Hypothesis spaces for deep learning | JoVE Visualize

Area of Science:

Machine Learning
Functional Analysis
Deep Learning Theory

Background:

Deep neural networks (DNNs) are complex functions, but their theoretical underpinnings, particularly their hypothesis space, remain an active area of research.
Understanding the mathematical structure of DNNs can lead to more principled learning algorithms and theoretical guarantees.
Existing work often focuses on specific network architectures or properties, lacking a unified functional space perspective.

Purpose of the Study:

To introduce and characterize a novel hypothesis space for deep learning models based on deep neural networks.
To establish the theoretical framework of a reproducing kernel Banach space (RKBS) for DNNs.
To investigate the implications of this RKBS framework for understanding and solving learning problems like regularized learning and minimum norm interpolation (MNI).

Main Methods:

Treating deep neural networks (DNNs) as functions of input and parameter variables.
Defining the parameter space using weight matrices and biases for a given network depth and width.
Constructing a Banach space by taking the weak* closure of the linear span of the DNN set.
Proving the resulting space is an RKBS and deriving its reproducing kernel.
Establishing representer theorems for regularized learning and MNI problems within the RKBS framework.

Main Results:

The hypothesis space of DNNs, under specific constructions, forms a reproducing kernel Banach space (RKBS).
A reproducing kernel for this RKBS is explicitly derived.
Representer theorems are established for regularized learning and minimum norm interpolation (MNI) problems.
Solutions to these learning problems are shown to be expressible as finite sums of kernel expansions dependent on training data.

Conclusions:

The proposed RKBS framework provides a rigorous mathematical foundation for understanding the hypothesis space of DNNs.
The derived reproducing kernel and representer theorems offer new analytical tools for DNNs.
This work bridges functional analysis and deep learning, potentially enabling the development of more efficient and interpretable learning algorithms.