Implicit Neural Representations with Periodic Activation Functions, Sitzmann, Martel, Bergman, Lindell, Wetzstein; 2020 - Summary
author: kraja99
score: 7 / 10


The authors introduce sinusoidal representation networks (SIRENS) as an improvement over current network architectures (namely ReLU MLPs) for implicit neural representations of signals. They perform experiments on a range of signals (image, video, sound) as well as on a number of equations rooted in physics applications. The main benefit of SIRENS is the ability to more easily compute their derivatives, as well as any higher order derivatives required.


Implicit neural representations serve as a method for continuous encoding of signals via neural networks. These representations provide benefits over traditional discrete representations since they can be more memory efficient and are differentiable.


The authors note that most recent work in implicit neural representations have utilized the ReLU MLP architecture. This architecture is limited in the fact that it lacks the capacity to model details well and its incapable of modeling information in higher-order derivatives (since its second derivative is 0).


The authors propose SIREN as a simple architecture that utilizes sine as a periodic activation function. It is formulated as shown below:

SIREN Formulation

where each phi term represents a layer of the network. SIREN is trained on the following loss

\[\mathrm{ L = \sum_{i \in D} \sum_{m=1}^M ||C_m(a(x_i), \Phi(x_i), \nabla\Phi(x_i), ...)|| }\]

where a function Phi tries to satisfy a set of M constraints across each point. To implement the network, the authors also introduce a principled initialization scheme that trys to preserve the distribution of activations through the network, so the final output at initialization doesn’t depend on the number of layers. SIRENs are shown to converge faster than baseline architectures and have some interesting properties; for example the derivative of a SIREN is also a SIREN (given that the derivative of sine is cosine, which in itself is a phase-shifted sine).

Basic Image Usecase

The authors provide an example of trying to parameterize a discrete image. In this case each image defines a dataset D = {(xi, f(xi))}i where xi is the pixel coordinate (xi, yi) and f(xi)) is the associated RGB color. Network phi (trying to predict f(xi)) is fit using an L2 loss function and trained over the dataset.

Image Example

Running this approach over a short video shows that SIREN has an average peak signal-to-noise ratio around 5 dB higher than a ReLU baseline, showing higher average image fidelity.

Here are some further results comparing SIREN versus a number of other activation functions.

Activation Comparison


The authors conduct several experiments, I’ve listed a few notable ones below.

Poisson Image Reconstruction

Images can be reconstructed by solving the Poisson equation using their derivatives. The authors supervise implicit neural representations using either just gradients or laplacians of the original function to reconstruct the original image. Here the loss can be thought of as:

\[\mathrm{ L_{grad.} = \int_\Omega ||\nabla_x\Phi(x_i) - \nabla_xf(x)||dx }\]

Poisson Example

Learning from a Space

The authors utilize a convolutional encoder to output a latent code that can be used by a hypernetwork to generate the parameters for a SIREN. This enables the learning of a prior over the CelebA image dataset. The authors observe performance using SIREN on an image inpainting task.

Inpainting Example

Space Example