n-Sphere manifold¶

In [1]:

import numpy as np
import matplotlib.pyplot as plt
import torch

try:
    import numqi
except ImportError:
    %pip install numqi
    import numqi

np_rng = np.random.default_rng()

import numpy as np import matplotlib.pyplot as plt import torch try: import numqi except ImportError: %pip install numqi import numqi np_rng = np.random.default_rng()

In [2]:

class DummyMinEigen(torch.nn.Module):
    def __init__(self, mat):
        super().__init__()
        self.mat = torch.tensor(mat, dtype=torch.float64)
        self.theta = torch.nn.Parameter(torch.randn(mat.shape[0], dtype=torch.float64))
        # self.manifold = numqi.manifold.Sphere(mat.shape[0], dtype=torch.float64)
        self.vec = None

    def forward(self):
        vec = self.theta / torch.linalg.norm(self.theta)
        # vec = self.manifold()
        self.vec = vec.detach()
        loss = torch.vdot(vec, (self.mat @ vec)).real
        return loss

class DummyMinEigen(torch.nn.Module): def __init__(self, mat): super().__init__() self.mat = torch.tensor(mat, dtype=torch.float64) self.theta = torch.nn.Parameter(torch.randn(mat.shape[0], dtype=torch.float64)) # self.manifold = numqi.manifold.Sphere(mat.shape[0], dtype=torch.float64) self.vec = None def forward(self): vec = self.theta / torch.linalg.norm(self.theta) # vec = self.manifold() self.vec = vec.detach() loss = torch.vdot(vec, (self.mat @ vec)).real return loss

In [3]:

N0 = 128
tmp0 = np_rng.normal(size=(N0,N0))
mat = (tmp0 + tmp0.T) / 2

model = DummyMinEigen(mat)
callback = numqi.optimize.MinimizeCallback(print_freq=1, extra_key='grad_norm', tag_print=False)
theta_optim = numqi.optimize.minimize(model, theta0='uniform', num_repeat=1, callback=callback, tol=1e-14)
EVL = theta_optim.fun
EVC = model.vec.numpy()
EVL_ = np.linalg.eigvalsh(mat)[0]
print('error(EVL)', np.abs(EVL-EVL_))
print('mae(EVC)', np.abs(mat @ EVC - EVC * EVL).max())

N0 = 128 tmp0 = np_rng.normal(size=(N0,N0)) mat = (tmp0 + tmp0.T) / 2 model = DummyMinEigen(mat) callback = numqi.optimize.MinimizeCallback(print_freq=1, extra_key='grad_norm', tag_print=False) theta_optim = numqi.optimize.minimize(model, theta0='uniform', num_repeat=1, callback=callback, tol=1e-14) EVL = theta_optim.fun EVC = model.vec.numpy() EVL_ = np.linalg.eigvalsh(mat)[0] print('error(EVL)', np.abs(EVL-EVL_)) print('mae(EVC)', np.abs(mat @ EVC - EVC * EVL).max())

[round=0] min(f)=-16.003449143494947, current(f)=-16.003449143494947
error(EVL) 6.039613253960852e-14
mae(EVC) 1.647571090668265e-07

In [4]:

fig,(ax0,ax1) = plt.subplots(1, 2, figsize=(8,4.5))
ax0.plot(np.array(callback.state['fval'])-EVL_)
ax0.grid()
ax0.set_yscale('log')
ax0.set_xlabel('step')
ax0.set_ylabel('loss-optimal')
ax1.plot(callback.state['grad_norm'])
ax1.set_yscale('log')
ax1.grid()
ax1.set_xlabel('step')
ax1.set_ylabel('gradient norm')
fig.tight_layout()

fig,(ax0,ax1) = plt.subplots(1, 2, figsize=(8,4.5)) ax0.plot(np.array(callback.state['fval'])-EVL_) ax0.grid() ax0.set_yscale('log') ax0.set_xlabel('step') ax0.set_ylabel('loss-optimal') ax1.plot(callback.state['grad_norm']) ax1.set_yscale('log') ax1.grid() ax1.set_xlabel('step') ax1.set_ylabel('gradient norm') fig.tight_layout()