From data to functa: Your data point is a function and you can treat it like one (2201.12204v3)

Abstract: It is common practice in deep learning to represent a measurement of the world on a discrete grid, e.g. a 2D grid of pixels. However, the underlying signal represented by these measurements is often continuous, e.g. the scene depicted in an image. A powerful continuous alternative is then to represent these measurements using an implicit neural representation, a neural function trained to output the appropriate measurement value for any input spatial location. In this paper, we take this idea to its next level: what would it take to perform deep learning on these functions instead, treating them as data? In this context we refer to the data as functa, and propose a framework for deep learning on functa. This view presents a number of challenges around efficient conversion from data to functa, compact representation of functa, and effectively solving downstream tasks on functa. We outline a recipe to overcome these challenges and apply it to a wide range of data modalities including images, 3D shapes, neural radiance fields (NeRF) and data on manifolds. We demonstrate that this approach has various compelling properties across data modalities, in particular on the canonical tasks of generative modeling, data imputation, novel view synthesis and classification. Code: https://github.com/deepmind/functa

• The paper introduces functa, a revolutionary method that transforms each data point into a continuous function using implicit neural representations.
• It leverages meta-learning to enable rapid adaptation to new data with minimal fine-tuning, simplifying representation across diverse modalities.
• Generative, inferential, and classification tasks demonstrate that functa offer improved scalability and efficiency over traditional discrete data representations.

From Data to Functa: A Framework for Deep Learning on Functional Representations

The research paper "From data to functa: Your data point is a function and you can treat it like one" presents a novel paradigm shift in the approach to deep learning by introducing the concept of functa—where data points are represented as functions. This framework leverages implicit neural representations (INRs), which are neural networks trained to map continuous inputs to their respective data values. The authors propose a method for deep learning tasks where these functions, called functa, serve as the primary data representation rather than traditional discrete data arrays.

Motivation and Conceptual Framework

The traditional approach in deep learning involves using discrete grids to represent continuous phenomena—for instance, pixels in images or voxels in 3D shapes. While effective, this discretization can lead to inefficiencies, especially when dealing with high-resolution or inherently continuous data structures that resist simple grid representation, like neural radiance fields (NeRFs). By using INRs, the authors propose a functional representation that models data in a continuous domain, allowing for potentially more effective handling of varying resolutions and data types. This leads to four major advances: scaling with resolution, managing data of varying resolutions, handling signals difficult to discretize, and simplifying multimodal learning.

Functa: Representation and Meta-Learning

The paper introduces functa as the unit of data in the proposed framework, harnessing the power of INRs. To address challenges in efficiently representing functa and converting large datasets into functa, the paper adopts meta-learning techniques. This involves creating a shared initialization for a neural network that can quickly adapt to new data points with minimal fine-tuning (e.g., a few gradient steps), allowing the learning of these functional representations in a scalable manner. This approach enables the training of large datasets (termed functasets) of functa across multiple data modalities.

Applications and Performance Analysis

The paper explores multiple applications, demonstrating the versatility and effectiveness of functa:

1. Generative Modeling: Both normalizing flows and diffusion models are applied to functa, showing promising results in generating high-quality samples across modalities like images and 3D shapes. The decoupling of encoding functa and training on them simplifies the generative modeling process and offers a unified approach across data types.
2. Inference: By utilizing a learned prior distribution over functa, tasks such as data imputation and novel view synthesis (key in virtual reality and graphics) become more efficient and accurate when compared to traditional methods.
3. Classification: The framework also shows efficiency in discriminative tasks, achieving high accuracy with relatively simple models, highlighting the potential for reducing computational complexity in large models.

Challenges and Further Directions

While promising, the paper acknowledges certain limitations. The removal of spatial structure in functa representation poses challenges for employing traditional neural architectures like CNNs, which rely on such inductive biases. Future work could explore spatially-aware functa architectures that balance generality with domain-specific biases. Additionally, the paper highlights the computational cost of meta-learning, a limitation that could be addressed by alternative learning strategies or architectural improvements.

Overall, the research outlines a new frontier for deep learning that rethinks data representation, allowing for a more natural alignment with the continuous nature of the real world. This approach has significant implications for improving learning efficiency and model effectiveness across domains where traditional discrete data representations are suboptimal.