Normalizing Flows with Coupling Layers#
🌟 Motivation#
Imagine you have data from a complicated distribution—such as images, speech signals, or intricate 2-D patterns like spirals, checkerboards, or moons. How could we generate new, realistic samples from this data? Or evaluate how “likely” a new sample is under this complex distribution?
One elegant solution is to use Normalizing Flows, a powerful class of generative models that:
Transform complex data into simple, well-known distributions (like a Gaussian).
Let you easily sample new data by transforming simple Gaussian samples back into the complex space.
Allow precise computation of the likelihood of any given data point.
This makes Normalizing Flows extremely useful for density estimation, generative modeling, and even anomaly detection.
📌 What You Will Do in This Project#
In this project, you’ll implement and experiment with Real Non-Volume-Preserving (RealNVP) flows, specifically using Affine Coupling Layers.
Your main goal is:
Implement a density estimator using Normalizing Flows to transform a complex 2-D distribution into a simple Gaussian distribution, and then back.
You’ll:
Build affine coupling layers from scratch in PyTorch.
Chain these layers together (at least four) using alternating masking patterns.
Train your model on well-known synthetic datasets (such as the two-moons or checkerboard data).
Visualize how your model learns to map complex data to Gaussian space and vice versa.
🔍 Key Concepts You’ll Master#
You’ll dive into several fascinating and practically relevant concepts:
Change of Variables: How can we map a complex distribution to a simpler one in a differentiable way, while keeping track of probabilities?
Log-Determinant of the Jacobian: Every transformation in a flow changes volumes in the data space. You’ll derive and compute how much “volume” changes under affine coupling layers.
Invertible Neural Networks: Coupling layers are carefully designed neural networks that can be easily inverted. You’ll understand why invertibility is guaranteed and critical.
Monte-Carlo Log-Likelihood Estimation: You’ll use samples to estimate and optimize the likelihood directly.
🚧 Core Tasks (Implementation Details)#
Your implementation will involve the following steps:
Implement one affine coupling layer (forward and inverse passes, plus log-det Jacobian) in PyTorch.
Stack at least four coupling layers into a Normalizing Flow, ensuring masks alternate.
Train on a simple yet rich dataset (two-moons or checkerboard).
Plot and analyze forward and inverse samples as well as training curves of the log-likelihood.
Your final result will showcase:
Complex-to-simple and simple-to-complex transformations visually.
Clear, well-documented Python code (NumPy and PyTorch) that others can easily follow.
📝 Reporting: Derivations and Insights#
Your short (~2 pages) report should clearly present:
A derivation of the log-determinant Jacobian for affine coupling layers.
A conceptual explanation for why the coupling layer is invertible by construction.
Visual evidence of your model’s capability and learning progress.
🚀 Stretch Goals (Optional, for Extra Insight)#
If you’re eager to go deeper, you might:
Add ActNorm layers or 1×1 convolutions (Glow) to enhance flow expressivity.
Experiment with a direct NumPy implementation and compare speed and performance to PyTorch.
📚 Resources and Support#
You are encouraged to leverage open-source resources and AI tools to assist you, as long as you clearly document their use.
Use provided starter notebooks for datasets and visualizations to focus your effort effectively.
✅ Why This Matters#
Beyond the project grade, this work equips you with practical experience in modern generative modeling, a core skill in today’s AI landscape. Normalizing Flows appear in research and real-world applications from image generation to anomaly detection, finance, physics, and bioinformatics.
This is your chance to:
Gain a deep, intuitive understanding of Normalizing Flows.
Showcase practical coding skills highly sought-after in industry and academia.
Produce a polished, impressive piece for your portfolio.
Project Summary: Normalizing Flows with Coupling Layers#
(NumPy + PyTorch for autodiff)
Item |
Details |
---|---|
Goal |
Build a density estimator that maps complex 2-D data to a unit Gaussian via affine coupling layers (RealNVP). |
Key ideas |
Change of variables, log-det Jacobian, invertible nets, Monte-Carlo log-likelihood. |
Core tasks |
|
Report focus (≈2 pages) |
Derive the log-det Jacobian for coupling layers; discuss why invertibility is guaranteed. |
Stretch ideas |
Add an ActNorm layer or glow-style 1×1 convolution; compare NumPy vs. PyTorch speed. |