Learning on manifolds without manifold learning (2402.12687v2)

Abstract: Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace-Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively ``rough'' functions.

• The paper introduces a novel method for direct function approximation on manifolds using spherical polynomials, eliminating the need for manifold learning.
• It establishes optimal error bounds that depend on the intrinsic dimensionality, effectively mitigating the curse of dimensionality.
• The approach enhances computational efficiency and robustness in applications such as computer vision and signal processing by leveraging low-dimensional data structures.

An Overview of "Learning on Manifolds without Manifold Learning"

The paper "Learning on Manifolds without Manifold Learning" by H. N. Mhaskar and Ryan O'Dowd addresses a pivotal challenge in the field of machine learning—function approximation on high-dimensional data sampled from a low-dimensional manifold. Traditional methods rely on manifold learning to derive characteristics of the manifold before performing function approximation, typically as a two-step process. These traditional approaches entail manifold learning efforts such as eigendecomposition of the Laplace-Beltrami operator or deriving coordinate charts, imposing an extra error due to manifold approximation.

Overview of the Methodology

The authors propose a method that avoids the manifold learning step altogether by directly targeting function approximation using spherical polynomials. This approach circumvents the need to explicitly learn manifold properties other than its dimension. The data is instead viewed as samples drawn from a sub-manifold of an ambient hypersphere, which allows the use of spherical polynomials for constructing a one-shot approximation. This methodology is rooted in approximation theory and leverages optimal error bounds assuming the manifold structure of the data.

Specifically, the paper introduces a kernel-based approximation on hyperspheres with a direct sum of spherical harmonics up to a certain degree. The polynomial expansion's coefficients are computed directly from the data, without the need to infer the manifold’s underlying structure.

Theoretical Framework and Results

The primary contribution of the paper is a theoretical framework that establishes optimal rates of approximation for relatively rough functions using spherical harmonics. The results show that for a function $f(x) = \mathbb{E}(z|y)$, where $(y_j, z_j)$ are samples, the approximation error $\| F_n(\mathcal{D}; \cdot) - ff_0 \|_{\mathbb{X}}$ scales as:

$$\lesssim \left( \sqrt{f_0}_{\mathbb{X}} + z_{\mathbb{X} \times \Omega} + (ff_0)_{W_\gamma(\mathbb{X})} \right) \left( \frac{\log(M/\delta)^{q+2\gamma}}{M} \right)^{\gamma/(q+2\gamma)}$$

Where $f_0$ represents the density function of the manifold distribution, and $M$ is the sample size. The approach demonstrates that the rate of convergence depends on the dimensionality of the manifold rather than the ambient space—effectively mitigating the curse of dimensionality.

Implications and Future Directions

The implications of this research are significant in machine learning and artificial intelligence, particularly in applications where data do not densely populate the ambient space but adhere to lower-dimensional manifolds, such as computer vision and signal processing. By eliminating the manifold learning step, the method becomes computationally more efficient and potentially more robust to diverse data distributions.

The notable outcomes invite several avenues for future research, including:

• Extending approximation techniques to cover more complex or non-compact manifolds.
• Exploring the integration of this approach within neural network architectures to enhance feature learning on manifold-structured data.
• Investigating the interplay of this direct approximation method with probabilistic modeling and inference to handle uncertainties in data distributions.

Overall, this paper presents a methodologically sound and theoretically supported alternative for manifold-related learning tasks, highlighting its potential to streamline and improve the efficiency of machine learning workflows that involve high-dimensional data approximation on unknown low-dimensional sub-manifolds.