Learning Likelihoods with Conditional Normalizing Flows (1912.00042v2)

Abstract: Normalizing Flows (NFs) are able to model complicated distributions p(y) with strong inter-dimensional correlations and high multimodality by transforming a simple base density p(z) through an invertible neural network under the change of variables formula. Such behavior is desirable in multivariate structured prediction tasks, where handcrafted per-pixel loss-based methods inadequately capture strong correlations between output dimensions. We present a study of conditional normalizing flows (CNFs), a class of NFs where the base density to output space mapping is conditioned on an input x, to model conditional densities p(y|x). CNFs are efficient in sampling and inference, they can be trained with a likelihood-based objective, and CNFs, being generative flows, do not suffer from mode collapse or training instabilities. We provide an effective method to train continuous CNFs for binary problems and in particular, we apply these CNFs to super-resolution and vessel segmentation tasks demonstrating competitive performance on standard benchmark datasets in terms of likelihood and conventional metrics.

• The paper introduces Conditional Normalizing Flows (CNFs) that model conditional densities using invertible transformations to minimize manual design choices.
• The paper demonstrates CNFs' effectiveness in super-resolution and vessel segmentation, achieving competitive likelihood and precision metrics on standard benchmarks.
• The paper innovates a variational dequantization technique to handle binary data, enhancing performance in medical imaging segmentation tasks.

Overview of "Learning Likelihoods with Conditional Normalizing Flows"

The paper presents a comprehensive paper on Conditional Normalizing Flows (CNFs), a sophisticated variant of Normalizing Flows (NFs) designed to model conditional distributions efficiently. Traditional Normalizing Flows transform simple base densities into complex distributions, facilitating high-dimensional distribution learning with strong inter-dimensional correlations and multimodality. CNFs extend this concept by conditioning the flow transformation on an additional input variable, which enables the modeling of conditional densities $p_{Y|X}(y|x)$.

Methodological Contributions

1. Conditional Normalizing Flows (CNFs): The paper proposes utilizing CNFs for modeling conditional distributions with minimal manual design choices, leveraging the invertible nature of Normalizing Flows. CNFs facilitate efficient sampling and inference and avoid typical problems like mode collapse observed in other generative models such as GANs.
2. Application to Multivariate Prediction Tasks: The paper implements CNFs in super-resolution and vessel segmentation tasks to demonstrate their effectiveness. CNFs are evaluated against standard benchmarks, showing competitive performance when compared to traditional methods that rely on handcrafted per-pixel losses.
3. Variational Dequantization for Binary Problems: The paper introduces a novel approach for handling binary variables using CNFs. This approach is particularly beneficial for binary image segmentation tasks like vessel segmentation, enhancing the capability of CNFs to model binary data distributions effectively.

Numerical Results and Implications

The experimental results underline the robustness of CNFs in capturing complex distributional characteristics in both continuous and binary settings. Noteworthy results include:

• Super-Resolution: CNFs achieve superior likelihood metrics compared to factorized baseline models on benchmarks like ImageNet32 and ImageNet64. They effectively generate high-quality, high-resolution images from low-resolution inputs, improving PSNR and SSIM metrics.
• Vessel Segmentation: CNFs efficiently handle binary segmentation tasks, showing high recall and precision in detecting retinal vessels, aligning closely with human-level performance. This suggests CNFs' potential in medical imaging tasks where precision is critical.

Implications and Future Directions

The paper paves the way for further exploration of CNFs in various domains where modeling conditional distributions are crucial. The success in super-resolution and medical imaging implies applicability in fields requiring detailed structural understanding, such as autonomous systems and advanced medical diagnostics. Moving forward, improving the efficiency of CNFs for extremely high-dimensional data or exploring modifications that further reduce computational requirements could be worthwhile areas of research.

The findings of this paper contribute significantly to the theoretical and practical understanding of generative flows conditioned on auxiliary information, bridging a gap between efficiency and complexity in probabilistic modeling tasks.