blas.dot(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:]))
# Solve dy1 := K^-1 * x[:N]
blas.copy(x, dy1, n=N)
chompack.solve(L, dy1, mode=0)
chompack.solve(L, dy1, mode=1)
# Find ux_y = dy1 - ux_b * dy2 s.t
#
# d3' * ( dy1 - ux_b * dy2 + ux_b ) = x[N]
#
# i.e. x[N] := ( x[N] - d3'* dy1 ) / ( d3'* ( 1 - dy2 ) ).
x[N] = ( x[N] - blas.dot(d3, dy1) ) / \
( blas.asum(d3) - blas.dot(d3, dy2) )
x[:N] = dy1 - x[N] * dy2
# ux_v = D4 * ( bx_v - D1^-1 (bz_z + D * (ux_y + ux_b))
# - D2^-1 * bz_w )
x[-N:] = mul(
d4, x[-N:] - div(z[:N] + mul(d, x[:N] + x[N]), d1) -
div(z[N:], d2))
# uz_z = - D1^-1 * ( bx_z - D * ( ux_y + ux_b ) - ux_v )
# uz_w = - D2^-1 * ( bx_w - uz_w )
z[:N] += base.mul(d, x[:N] + x[N]) + x[-N:]
z[-N:] += x[-N:]
blas.scal(-1.0, z)
# Return W['di'] * uz
blas.tbmv(W['di'], z, n=2 * N, k=0, ldA=1)
def Fkkt(W):
"""
Custom KKT solver for
[ Qinv 0 0 -D 0 ] [ ux_y ] [ bx_y ]
[ 0 0 0 -d' 0 ] [ ux_b ] [ bx_b ]
[ 0 0 0 -I -I ] [ ux_v ] = [ bx_v ]
[ -D -d -I -D1 0 ] [ uz_z ] [ bz_z ]
[ 0 0 -I 0 -D2 ] [ uz_w ] [ bz_w ]
with D1 = diag(d1), D2 = diag(d2), d1 = W['d'][:N]**2,
d2 = W['d'][N:])**2.
"""
d1, d2 = W['d'][:N]**2, W['d'][N:]**2
d3, d4 = (d1 + d2)**-1, (d1**-1 + d2**-1)**-1
# Factor the chordal matrix K = Qinv + (D_1+D_2)^-1.
K.V = Qinv.V
K[::N + 1] = K[::N + 1] + d3
L = chompack.project(Qc, K)
L = chompack.cholesky(L)
# Solve (Qinv + (D1+D2)^-1) * dy2 = (D1 + D2)^{-1} * 1
blas.copy(d3, dy2)
chompack.solve(L, dy2, mode=0)
chompack.solve(L, dy2, mode=1)
def g(x, y, z):
# Solve
#
# [ K d3 ] [ ux_y ]
# [ ] [ ] =
# [ d3' 1'*d3 ] [ ux_b ]
#
# [ bx_y ] [ D ]
# [ ] - [ ] * D3 * (D2 * bx_v + bx_z - bx_w).
# [ bx_b ] [ d' ]
x[:N] -= mul(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:]))
x[N] -= blas.dot(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:]))
# Solve dy1 := K^-1 * x[:N]
blas.copy(x, dy1, n=N)
chompack.solve(L, dy1, mode=0)
chompack.solve(L, dy1, mode=1)
# Find ux_y = dy1 - ux_b * dy2 s.t
#
# d3' * ( dy1 - ux_b * dy2 + ux_b ) = x[N]
#
# i.e. x[N] := ( x[N] - d3'* dy1 ) / ( d3'* ( 1 - dy2 ) ).
x[N] = ( x[N] - blas.dot(d3, dy1) ) / \
( blas.asum(d3) - blas.dot(d3, dy2) )
x[:N] = dy1 - x[N] * dy2
# ux_v = D4 * ( bx_v - D1^-1 (bz_z + D * (ux_y + ux_b))
# - D2^-1 * bz_w )
x[-N:] = mul(
d4, x[-N:] - div(z[:N] + mul(d, x[:N] + x[N]), d1) -
div(z[N:], d2))
# uz_z = - D1^-1 * ( bx_z - D * ( ux_y + ux_b ) - ux_v )
# uz_w = - D2^-1 * ( bx_w - uz_w )
z[:N] += base.mul(d, x[:N] + x[N]) + x[-N:]
z[-N:] += x[-N:]
blas.scal(-1.0, z)
# Return W['di'] * uz
blas.tbmv(W['di'], z, n=2 * N, k=0, ldA=1)
return g