“coarse-to-fine” is a powerful paradigm in a lot of domains. as some concrete examples:
- engineering: developing a “minimal viable product” that addresses the major use cases first and then refining the design, experience, and adding features to address long-tail use cases over time
- creative expression: roughing out shapes on a canvas before adding detail, constructing an outline before writing an essay and then fleshing out the essay by incrementally adding detail to the outline * probabilistic modeling: DDPMs resolve low frequency details in early denoising steps and then high frequency details in later steps, hierarchical models with variables corresponding to scales of resolution
- ontologies - hierarchies of knowledge structured over layers of abstraction RG is explicitly a fine-to-coarse process. In practice, a lot of these strategies are probably learned from processes that resemble RG’s coarse-graining from starting at the finer resolution and working up to lower resolution. Has there been any attempt to formalize something like an inverse RG? I’m especially interested in characterizing coarse-to-fine phenomena
The idea of a “coarse-to-fine” strategy, as you’ve outlined, indeed finds resonance across various domains, offering a systematic approach to problem-solving, design, and analysis by starting from a broad overview and progressively adding detail and complexity. This contrasts with the Renormalization Group (RG) approach in physics, which systematically reduces degrees of freedom to study phenomena at larger scales, effectively moving from fine to coarse resolutions. The concept of an “inverse RG” or a formalism that captures the essence of coarse-to-fine processes is intriguing and multifaceted.
Attempts to Formalize Coarse-to-Fine Processes
While the traditional RG approach is well-established in statistical physics for studying systems across scales, particularly in the context of critical phenomena and phase transitions, the notion of an inverse process—systematically adding degrees of freedom or detail to study phenomena at smaller scales—is less straightforward. However, there are areas where concepts analogous to an “inverse RG” have been explored, albeit under different terminologies and frameworks:
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Multi-Scale Modeling and Simulation: In fields such as materials science and computational biology, multi-scale modeling techniques aim to bridge the gap between coarse-grained and fine-grained descriptions of systems. These approaches often start with a broad, simplified model and incrementally integrate finer-scale details to capture more complex behaviors. This process mirrors the coarse-to-fine paradigm, although it may not always be formalized in the same theoretical framework as RG.
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Machine Learning and Computer Vision: In machine learning, particularly in computer vision, coarse-to-fine strategies are employed for tasks such as image segmentation, object detection, and image synthesis. Deep Learning models, including certain types of generative models like Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs), often learn to capture global, low-frequency features before refining predictions with local, high-frequency details. While not directly termed as “inverse RG,” these approaches conceptually align with moving from coarse to fine resolutions.
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Adaptive Mesh Refinement in Numerical Simulations: In computational physics and engineering, adaptive mesh refinement (AMR) techniques adjust the resolution of the computational grid dynamically, starting with a coarse mesh for the overall problem domain and adding finer grids in regions requiring higher resolution. This methodological framework could be seen as an application of the coarse-to-fine paradigm, enhancing computational efficiency while maintaining accuracy where needed.
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Hierarchical and Multi-Resolution Models: In probabilistic modeling and statistics, hierarchical models and those employing multi-resolution techniques often build understanding from coarse to fine layers. These models can start with broad assumptions or parameters and refine these as more data or finer scale information becomes available. This approach is particularly relevant in Bayesian modeling, where prior knowledge is updated with incoming data in a layered or sequential manner.
Formalization and Challenges
The formalization of a coarse-to-fine process akin to an “inverse RG” faces several challenges:
- Complexity and Detail Integration: Systematically adding detail and complexity without a predefined finer-scale model poses conceptual and practical challenges, particularly in ensuring consistency across scales.
- Scale Bridging: Identifying and applying the correct mechanisms for bridging between scales, especially in a way that accurately captures emergent phenomena at finer resolutions, is non-trivial.
- Computational Efficiency: In computational contexts, maintaining efficiency while increasing resolution or detail requires innovative approaches to model architecture and algorithm design.
Despite these challenges, the exploration of coarse-to-fine strategies and their potential formalization offers promising avenues for advancing our understanding and capabilities across scientific and engineering domains. The ongoing developments in computational methods, machine learning, and theoretical modeling continue to contribute to this exploration, potentially leading to new frameworks that can systematically capture the essence of coarse-to-fine phenomena.
Systematically adding detail and complexity without a predefined finer-scale model poses conceptual and practical challenges, particularly in ensuring consistency across scales. Identifying and applying the correct mechanisms for bridging between scales, especially in a way that accurately captures emergent phenomena at finer resolutions, is non-trivial. i’m not sure this is actually a problem. By starting from the scale of the desired emergent phenomena and working backwards, we can ensure that any details added at finer resolutions are consistent with the phenomena we specified at higher resolutions.
You’ve highlighted a crucial point that inverts the traditional concern in multi-scale modeling and analysis. By anchoring the process on the scale of the desired emergent phenomena and then working backwards towards finer resolutions, the approach inherently aligns the finer-scale details with the overarching phenomena observed at the coarser scale. This strategy offers a direct pathway to maintain consistency across scales, as the emergent properties and behaviors at the higher levels serve as constraints for the development and integration of details at lower levels. Let’s explore how this strategy can address the challenges:
Consistency Across Scales
By using emergent phenomena as the starting point and working backwards:
- Constraint-Based Modeling: This approach can be seen as establishing a set of constraints derived from observed or desired emergent properties. When adding details at finer resolutions, these constraints ensure that any new features or behaviors are compatible with the higher-level phenomena. This method can be particularly effective in systems where the emergent properties are well understood, even if the underlying microscale mechanisms are not.
- Iterative Refinement and Feedback: A backward design process allows for iterative refinement. As finer-scale details are added, their impact on the emergent phenomena can be continuously assessed. This iterative loop ensures that adjustments at the microscale are always informed by the macroscopic goals, facilitating a more targeted and efficient exploration of model parameters and behaviors.
Bridging Between Scales
Starting from the emergent phenomena simplifies the identification of relevant mechanisms and scales by:
- Focused Scale Interaction: Knowing the target emergent properties allows modelers to selectively focus on the scale interactions most relevant to those properties. This can help in identifying which finer-scale details are likely to have a significant impact on the emergent phenomena, thus prioritizing modeling efforts.
- Guided by Emergent Needs: This strategy can guide the selection of modeling techniques and the development of algorithms that are best suited for capturing the critical interactions leading to the desired phenomena. For example, in computational simulations, this could influence the choice between agent-based models, continuum models, or hybrid approaches based on the specific emergent properties of interest.
Practical Implementation
Implementing this strategy effectively may involve:
- Top-Down Analysis and Design: Starting with a comprehensive analysis of the emergent phenomena to identify key properties and behaviors. This analysis then guides the development of the finer-scale model components, ensuring they are designed with the necessary characteristics to reproduce the emergent phenomena.
- Adaptive and Multi-Resolution Methods: Utilizing adaptive techniques that allow for dynamic resolution adjustments based on the needs of the emergent phenomena. Such methods can allocate modeling resources more efficiently, focusing detail where it impacts emergent properties the most.
Conclusion
This reverse engineering approach to modeling and analysis, anchored in the desired emergent phenomena, offers a promising pathway to tackle the complexity of multi-scale systems. It provides a coherent framework for ensuring consistency across scales and effectively bridges the gap between microscale details and macroscale behaviors. While this strategy may not be applicable in all contexts, especially where emergent phenomena are poorly understood or extremely complex, it represents a valuable paradigm in the toolkit of multi-scale modeling and analysis, particularly when the goal is to capture specific emergent properties with high fidelity.
Misc Thoughts
coarse → fine ⇒ constraint development starts with specification. coarse grained starting point is a constraint the higher resolution needs to satisfy
fine → coarse ⇒ characterization development starts by considering the state of the system and developing decreasing resolutions of abstraction with which to characterize the system