torchortho 📐

torchortho is a PyTorch library for learnable activation functions based on:

• Hermite Polynomials 🧙‍♂️
• Fourier Series 〰
• Tropical Polynomials & Rational Functions 🌴

These adaptive activations dynamically adjust during training, offering improved expressivity, better gradient flow, and enhanced generalization for vision and language models.

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📜 Paper Reference

This library is based on the paper:
📄 Polynomial, Trigonometric, and Tropical Activations (Khalfaoui-Hassani & Kesselheim, 2025).

For experimental results, check our repos:

• Vision models (ConvNeXt with torchortho activations): 🔗 GitHub
• Language models (GPT-2 with torchortho activations): 🔗 GitHub

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📦 Installation

Install from PyPI:

pip install torchortho

or install directly from GitHub:

pip install git+https://github.com/K-H-Ismail/torchortho.git

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📝 Usage

You can use torchortho activations just like any other PyTorch activation:

Example: Using Hermite Activation

import torch
from torchortho import HermiteActivation

# Define a learnable Hermite activation
degree = 5
activation = HermiteActivation(degree)

# Forward pass
x = torch.rand(7, 4, 3, 2)
y = activation(x)

# Compute gradients
loss = y.sum()
loss.backward()

print("Gradients of activation coefficients:", activation.coefficients.grad)
print("Output:", y)

Example: Using Fourier Activation in a Neural Network

import torch
import torch.nn as nn
from torchortho import FourierActivation

class CustomMLP(nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super().__init__()
        self.fc1 = nn.Linear(input_dim, hidden_dim)
        self.activation = FourierActivation(degree=4)  # Learnable Fourier activation
        self.fc2 = nn.Linear(hidden_dim, output_dim)

    def forward(self, x):
        x = self.fc1(x)
        x = self.activation(x)
        return self.fc2(x)

# Initialize the model
model = CustomMLP(input_dim=10, hidden_dim=32, output_dim=1)
x = torch.randn(5, 10)
output = model(x)
print("Model Output:", output)

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⚡ Why Use torchortho?

1️⃣ Adaptive and Learnable Activations

Unlike static activations (ReLU, GELU), torchortho functions dynamically adapt during training, allowing models to learn optimal activation functions for different tasks.

Activation Type	Strengths
Hermite Activation	Adaptive polynomial approximation, variance-preserving, smooth optimization
Fourier Activation	Captures periodic structures in data (useful for NLP, physics-based models, and time-series)
Tropical Polynomial Activation	Convex activation for structured learning (e.g., decision boundaries, optimization landscapes)
Rational Activation	Generalizes standard Tropical activation functions for non-convex function fitting

2️⃣ Improved Expressivity and Gradient Flow

• Better function approximation → Increases expressivity for deep networks.
• Variance-preserving initialization → Ensures stable training, avoiding vanishing/exploding gradients.
• More flexible than ReLU/SwiGLU → Adapts activation behavior based on data.

3️⃣ Benchmarked on Real-World Models

The effectiveness of torchortho activations has been validated on large-scale deep learning benchmarks:

✅ Image Classification (ConvNeXt-T on ImageNet-1K)

• Replacing GELU with torchortho activations improves top-1 accuracy.

✅ Language Modeling (GPT-2 on OpenWebText)

• Learnable activations reduce perplexity compared to GELU-based models.

For full benchmarks, see:

• Vision repo
• Language repo

📜 License

This project is licensed under the GPL-3.0 License. See LICENSE for details.

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🙌 Contributing

We welcome contributions! Feel free to submit issues, open PRs, or suggest improvements.

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📬 Contact

For questions or collaborations, reach out via GitHub Issues.

📚 Citation

If you use torchortho in your research, please cite the following paper:

@inproceedings{khalfaoui-hassani2026polynomial,
    title={Polynomial, trigonometric, and tropical activations},
    author={Ismail Khalfaoui-Hassani and Stefan Kesselheim},
    booktitle={The Fourteenth International Conference on Learning Representations},
    year={2026},
    url={https://openreview.net/forum?id=QywpTFx86x}
}

or

@article{khalfaoui2025polynomial,
  title={Polynomial, trigonometric, and tropical activations},
  author={Khalfaoui-Hassani, Ismail and Kesselheim, Stefan},
  journal={arXiv preprint arXiv:2502.01247},
  year={2025}
}