Source Code for Module geometry.manifolds.manifold_embedding_propagation

  1  from . import DifferentiableManifold 
  2   
  3  SYMBOL_ISOMORPHISM = '~' # XXX: not used 
  4  SYMBOL_EMBEDDING = '<' 
  5  SYMBOL_PROJECTION = '>' 
  6   
  7   
  8 -def compute_manifold_relations(manifolds): 
  9      isomorphisms, embeddings = find_embedding_relations(manifolds) 
 10      create_isomorphisms(isomorphisms) 
 11      create_embeddings(embeddings) 
 12   
 13      if False: 
 14          for m1 in manifolds: 
 15              for m2 in m1._isomorphisms: 
 16                  print('%5s ~ %5s  via %s' % (m1, m2, 
 17                                               m1._isomorphisms[m2].steps)) 
 18   
 19              for m2 in m1._embedding: 
 20                  print('%5s < %5s  via %s' % (m1, m2, 
 21                                               m1._embedding[m2].steps)) 
 22   
 23   
 24 -def find_embedding_relations(manifolds): 
 25      visited = set() 
 26   
 27      class AutoVivification(dict): 
 28          def __getitem__(self, item): 
 29              try: 
 30                  return dict.__getitem__(self, item) 
 31              except KeyError: 
 32                  value = self[item] = type(self)() 
 33                  return value 
 34   
 35      isomorphisms = AutoVivification() 
 36      embeddings = AutoVivification() 
 37   
 38      def visit(m1): 
 39          if m1 in visited: 
 40              return 
 41          visited.add(m1) 
 42          # first level 
 43          for m2 in m1._isomorphisms: 
 44              isomorphisms[m1][m2] = [(m1, SYMBOL_ISOMORPHISM, m2)] 
 45          # second level 
 46          for m2 in m1._isomorphisms: 
 47              visit(m2) 
 48              for m3 in isomorphisms[m2]: 
 49                  if not m3 in isomorphisms[m1] and m3 != m1: 
 50                      isomorphisms[m1][m3] = (isomorphisms[m1][m2] + 
 51                                              isomorphisms[m2][m3]) 
 52          # first level 
 53          for m2 in m1._embedding: 
 54              embeddings[m1][m2] = [(m1, SYMBOL_EMBEDDING, m2)] 
 55          # second level (direct) 
 56          for m2 in list(embeddings[m1].keys()): 
 57              visit(m2) 
 58              for m3 in embeddings[m2]: 
 59                  if not m3 in embeddings[m1]: 
 60                      embeddings[m1][m3] = (embeddings[m1][m2] + 
 61                                            embeddings[m2][m3]) 
 62          # second level (with iso) 
 63          for m2 in list(isomorphisms[m1].keys()): 
 64              visit(m2) 
 65              for m3 in embeddings[m2]: 
 66                  if not m3 in embeddings[m1]: 
 67                      embeddings[m1][m3] = (isomorphisms[m1][m2] + 
 68                                            embeddings[m2][m3]) 
 69  #            # finally, each isomorphism is an embedding 
 70  #            if not m2 in embeddings[m1]: 
 71  #                embeddings[m1][m2] = [ (m1, SYMBOL_ISOMORPHISM, m2) ] 
 72   
 73      for m1 in manifolds: 
 74          visit(m1) 
 75   
 76      return isomorphisms, embeddings 
 77   
 78   
 79 -def create_isomorphisms(isomorphisms): 
 80      for m1 in isomorphisms: 
 81          for m2 in isomorphisms[m1]: 
 82              steps = isomorphisms[m1][m2] 
 83   
 84              m1_to_m2 = steps2function(steps) 
 85              m2_to_m1 = steps2function(reverse_steps(steps)) 
 86              desc = str(steps) 
 87   
 88              # also create embedding 
 89              if len(steps) == 1: 
 90                  DifferentiableManifold.embedding( 
 91                     m1, m2, m1_to_m2, m2_to_m1, itype='isomorphism', 
 92                     steps=steps, desc=desc) 
 93                  continue 
 94              DifferentiableManifold.isomorphism( 
 95                 m1, m2, m1_to_m2, m2_to_m1, itype='derived', 
 96                 steps=steps, desc=desc) 
 97   
 98   
 99 -def create_embeddings(embeddings): 
100      for m1 in embeddings: 
101          for m2 in embeddings[m1]: 
102              steps = embeddings[m1][m2] 
103              if len(steps) == 1: 
104                  continue 
105   
106              m1_to_m2 = steps2function(steps) 
107              m2_to_m1 = steps2function(reverse_steps(steps)) 
108              desc = str(steps) 
109              DifferentiableManifold.embedding( 
110                 m1, m2, m1_to_m2, m2_to_m1, itype='derived', 
111                 steps=steps, desc=desc) 
112   
113   
114 -def reverse_steps(steps): 
115      def reverse_step(step): 
116          A, op, B = step 
117          rev_op = {SYMBOL_EMBEDDING: SYMBOL_PROJECTION, 
118                    SYMBOL_PROJECTION: SYMBOL_EMBEDDING, 
119                    SYMBOL_ISOMORPHISM: SYMBOL_ISOMORPHISM}[op] 
120          return B, rev_op, A 
121      return [reverse_step(step) for step in reversed(steps)] 
122   
123   
124 -def steps2function(steps): 
125      functions = [] 
126      for m1, operation, m2 in steps: 
127          if operation == SYMBOL_EMBEDDING: 
128              functions.append(m1._embedding[m2].A_to_B) 
129          elif operation == SYMBOL_PROJECTION: 
130              functions.append(m1._projection[m2].A_to_B) 
131          elif operation == SYMBOL_ISOMORPHISM: 
132              functions.append(m1._isomorphisms[m2].A_to_B) 
133   
134      if len(functions) == 1: 
135          return functions[0] 
136   
137      def conversion(x): 
138          try: 
139              for f in functions: 
140                  x = f(x) 
141              return x 
142          except: 
143              print('error during %s' % steps) 
144              print(' functions %s' % functions) 
145              raise 
146   
147      return conversion 
148