Although deep networks are typically used to approximate functions over high dimensional inputs, recent work has increased interest in neural networks as function approximators for low-dimensional-but-complex functions, such as representing images as a function of pixel coordinates, solving differential equations, or representing signed distance fields or neural radiance fields. Key to these recent successes has been the use of new elements such as sinusoidal nonlinearities, or Fourier features in positional encodings, which vastly outperform simple ReLU networks. In this paper, we propose and empirically demonstrate that an arguably simpler class of function approximators can work just as well for such problems: multiplicative filter networks. In these networks, we avoid traditional compositional depth altogether, and simply multiply together (linear functions of) sinusoidal or Gabor wavelet functions applied to the input. This representation has the notable advantage that the entire function can simply be viewed as a linear function approximator over an exponential number of Fourier or Gabor basis functions, respectively. Despite this simplicity, when compared to recent approaches that use Fourier features with ReLU networks or sinusoidal activation networks, we show that these multiplicative filter networks largely outperform or match the performance of these recent approaches on the domains highlighted in these past works.
Source: Multiplicative Filter Networks (2020-09-28). See: paper link.
import torch
import torch.nn as nn
import skimage
import numpy as np
from tqdm import tqdm
import matplotlib.pyplot as plt
import torchvision
class SineLayer(nn.Module):
def __init__(self, w0):
super(SineLayer, self).__init__()
self.w0 = w0
def forward(self, x):
return torch.sin(self.w0 * x)
class Siren(nn.Module):
def __init__(self, w0=30, in_dim=2, hidden_dim=256, out_dim=1):
super(Siren, self).__init__()
self.net = nn.Sequential(nn.Linear(in_dim, hidden_dim), SineLayer(w0),
nn.Linear(hidden_dim, hidden_dim), SineLayer(w0),
nn.Linear(hidden_dim, hidden_dim), SineLayer(w0),
nn.Linear(hidden_dim, hidden_dim), SineLayer(w0),
nn.Linear(hidden_dim, out_dim))
# Init weights
with torch.no_grad():
self.net[0].weight.uniform_(-1. / in_dim, 1. / in_dim)
self.net[2].weight.uniform_(-np.sqrt(6. / hidden_dim) / w0, np.sqrt(6. / hidden_dim) / w0)
self.net[4].weight.uniform_(-np.sqrt(6. / hidden_dim) / w0, np.sqrt(6. / hidden_dim) / w0)
self.net[6].weight.uniform_(-np.sqrt(6. / hidden_dim) / w0, np.sqrt(6. / hidden_dim) / w0)
self.net[8].weight.uniform_(-np.sqrt(6. / hidden_dim) / w0, np.sqrt(6. / hidden_dim) / w0)
def forward(self, x):
return self.net(x)
class GaborFilter(nn.Module):
def __init__(self, in_dim, out_dim, alpha, beta=1.0):
super(GaborFilter, self).__init__()
self.mu = nn.Parameter(torch.rand((out_dim, in_dim)) * 2 - 1)
self.gamma = nn.Parameter(torch.distributions.gamma.Gamma(alpha, beta).sample((out_dim, )))
self.linear = torch.nn.Linear(in_dim, out_dim)
# Init weights
self.linear.weight.data *= 128. * torch.sqrt(self.gamma.unsqueeze(-1))
self.linear.bias.data.uniform_(-np.pi, np.pi)
def forward(self, x):
# torch.norm((x.unsqueeze(1) - mu), dim=2)**2
norm = (x ** 2).sum(dim=1).unsqueeze(-1) + (self.mu ** 2).sum(dim=1).unsqueeze(0) - 2 * x @ self.mu.T
return torch.exp(- self.gamma.unsqueeze(0) / 2. * norm) * torch.sin(self.linear(x))
class GaborNet(nn.Module):
def __init__(self, in_dim=2, hidden_dim=256, out_dim=1, k=4):
super(GaborNet, self).__init__()
self.k = k
self.gabon_filters = nn.ModuleList([GaborFilter(in_dim, hidden_dim, alpha=6.0 / k) for _ in range(k)])
self.linear = nn.ModuleList(
[torch.nn.Linear(hidden_dim, hidden_dim) for _ in range(k - 1)] + [torch.nn.Linear(hidden_dim, out_dim)])
for lin in self.linear[:k - 1]:
lin.weight.data.uniform_(-np.sqrt(1.0 / hidden_dim), np.sqrt(1.0 / hidden_dim))
def forward(self, x):
# Recursion - Equation 3
zi = self.gabon_filters[0](x) # Eq 3.a
for i in range(self.k - 1):
zi = self.linear[i](zi) * self.gabon_filters[i + 1](x) # Eq 3.b
return self.linear[self.k - 1](zi) # Eq 3.c
def train(model, optim, nb_epochs=15000):
psnrs = []
for _ in tqdm(range(nb_epochs)):
model_output = model(pixel_coordinates)
loss = ((model_output - pixel_values) ** 2).mean()
psnrs.append(20 * np.log10(1.0 / np.sqrt(loss.item())))
optim.zero_grad()
loss.backward()
optim.step()
return psnrs, model_output
if __name__ == "__main__":
device = 'cuda'
siren = Siren().to(device)
gabor_net = GaborNet().to(device)
# Target
img = ((torch.from_numpy(skimage.data.camera()) - 127.5) / 127.5)
img = torchvision.transforms.Resize(256)(img.unsqueeze(0))[0]
pixel_values = img.reshape(-1, 1).to(device)
# Input
resolution = img.shape[0]
tmp = torch.linspace(-1, 1, steps=resolution)
x, y = torch.meshgrid(tmp, tmp)
pixel_coordinates = torch.cat((x.reshape(-1, 1), y.reshape(-1, 1)), dim=1).to(device)
fig, axes = plt.subplots(1, 5, figsize=(15, 3))
axes[0].imshow(img, cmap='gray')
axes[0].set_title('Ground Truth', fontsize=13)
for i, model in enumerate([siren, gabor_net]):
# Training
optim = torch.optim.Adam(lr=1e-4 if (i == 0) else 1e-2, params=model.parameters())
psnrs, model_output = train(model, optim, nb_epochs=1000)
axes[i + 1].imshow(model_output.cpu().view(resolution, resolution).detach().numpy(), cmap='gray')
axes[i + 1].set_title('SIREN' if (i == 0) else 'GaborNet', fontsize=13)
axes[4].plot(psnrs, label='SIREN' if (i == 0) else 'GaborNet', c='green' if (i == 0) else 'purple')
axes[4].set_xlabel('Iterations', fontsize=14)
axes[4].set_ylabel('PSNR', fontsize=14)
axes[4].legend(fontsize=13)
for i in range(4):
axes[i].set_xticks([])
axes[i].set_yticks([])
axes[3].axis('off')
plt.savefig('Multiplicative_Filter_Networks.png')
plt.close()
python implementation Multiplicative Filter Networks in 100 lines
2020-09-28