From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 5 Jun 2015 22:49:52 +0000 (-0400)
Subject: Commit a big fucking mess while I refactor the span restriction.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=446018aef1279ff14866ae5e1d803a4a2a7c8024;p=sage.d.git

Commit a big fucking mess while I refactor the span restriction.
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index ff7d195..6fb15ae 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,55 +8,263 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
-def project_span(K):
+def drop_dependent(vs):
     r"""
-    Project ``K`` into its own span.
+    Return the largest linearly-independent subset of ``vs``.
+    """
+    if len(vs) == 0:
+        # ...for lazy enough definitions of linearly-independent
+        return vs
+
+    result = []
+    old_V = VectorSpace(vs[0].parent().base_field(), 0)
+
+    for v in vs:
+        new_V = span(result + [v])
+        if new_V.dimension() > old_V.dimension():
+            result.append(v)
+            old_V = new_V
+
+    return result
+
+
+def iso_space(K):
+    r"""
+    Construct the space `W \times W^{\perp}` isomorphic to the ambient space
+    of ``K`` where `W` is equal to the span of ``K``.
+    """
+    V = K.lattice().vector_space()
+
+    # Create the space W \times W^{\perp} isomorphic to V.
+    W_basis = drop_dependent(K.rays())
+    W = V.subspace_with_basis(W_basis)
+    W_perp = W.complement()
+
+    return W.cartesian_product(W_perp)
+
+
+def ips_iso(K):
+    r"""
+    Construct the IPS isomorphism and its inverse from our paper.
+
+    Given a cone ``K``, the returned isomorphism will split its ambient
+    vector space `V` into a cartesian product `W \times W^{\perp}` where
+    `W` equals the span of ``K``.
+    """
+    V = K.lattice().vector_space()
+    V_iso = iso_space(K)
+    (W, W_perp) = V_iso.cartesian_factors()
+
+    # A space equivalent to V, but using our basis.
+    V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
+
+    def phi(v):
+        # Write v in terms of our custom basis, where the first dim(W)
+        # coordinates are for the W-part of the basis.
+        cs = V_user.coordinates(v)
+
+        w1 = sum([ V_user.basis()[idx]*cs[idx]
+                    for idx in range(0, W.dimension()) ])
+        w2 = sum([ V_user.basis()[idx]*cs[idx]
+                    for idx in range(W.dimension(), V.dimension()) ])
+
+        return V_iso( (w1, w2) )
+
+
+    def phi_inv( pair ):
+        # Crash if the arguments are in the wrong spaces.
+        V_iso(pair)
+
+        #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
+        #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
+        #             for idx in range(0,n-m) ])
+
+        return sum( pair.cartesian_factors() )
+
+
+    return (phi,phi_inv)
+
+
+
+def unrestrict_span(K, K2=None):
+    if K2 is None:
+        K2 = K
+
+    _,phi_inv = ips_iso(K2)
+    V_iso = iso_space(K2)
+    (W, W_perp) = V_iso.cartesian_factors()
+
+    rays = []
+    for r in K.rays():
+        w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ])
+        pair = V_iso( (w, W_perp.zero()) )
+        rays.append( phi_inv(pair) )
+
+    L = ToricLattice(W.dimension() + W_perp.dimension())
+
+    return Cone(rays, lattice=L)
+
+
+
+def intersect_span(K1, K2):
+    r"""
+    Return a new cone obtained by intersecting ``K1`` with the span of ``K2``.
+    """
+    L = K1.lattice()
+
+    if L.rank() != K2.lattice().rank():
+        raise ValueError('K1 and K2 must belong to lattices of the same rank.')
+
+    SL_gens = list(K2.rays())
+    span_K2_gens = SL_gens + [ -g for g in SL_gens ]
+
+    # The lattices have the same rank (see above) so this should work.
+    span_K2 = Cone(span_K2_gens, L)
+    return K1.intersection(span_K2)
+
+
+
+def restrict_span(K, K2=None):
+    r"""
+    Restrict ``K`` into its own span, or the span of another cone.
+
+    INPUT:
+
+    - ``K2`` -- another cone whose lattice has the same rank as this cone.
+
+    OUTPUT:
+
+    A new cone in a sublattice.
 
     EXAMPLES::
 
         sage: K = Cone([(1,)])
-        sage: project_span(K) == K
+        sage: restrict_span(K) == K
         True
 
         sage: K2 = Cone([(1,0)])
-        sage: project_span(K2).rays()
+        sage: restrict_span(K2).rays()
         N(1)
         in 1-d lattice N
         sage: K3 = Cone([(1,0,0)])
-        sage: project_span(K3).rays()
+        sage: restrict_span(K3).rays()
         N(1)
         in 1-d lattice N
-        sage: project_span(K2) == project_span(K3)
+        sage: restrict_span(K2) == restrict_span(K3)
         True
 
     TESTS:
 
     The projected cone should always be solid::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
+        sage: K_S = restrict_span(K)
         sage: K_S.is_solid()
         True
 
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10)
+        sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() )
+        sage: K_S.lattice_dim() == K.dual().dim()
+        True
+
+    This function has ``unrestrict_span()`` as its inverse::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10, solid=True)
+        sage: J = restrict_span(K)
+        sage: K == unrestrict_span(J,K)
+        True
+
+    This function should not affect the dimension of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10)
+        sage: K.dim() == restrict_span(K).dim()
+        True
+
+    Nor should it affect the lineality of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10)
+        sage: lineality(K) == lineality(restrict_span(K))
+        True
+
+    No matter which space we restrict to, the lineality should not
+    increase::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10)
+        sage: J = intersect_span(K, K.dual())
+        sage: lineality(K) >= lineality(restrict_span(J, K.dual()))
+        True
+
     If we do this according to our paper, then the result is proper::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
         sage: P.is_proper()
         True
 
+    If ``K`` is strictly convex, then both ``K_W`` and
+    ``K_star_W.dual()`` should equal ``K`` (after we unrestrict)::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10, strictly_convex=True)
+        sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+        sage: K_star_W_star = restrict_span(K.dual()).dual()
+        sage: j1 = unrestrict_span(K_W, K.dual())
+        sage: j2 = unrestrict_span(K_star_W_star, K.dual())
+        sage: j1 == j2
+        True
+        sage: j1 == K
+        True
+        sage: K; [ list(r) for r in K.rays() ]
+
+    Test the proposition in our paper concerning the duals, where the
+    subspace `W` is the span of `K^{*}`::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False)
+        sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual())
+        sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual()
+        sage: K_W.nrays() == K_star_W_star.nrays()
+        True
+        sage: K_W.dim() == K_star_W_star.dim()
+        True
+        sage: lineality(K_W) == lineality(K_star_W_star)
+        True
+        sage: K_W.is_solid() == K_star_W_star.is_solid()
+        True
+        sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex()
+        True
+
     """
-    L = K.lattice()
-    F = L.base_field()
-    Q = L.quotient(K.sublattice_complement())
-    vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+    if K2 is None:
+        K2 = K
+
+    phi,_ = ips_iso(K2)
+    (W, W_perp) = iso_space(K2).cartesian_factors()
+
+    ray_pairs = [ phi(r) for r in K.rays() ]
+
+    if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
+        msg = 'Cone has nonzero components in W-perp!'
+        raise ValueError(msg)
 
-    newL = None
-    if len(vecs) == 0:
-        newL = ToricLattice(0)
+    # Represent the cone in terms of a basis for W, i.e. with smaller
+    # vectors.
+    ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
 
-    return Cone(vecs, lattice=newL)
+    L = ToricLattice(W.dimension())
+
+    return Cone(ws, lattice=L)
 
 
 
@@ -112,6 +320,7 @@ def lineality(K):
     The lineality of a cone should be an integer between zero and the
     dimension of the ambient space, inclusive::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10)
         sage: l = lineality(K)
         sage: l in ZZ
@@ -121,6 +330,7 @@ def lineality(K):
 
     A strictly convex cone should have lineality zero::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10, strictly_convex = True)
         sage: lineality(K)
         0
@@ -191,6 +401,7 @@ def codim(K):
     The codimension of a cone should be an integer between zero and
     the dimension of the ambient space, inclusive::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10)
         sage: c = codim(K)
         sage: c in ZZ
@@ -200,12 +411,14 @@ def codim(K):
 
     A solid cone should have codimension zero::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10, solid = True)
         sage: codim(K)
         0
 
     The codimension of a cone is equal to the lineality of its dual::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim = 10, solid = True)
         sage: codim(K) == lineality(K.dual())
         True
@@ -268,11 +481,12 @@ def discrete_complementarity_set(K):
     The complementarity set of the dual can be obtained by switching the
     components of the complementarity set of the original cone::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K1 = random_cone(max_dim=6)
         sage: K2 = K1.dual()
         sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
         sage: actual = discrete_complementarity_set(K1)
-        sage: actual == expected
+        sage: sorted(actual) == sorted(expected)
         True
 
     """
@@ -352,18 +566,33 @@ def LL(K):
     every pair `\left( x,s \right)` in the discrete complementarity set
     of the cone::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=8, max_rays=10)
         sage: C_of_K = discrete_complementarity_set(K)
         sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
         sage: sum(map(abs, l))
         0
 
+    The Lyapunov-like transformations on a cone and its dual are related
+    by transposition, but we're not guaranteed to compute transposed
+    elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
+    \right)`
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
+        sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
+        sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
+        sage: LL2_vecs = [ V(m.list()) for m in LL2   ]
+        sage: V.span(LL1_vecs) == V.span(LL2_vecs)
+        True
+
     """
     V = K.lattice().vector_space()
 
     C_of_K = discrete_complementarity_set(K)
 
-    tensor_products = [s.tensor_product(x) for (x,s) in C_of_K]
+    tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
 
     # Sage doesn't think matrices are vectors, so we have to convert
     # our matrices to vectors explicitly before we can figure out how
@@ -465,8 +694,8 @@ def lyapunov_rank(K):
     [Orlitzky/Gowda]_::
 
         sage: R5 = VectorSpace(QQ, 5)
-        sage: gens = R5.basis() + [ -r for r in R5.basis() ]
-        sage: K = Cone(gens)
+        sage: gs = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gs)
         sage: lyapunov_rank(K)
         25
 
@@ -499,8 +728,8 @@ def lyapunov_rank(K):
         sage: neg_e1 = (-1,0,0,0,0)
         sage: e2 = (0,1,0,0,0)
         sage: neg_e2 = (0,-1,0,0,0)
-        sage: zero = (0,0,0,0,0)
-        sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero])
+        sage: z = (0,0,0,0,0)
+        sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
         sage: lyapunov_rank(K)
         19
         sage: K.lattice_dim()**2 - K.dim()*codim(K)
@@ -535,6 +764,7 @@ def lyapunov_rank(K):
     The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
+        sage: set_random_seed()
         sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: K = K1.cartesian_product(K2)
@@ -544,16 +774,25 @@ def lyapunov_rank(K):
     The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
     itself [Rudolf et al.]_::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=10, max_rays=10)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
+    Make sure we exercise the non-strictly-convex/non-solid case::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
     The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
     be any number between `1` and `n` inclusive, excluding `n-1`
     [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
     trivial cone in a trivial space as well. However, in zero dimensions,
     the Lyapunov rank of the trivial cone will be zero::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
@@ -565,6 +804,7 @@ def lyapunov_rank(K):
     In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
     Lyapunov rank `n-1` in `n` dimensions::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=10)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
@@ -574,10 +814,11 @@ def lyapunov_rank(K):
     The calculation of the Lyapunov rank of an improper cone can be
     reduced to that of a proper cone [Orlitzky/Gowda]_::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=10)
         sage: actual = lyapunov_rank(K)
-        sage: K_S = project_span(K)
-        sage: P = project_span(K_S.dual()).dual()
+        sage: K_S = restrict_span(K)
+        sage: P = restrict_span(K_S.dual()).dual()
         sage: l = lineality(K)
         sage: c = codim(K)
         sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
@@ -586,11 +827,13 @@ def lyapunov_rank(K):
 
     The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
 
+        sage: set_random_seed()
         sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: lyapunov_rank(K) == len(LL(K))
         True
 
     """
+    K_orig = K
     beta = 0
 
     m = K.dim()
@@ -599,14 +842,23 @@ def lyapunov_rank(K):
 
     if m < n:
         # K is not solid, project onto its span.
-        K = project_span(K)
+        K = restrict_span(K)
 
         # Lemma 2
         beta += m*(n - m) + (n - m)**2
 
     if l > 0:
         # K is not pointed, project its dual onto its span.
-        K = project_span(K.dual()).dual()
+        # Uses a proposition from our paper, i.e. this is
+        # equivalent to K = restrict_span(K.dual()).dual()
+        K = restrict_span(intersect_span(K,K.dual()), K.dual())
+        #K = restrict_span(K.dual()).dual()
+
+        #Ks = [ list(r) for r in sorted(K.rays()) ]
+        #Js = [ list(r) for r in sorted(J.rays()) ]
+
+        #if Ks != Js:
+        #    print [ list(r) for r in K_orig.rays() ]
 
         # Lemma 3
         beta += m * l