Decodable and Sample Invariant Continuous Object Encoder

TL;DR

The paper tackles encoding continuous objects (functions) into fixed-size representations that are invariant to sampling and density, decodable back to the original object, and suitable as neural-network inputs. It proposes Hyper-Dimensional Function Encoding (HDFE), a training-free framework that maps samples to a complex vector in $\mathbb{C}^N$ using explicit binding/unbinding operations and Fractional Power Encoding (FPE) mappings, with a procedure to ensure asymptotic sample invariance. The authors prove (and empirically validate) that HDFE is asymptotically sample-invariant and distance-preserving (isometric) and demonstrate its effectiveness on both mesh-grid function mappings and sparse data, including substantial improvements in point-cloud normal estimation when replacing or augmenting PointNet. In PDE-solving tasks, HDFE achieves competitive results with state-of-the-art neural operators, while offering an explicit function representation and applicability to sparse inputs. Collectively, HDFE provides a versatile, training-free interface for processing continuous objects in downstream learning tasks, with potential extensions to robustness to rotations and higher-capacity encodings.

Abstract

We propose Hyper-Dimensional Function Encoding (HDFE). Given samples of a continuous object (e.g. a function), HDFE produces an explicit vector representation of the given object, invariant to the sample distribution and density. Sample distribution and density invariance enables HDFE to consistently encode continuous objects regardless of their sampling, and therefore allows neural networks to receive continuous objects as inputs for machine learning tasks, such as classification and regression. Besides, HDFE does not require any training and is proved to map the object into an organized embedding space, which facilitates the training of the downstream tasks. In addition, the encoding is decodable, which enables neural networks to regress continuous objects by regressing their encodings. Therefore, HDFE serves as an interface for processing continuous objects. We apply HDFE to function-to-function mapping, where vanilla HDFE achieves competitive performance as the state-of-the-art algorithm. We apply HDFE to point cloud surface normal estimation, where a simple replacement from PointNet to HDFE leads to immediate 12% and 15% error reductions in two benchmarks. In addition, by integrating HDFE into the PointNet-based SOTA network, we improve the SOTA baseline by 2.5% and 1.7% in the same benchmarks.

Figures (13)

• Figure 1: Left: HDFE encodes continuous objects into fixed-length vectors without any training. The encoding is not affected by the distribution and size with which the object is sampled. The encoding can be decoded to reconstruct the continuous object. Right: Applications of HDFE. HDFE can be used to perform machine learning tasks (e.g. classification, regression) on continuous objects. HDFE also enables neural networks to regress continuous objects by predicting their encodings.
• Figure 2: (a): How $\alpha$ and $\beta$ in equation \ref{['eqn:FPE']} affects the receptive field of HDFE. (b)-(d): Functions can be reconstructed accurately given a suitable receptive field and encoding dimension. To capture the high-frequency component of the function, a small receptive field and a high dimension are required.
• Figure 3: HDFE solves a PDE by predicting the encoding of its solution and then reconstructing at points, so the error consists of a function encoding prediction error and a reconstruction error. Left: Prediction error of different methods under different testing resolutions, evaluated on the 1d Burgers' equation. Mid: The reconstruction error (in HDFE) dominates the function encoding prediction error, while the reconstruction error can be reduced by increasing the dimensionality of the embedding. Right: Prediction error of different methods evaluated on 2d Darcy Flow.
• Figure 4: PointNet fails to learn an effective function encoder. Left: Training curve of the PointNet function encoder. The error is still significantly higher than HDFE though training for 700 epochs. Right: The PointNet function encoder does not produce a reasonable reconstruction.
• Figure 5: VFA fails to reconstruct the function but HDFE succeeds.

Theorems & Definitions (11)

• Definition 1: Asymptotic Sample Invariance
• Theorem 1: Sample Invariance
• Theorem 2: Distance Preserving
• Theorem 3
• proof
• Definition : Asymptotic Sample Invariance
• proof
• Theorem
• Lemma 4
• proof