**modReLU** is an activation that is a modification of a [ReLU](https://paperswithcode.com/method/relu). It is a pointwise nonlinearity, $\sigma\_{modReLU}\left(z\right) : C \rightarrow C$, which affects only the absolute value of a complex number, defined as:

$$ \sigma\_{modReLU}\left(z\right) = \left(|z| + b\right)\frac{z}{|z|} \text{ if } |z| + b \geq 0 $$
$$ \sigma\_{modReLU}\left(z\right) = 0 \text{ if } |z| + b \leq 0 $$

where $b \in \mathbb{R}$ is a bias parameter of the nonlinearity. For a $n\_{h}$ dimensional hidden space we learn $n\_{h}$ nonlinearity bias parameters, one per dimension.

**$L_{1}$ Regularization** is a regularization technique applied to the weights of a neural network. We minimize a loss function compromising both the primary loss function and a penalty on the $L\_{1}$ Norm of the weights:

$$L\_{new}\left(w\right) = L\_{original}\left(w\right) + \lambda{||w||}\_{1}$$

where $\lambda$ is a value determining the strength of the penalty. In contrast to [weight decay](https://paperswithcode.com/method/weight-decay), $L_{1}$ regularization promotes sparsity; i.e. some parameters have an optimal value of zero.

Image Source: [Wikipedia](https://en.wikipedia.org/wiki/Regularization_(mathematics)#/media/File:Sparsityl1.png)

L1 Regularization

modReLU

Unitary Evolution Recurrent Neural Networks

**Mode Normalization** extends normalization to more than a single mean and variance, allowing for detection of modes of data on-the-fly, jointly normalizing samples that share common features. It first assigns samples in a mini-batch to different modes via a gating network, and then normalizes each sample with estimators for its corresponding mode.

Source	Unitary Evolution Recurrent Neural Networks
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com