Provable approximation properties for deep neural networks (1509.07385v3)

Abstract: We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $\Gamma$, the complexity of $f$, in terms of its wavelet description, and only weakly on the ambient dimension $m$. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU)

• The paper introduces a depth-4 neural network with provable error bounds for approximating functions on smooth, low-dimensional manifolds.
• It employs ReLU-based wavelet constructions and local coordinate charts to capture function complexity with sparse connections.
• The approach achieves L2 and pointwise error guarantees, offering actionable insights for efficient DNN architecture design in high-dimensional applications.

Provable Approximation Properties for Deep Neural Networks: An Analytical Overview

The paper entitled "Provable approximation properties for deep neural networks" by Uri Shaham, Alexander Cloninger, and Ronald R. Coifman presents an analytical framework for approximating functions on smooth, low-dimensional manifolds embedded in higher-dimensional spaces using deep neural networks (DNNs). The core contribution of this work is the construction of sparsely-connected depth-4 neural networks capable of achieving specified approximation errors for given target functions over such manifolds. This addresses a fundamental challenge in the theoretical understanding of DNNs: determining how network topology and other hyperparameters should be defined to obtain desired approximation properties.

Key Contributions

The paper articulates a theoretical approach in which the function to be approximated is defined on a d-dimensional manifold $\Gamma \subset \mathbb{R}^m$. The authors construct a depth-4 neural network where the number of units, $N$, is determined by several factors: the wavelet-based complexity of the target function $f$, the curvature and dimension of the manifold $\Gamma$, and, notably, only weakly on the ambient dimension $m$. The construction relies on decomposing $f$ into wavelet functions computed with Rectified Linear Units (ReLUs).

The authors methodologically cover several aspects:

1. Wavelet Frame Construction: The development of wavelet frames from Rectified Linear Units (ReLU) is a cornerstone of the approach. This allows for the creation of an analytic representation of functions with desired approximation capabilities.
2. Local Coordinate Systems via Atlases: The manifold is covered by charts (local coordinate systems), facilitating the approximation of functions locally and the reduction of the ambient dimensionality's complexity.
3. Extension of Wavelet Terms: Each wavelet term is meticulously extended from the local chart to the entire ambient space $\mathbb{R}^m$. This step relies on the uniqueness of manifold curvature, particularly when $d \ll m$.
4. Network Topology Specification: The explicit construction of the network's topology concerning the curvature of the manifold and complexity of $f$ offers valuable guidance for theoretical and practical application.

Important Numerical and Theoretical Insights

• Sparse Wavelet Coefficients: If the function $f$ has wavelet coefficients in $l_1$, it is shown that there exists a depth-4 network achieving an $L_2$ approximation error rate such that $\|f-f_N\|_2^2 \le \frac{c}{N}$, where $c$ reflects characteristics like the curvature of $\Gamma$.
• Twice Differentiable Functions: For functions in $C^2$ with bounded Hessian, a pointwise approximation error of $\|f-f_N\|_\infty = O\left(N^{-\frac{2}{d}}\right)$ is achieved. This implies a markedly different dependence on the dimensionality of the manifold compared to the ambient space.

Implications and Future Directions

The paper's contributions are two-fold: it fortifies the theoretical framework of DNNs by providing insights into how manifold-induced constraints can be leveraged in neural architecture design. Practically, it offers potential for more efficient DNN applications, especially in scenarios where data lies on or near low-dimensional manifolds embedded in high-dimensional spaces.

Looking forward, future research can explore the applications of these provable approximation properties in real-world machine learning tasks, particularly in dimensionality reduction contexts like high-dimensional signal processing and computer vision. Extensions to anisotropic wavelets or adaptively tuned networks could further enhance approximation efficiency. Moreover, empirical validation and comparison against standard DNN training pipelines will be invaluable, lending practical credibility to the proposed theoretical constructs.

In conclusion, the paper's findings underscore the profound interplay between function complexity, network topology, and manifold geometry, anchoring an intricate layer of theory atop the empirical successes of deep learning.