On Surjectivity of Neural Networks: Can you elicit any behavior from your model? (2508.19445v1)

Abstract: Given a trained neural network, can any specified output be generated by some input? Equivalently, does the network correspond to a function that is surjective? In generative models, surjectivity implies that any output, including harmful or undesirable content, can in principle be generated by the networks, raising concerns about model safety and jailbreak vulnerabilities. In this paper, we prove that many fundamental building blocks of modern neural architectures, such as networks with pre-layer normalization and linear-attention modules, are almost always surjective. As corollaries, widely used generative frameworks, including GPT-style transformers and diffusion models with deterministic ODE solvers, admit inverse mappings for arbitrary outputs. By studying surjectivity of these modern and commonly used neural architectures, we contribute a formalism that sheds light on their unavoidable vulnerability to a broad class of adversarial attacks.

• The paper demonstrates that neural networks using Pre-Layer Normalization and LeakyReLU-based MLPs achieve surjectivity, enabling any output mapping with the appropriate input.
• Methodologically, it leverages differential topology—via homotopy and Brouwer degree theory—to rigorously prove surjectivity in architectures like Transformers and diffusion models.
• The study underscores that surjectivity poses security risks, as it allows adversarial manipulations in generative, language, vision, and robotics applications.

Surjectivity in Neural Networks: Implications and Techniques

The paper "On Surjectivity of Neural Networks: Can you elicit any behavior from your model?" explores the concept of surjectivity in neural networks, focusing on generative models' potential to produce any specified output. It raises concerns about the security and safety implications, especially in light of the vulnerabilities to adversarial attacks and content generation risks. Here, the concept of surjectivity is examined through differential topology, providing insights into neural architectures' inherent properties.

Surjectivity in Neural Networks

Surjectivity in the context of neural networks refers to the property where a function represented by the network can map any input to any output within its output space. This property implies a certain level of expressiveness and flexibility, suggesting that any desired output can potentially be generated given an appropriate input. The research demonstrates that modern neural architectures, particularly those used in Transformers and diffusion models, often meet the criteria for surjectivity due to their design and the mathematical properties of their building blocks.

Key Theoretical Contributions

1. Pre-Layer Normalization: The paper shows that neural networks employing Pre-Layer Normalization are predominantly surjective. This is significant as it implies that architectures built with this component can potentially map inputs to a wide range of outputs due to the bounding and smoothing effects of Layer Normalization.
2. Analysis of MLPs and Attention Mechanisms:
   • Multi-Layer Perceptrons (MLPs) with LeakyReLU activation are almost always surjective when the hidden dimensionality exceeds or matches the input dimensionality. This finding leverages the properties of LeakyReLU that allow small gradients even for negative inputs, maintaining flow through the network.
   • Linear Attention mechanisms, favored for their computational efficiency over the traditional soft-max based Attention, are also shown to be almost always surjective, unlike their non-linear counterparts. This emphasizes the importance of component selection in maintaining network surjectivity.
3. Differential Topology Application: The application of differential topology provides the tools to formally prove surjectivity by examining the smooth manifolds defined by these networks. The researchers effectively use homotopy and Brouwer degree theory to analyze the network behavior analytically.

Implications for Safety and Security

Surjectivity implies that a model is vulnerable in principle to generating any conceivable output if the right input is found. This poses significant challenges for AI safety and security, especially in generative applications.

Generative Models

1. LLMs: Transformers, by design, are shown to be almost always surjective. This means, theoretically, with exact embedding manipulation, any word sequence, including proprietary or harmful content, can be produced. This raises issues about copyright and safety, suggesting that outputs alone cannot be used as evidence for data misuse.
2. Vision Models: Surjectivity in diffusion models, used extensively in image generation, implies potential vulnerabilities. Adversarial attacks can leverage the deterministic nature of these models to craft specific inputs that result in pre-determined, possibly harmful outputs, despite extensive safety training and regularization.
3. Robotics: In robotics, surjectivity implies that policy networks can be coaxed into producing any sequence of actions, given the correct sequence of observations. This indicates risks in autonomy, where robots could be manipulated under specific conditions.

Future Directions

The findings advocate for a dual approach to AI safety: enhancing architectural design to mitigate unwanted surjectivity, and developing robust attack detection and response mechanisms. Additionally, further research is needed to balance model capability with safety, possibly by exploring less-expressive but safer architectural designs or employing real-time monitoring and intervention strategies.

Conclusion

This research highlights the intrinsic surjectivity of many modern neural networks and its implications for AI safety. By advancing our understanding of how neural architectures can potentially produce any output, it underscores the need for careful consideration in AI deployment, aiming to enhance security measures and refine the design of neural models to balance expressiveness and safety effectively.