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Tags:
research-artifact
synthetic-data
mathematical-proofs
system-identification
inverse-problems
nonlinear-dynamics
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b6abf2d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | """Mathematical identities, counterexamples, and scope guards."""
import unittest
import numpy as np
from scipy.linalg import expm,block_diag
from scipy.optimize import minimize
from ot.core import (Model,lattice,query_support,kernel_from_queries,reconstruct,
minimum_joint_cost,pair_log,stability_certificate)
class ResearchChecks(unittest.TestCase):
def setUp(self):
self.A=np.array([[-.2,-.7],[.7,-.3]])
self.C=np.array([[1.,.2]])
self.Q=np.array([[1.1,.1],[.1,.8]])
self.model=Model(self.A,self.C,self.Q)
def test_support_polarization_and_optimizer(self):
O=self.model.design([0.,.3,.9]);a=np.array([.2,-.7,.3]);a/=np.linalg.norm(a)
v=O.T@a
oracle=float(v@self.Q@v)
result=minimize(lambda x:float(x@np.linalg.solve(self.Q,x)),np.ones(2),
constraints={'type':'eq','fun':lambda x:float(v@x)-1.},
method='SLSQP',options={'ftol':1e-12,'maxiter':300})
self.assertTrue(result.success)
self.assertAlmostEqual(result.fun,1/oracle,places=8)
rows=query_support(O,self.Q)
np.testing.assert_allclose(kernel_from_queries(rows,3),O@self.Q@O.T,atol=1e-14)
def test_joint_infeasibility_has_null_certificate(self):
K=np.ones((2,2));a=np.array([1.,-1.]);y=np.array([1.,-1.])
self.assertEqual(minimum_joint_cost(K,y),float('inf'))
self.assertAlmostEqual(float(a@K@a),0.)
self.assertNotEqual(float(a@y),0.)
self.assertAlmostEqual(minimum_joint_cost(K,[2.,2.]),4.)
def test_coordinate_gauge_invariance(self):
S=np.array([[2.,.4],[.2,.8]])
m2=Model(S@self.A@np.linalg.inv(S),self.C@np.linalg.inv(S),S@self.Q@S.T)
times=[0.,.17,.81,1.3]
O=self.model.design(times);P=m2.design(times)
np.testing.assert_allclose(O@self.Q@O.T,P@m2.Q@P.T,atol=2e-14)
def test_unobservable_extension_not_identified(self):
m2=Model(block_diag(self.A,[[13.]]),np.column_stack([self.C,[0.]]),block_diag(self.Q,[[1.]]))
O=self.model.design([0.,.1,.3]);P=m2.design([0.,.1,.3])
np.testing.assert_allclose(O@self.Q@O.T,P@m2.Q@P.T,atol=1e-14)
def test_feasibility_does_not_calibrate_resource(self):
O=self.model.design([0.,.2,.8]);K=O@self.Q@O.T;y=O@np.array([.4,.8])
self.assertAlmostEqual(minimum_joint_cost(2*K,y),minimum_joint_cost(K,y)/2,places=10)
np.testing.assert_allclose(np.linalg.pinv(K)@K,np.linalg.pinv(2*K)@(2*K),atol=1e-13)
def test_singleton_data_and_uniform_grid_alias(self):
times=np.arange(5);w1=1.;w2=1.+2*np.pi
np.testing.assert_allclose(np.cos(w1*(times[:,None]-times)),np.cos(w2*(times[:,None]-times)),atol=1e-14)
self.assertGreater(abs(np.cos(w1*np.sqrt(2))-np.cos(w2*np.sqrt(2))),.1)
def test_two_clock_branch_recovery(self):
w=7.3;A=np.array([[0.,-w],[w,0.]])
F=expm(A);J=expm(np.sqrt(2)*A)
got,info=pair_log(F,J,1.,np.sqrt(2),40.)
np.testing.assert_allclose(got,A,atol=1e-11)
self.assertGreater(info['min_margin'],.1)
def test_branch_solver_rejects_degenerate_spectrum(self):
with self.assertRaises(ValueError):pair_log(np.eye(2),np.eye(2),1.,np.sqrt(2),40.)
def test_noisy_entry_bound(self):
O=self.model.design([0.,.2,.7]);delta=.002
raw=query_support(O,self.Q,np.random.default_rng(5),delta)
Kh=kernel_from_queries(raw,3);K=O@self.Q@O.T
self.assertLessEqual(np.max(abs(Kh-K)),delta+1e-14)
self.assertLessEqual(np.linalg.norm(Kh-K,2),3*delta+1e-14)
def test_clock_shift_orientation(self):
tau=.2;sigma=tau*np.sqrt(2);times,labels=lattice(2,1,tau,sigma)
O=self.model.design(times)
rec=reconstruct(kernel_from_queries(query_support(O,self.Q),len(O)),labels,2,tau,sigma)
B=O[rec['basis']]
np.testing.assert_allclose(rec['F'],B@expm(self.A*tau)@np.linalg.inv(B),atol=1e-12)
np.testing.assert_allclose(rec['A'],B@self.A@np.linalg.inv(B),atol=1e-11)
def test_independent_and_shared_preparation_differ(self):
# Same scalar marginals, different joint budgets when preparations reset.
Kshared=np.ones((2,2));Kindependent=np.eye(2)
self.assertEqual(minimum_joint_cost(Kshared,[1.,-1.]),float('inf'))
self.assertAlmostEqual(minimum_joint_cost(Kindependent,[1.,-1.]),2.)
def test_chain_precision_pair_exact_difference(self):
eps=.13;s=np.arange(4);O3=np.column_stack([np.ones(4),eps*s,(eps*s)**2/2])
O2=O3[:,:2];h=O3[:,2]
np.testing.assert_allclose(O3@O3.T-O2@O2.T,np.outer(h,h),atol=1e-15)
self.assertAlmostEqual(np.linalg.norm(np.outer(h,h),2),float(h@h),places=13)
def test_incomplete_queries_rejected(self):
O=self.model.design([0.,.2]);rows=query_support(O,self.Q)
with self.assertRaises(ValueError):kernel_from_queries(rows[:-1],2)
if __name__=='__main__':unittest.main(verbosity=2)
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