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"""Exact rational certificates for observable rank and quotient guarantees.
This module implements the constructive algebraic parts of the observable-rank
proof and the norm/mismatch certificates. The maximum-volume basis enumerates
rank-sized subsets: its running time is polynomial for FIXED variation rank.
The multivariate matching optimizer is specified in the manuscript; this module
does not replace it by a permutation enumeration or claim to implement it.
"""
from fractions import Fraction as F
from itertools import combinations
def dot(a,b):
return sum((F(x)*F(y) for x,y in zip(a,b)),F(0))
def matvec(a,x):
return [dot(row,x) for row in a]
def transpose(a):
return [list(row) for row in zip(*a)]
def matmul(a,b):
bt=transpose(b)
return [[dot(row,col) for col in bt] for row in a]
def subtract(a,b):
return [[F(x)-F(y) for x,y in zip(ra,rb)] for ra,rb in zip(a,b)]
def determinant(a):
a=[[F(x) for x in row] for row in a]
n=len(a); value=F(1)
for i in range(n):
pivot=next((j for j in range(i,n) if a[j][i]),None)
if pivot is None:return F(0)
if pivot!=i:
a[i],a[pivot]=a[pivot],a[i];value=-value
d=a[i][i];value*=d
for j in range(i+1,n):
f=a[j][i]/d
for k in range(i+1,n):a[j][k]-=f*a[i][k]
a[j][i]=F(0)
return value
def solve(a,b):
n=len(a)
if not n:return []
a=[[F(x) for x in row]+[F(v)] for row,v in zip(a,b)]
for i in range(n):
pivot=next((j for j in range(i,n) if a[j][i]),None)
if pivot is None:raise ValueError('singular linear system')
a[i],a[pivot]=a[pivot],a[i]
d=a[i][i];a[i]=[x/d for x in a[i]]
for j in range(n):
if j==i:continue
f=a[j][i]
a[j]=[x-f*y for x,y in zip(a[j],a[i])]
return [row[-1] for row in a]
def gram(vectors):
return [[dot(x,y) for y in vectors] for x in vectors]
def is_psd(a):
"""Exact PSD test by symmetric positive-pivot Schur complements."""
a=[[F(x) for x in row] for row in a]
n=len(a)
if any(len(row)!=n for row in a):return False
if any(a[i][j]!=a[j][i] for i in range(n) for j in range(n)):
return False
while a:
n=len(a)
if any(a[i][i]<0 for i in range(n)):return False
pivot=next((i for i in range(n) if a[i][i]>0),None)
if pivot is None:return all(not x for row in a for x in row)
order=[pivot]+[i for i in range(n) if i!=pivot]
a=[[a[i][j] for j in order] for i in order]
d=a[0][0]
a=[[a[i][j]-a[i][0]*a[0][j]/d
for j in range(1,n)] for i in range(1,n)]
return True
def norm_bound(x,bound):
bound=F(bound)
return bound>=0 and dot(x,x)<=bound*bound
def operator_norm_bound(a,bound):
bound=F(bound)
if bound<0 or not a:return False
n=len(a[0])
if any(len(row)!=n for row in a):return False
ata=matmul(transpose(a),a)
residual=[[bound*bound*(i==j)-ata[i][j]
for j in range(n)] for i in range(n)]
return is_psd(residual)
def maximum_gram_basis(vectors):
"""Return a rational maximum-Gram-volume basis and all coordinates."""
vectors=[[F(x) for x in v] for v in vectors]
if not vectors:raise ValueError('nonempty vector list required')
m=len(vectors[0])
if any(len(v)!=m for v in vectors):raise ValueError('dimension mismatch')
independent=[]
for v in vectors:
if determinant(gram(independent+[v]))>0:independent.append(v)
r=len(independent)
if not r:return dict(rank=0,basis=[],coordinates=[[] for _ in vectors],
gram=[],maximum_gram_determinant=F(1))
candidates=list(dict.fromkeys(tuple(v) for v in vectors if any(v)))
best=None;bestdet=F(-1)
for choice in combinations(candidates,r):
value=determinant(gram(choice))
if value>bestdet:best=list(choice);bestdet=value
g=gram(best)
coordinates=[solve(g,[dot(b,v) for b in best]) for v in vectors]
assert all(abs(x)<=1 for c in coordinates for x in c)
return dict(rank=r,basis=[list(b) for b in best],coordinates=coordinates,
gram=g,maximum_gram_determinant=bestdet)
def compress_assignment(a,edges):
"""Compress observed edge variations; preserve exact squared objectives."""
a=[F(x) for x in a];n=len(edges);m=len(a)
if not n or any(len(row)!=n for row in edges):
raise ValueError('edges must be a nonempty square array of vectors')
edges=[[[F(x) for x in v] for v in row] for row in edges]
if any(len(v)!=m for row in edges for v in row):
raise ValueError('vector dimension mismatch')
if any(not norm_bound(v,1) for row in edges for v in row):
raise ValueError('the theorem requires edge-vector norms at most one')
reference=[row[0] for row in edges]
variations=[[x-y for x,y in zip(v,reference[p])]
for p,row in enumerate(edges) for v in row]
result=maximum_gram_basis(variations)
constant=[a[j]+sum(reference[p][j] for p in range(n)) for j in range(m)]
flat=result.pop('coordinates')
coordinates=[flat[p*n:(p+1)*n] for p in range(n)]
result.update(size=n,ambient_dimension=m,constant=constant,
coordinates=coordinates,
constant_squared=dot(constant,constant),
cross=[dot(b,constant) for b in result['basis']])
return result
def compressed_objective(model,order):
r=model['rank']
total=[sum(model['coordinates'][p][i][j] for p,i in enumerate(order))
for j in range(r)]
return (model['constant_squared']+2*dot(model['cross'],total)
+dot(total,matvec(model['gram'],total)))
def assignment_residual(a,edges,order):
return [F(x)+sum(F(edges[p][i][j]) for p,i in enumerate(order))
for j,x in enumerate(a)]
def rounded_encoding(model,epsilon):
"""Carry-free scalar costs for the matching primitive in the proof."""
epsilon=F(epsilon);n=model['size'];r=model['rank']
if not 0<epsilon<=1:raise ValueError('epsilon must be in (0,1]')
if not r:return dict(rank=0,delta=F(0),base=1,costs=[[0]*n for _ in range(n)],
minima=[],vector_costs=[[[] for _ in range(n)] for _ in range(n)],
degree=0,coordinate_degree=[])
delta=epsilon/(32*r*n)
rounded=[[[int(F(x)//delta) for x in coord] for coord in row]
for row in model['coordinates']]
minima=[[min(rounded[p][i][j] for i in range(n)) for j in range(r)]
for p in range(n)]
costs=[[[rounded[p][i][j]-minima[p][j] for j in range(r)]
for i in range(n)] for p in range(n)]
degrees=[sum(max(costs[p][i][j] for i in range(n)) for p in range(n))
for j in range(r)]
base=max(degrees)+1
encoded=[[sum(costs[p][i][j]*base**j for j in range(r))
for i in range(n)] for p in range(n)]
return dict(rank=r,delta=delta,base=base,costs=encoded,
minima=[sum(row[j] for row in minima) for j in range(r)],
vector_costs=costs,degree=sum(max(row) for row in encoded),
coordinate_degree=degrees)
def decode_cost(value,base,rank):
digits=[]
for _ in range(rank):
digits.append(value%base);value//=base
if value:raise ValueError('encoded total exceeds supplied rank')
return digits
def rounded_residual(model,encoding,order):
r=model['rank'];out=model['constant'][:]
if not r:return out
q=sum(encoding['costs'][p][i] for p,i in enumerate(order))
digits=decode_cost(q,encoding['base'],r)
coordinates=[encoding['delta']*(x+y)
for x,y in zip(encoding['minima'],digits)]
for j,b in enumerate(model['basis']):
out=[x+coordinates[j]*y for x,y in zip(out,b)]
return out
def execute(matrices,increments,initial,order):
x=[F(v) for v in initial]
for i in order:
x=[u+F(v) for u,v in zip(matvec(matrices[i],x),increments[i])]
return x
def quotient_certificate(matrices,increments,response,quotient_increments,
readout,initial,quotient_initial,target,rho,
kappa=F(0),eta=F(0),eta0=F(0)):
"""Validate supplied rational bounds and return the uniform certificate.
q_N bounds every observation mismatch. 4*q_N bounds objective mismatch;
eps+8*q_N bounds true regret after eps-additive quotient optimization.
"""
rho=F(rho);kappa=F(kappa);eta=F(eta);eta0=F(eta0)
n=len(matrices)
if not n or not 0<=rho<1 or min(kappa,eta,eta0)<0:
raise ValueError('invalid contraction or mismatch constants')
if len(increments)!=n or len(quotient_increments)!=n:
raise ValueError('request count mismatch')
d=len(initial);r=len(quotient_initial)
if len(target)!=r or len(readout)!=r or any(len(row)!=d for row in readout):
raise ValueError('readout dimension mismatch')
if len(response)!=r or any(len(row)!=r for row in response):
raise ValueError('quotient response dimension mismatch')
if any(len(m)!=d or any(len(row)!=d for row in m) for m in matrices):
raise ValueError('hidden response dimension mismatch')
if any(len(b)!=d for b in increments) or any(len(c)!=r for c in quotient_increments):
raise ValueError('increment dimension mismatch')
if not operator_norm_bound(readout,1) or not operator_norm_bound(response,rho):
raise ValueError('readout or quotient norm bound failed')
if not all(norm_bound(x,1) for x in (initial,quotient_initial,target)):
raise ValueError('initial-state or target norm bound failed')
ac=matmul(response,readout)
for m,b,c in zip(matrices,increments,quotient_increments):
if not operator_norm_bound(m,rho):raise ValueError('hidden norm bound failed')
if not norm_bound(b,1-rho) or not norm_bound(c,1-rho):
raise ValueError('increment norm bound failed')
if not operator_norm_bound(subtract(matmul(readout,m),ac),kappa):
raise ValueError('observable response defect bound failed')
if not norm_bound([x-F(y) for x,y in zip(matvec(readout,b),c)],eta):
raise ValueError('observable increment defect bound failed')
if not norm_bound([x-F(y) for x,y in zip(matvec(readout,initial),quotient_initial)],eta0):
raise ValueError('initial observation mismatch bound failed')
q=rho**n*eta0+(kappa+eta)*(1-rho**n)/(1-rho)
return dict(requests=n,hidden_dimension=d,observed_dimension=r,
terminal_mismatch=q,objective_mismatch=4*q,
regret_transfer=8*q,exact_closure=(kappa==eta==eta0==0))
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