Changcun Huang

Changcun Huang

This paper proposes a theoretical framework on the mechanism of autoencoders. To the encoder part, under the main use of dimensionality reduction, we investigate its two fundamental properties: bijective maps and data disentangling. The general construction methods of an encoder that satisfies either or both of the above two properties are given. The generalization mechanism of autoencoders is modeled. Based on the theoretical framework above, we explain some experimental results of variational autoencoders, denoising autoencoders, and linear-unit autoencoders, with emphasis on the interpretation of the lower-dimensional representation of data via encoders; and the mechanism of image restoration through autoencoders is natural to be understood by those explanations. Compared to PCA and decision trees, the advantages of (generalized) autoencoders on dimensionality reduction and classification are demonstrated, respectively. Convolutional neural networks and randomly weighted neural networks are also interpreted by this framework.

Changcun Huang

This paper provides a theoretical framework on the solution of feedforward ReLU networks for interpolations, in terms of what is called an interpolation matrix, which is the summary, extension and generalization of our three preceding works, with the expectation that the solution of engineering could be included in this framework and finally understood. To three-layer networks, we classify different kinds of solutions and model them in a normalized form; the solution finding is investigated by three dimensions, including data, networks and the training; the mechanism of a type of overparameterization solution is interpreted. To deep-layer networks, we present a general result called sparse-matrix principle, which could describe some basic behavior of deep layers and explain the phenomenon of the sparse-activation mode that appears in engineering applications associated with brain science; an advantage of deep layers compared to shallower ones is manifested in this principle. As applications, a general solution of deep neural networks for classifications is constructed by that principle; and we also use the principle to study the data-disentangling property of encoders. Analogous to the three-layer case, the solution of deep layers is also explored through several dimensions. The mechanism of multi-output neural networks is explained from the perspective of interpolation matrices.

Changcun Huang

This paper first constructs a typical solution of ResNets for multi-category classifications by the principle of gate-network controls and deep-layer classifications, from which a general interpretation of the ResNet architecture is given and the performance mechanism is explained. We then use more solutions to further demonstrate the generality of that interpretation. The universal-approximation capability of ResNets is proved.

Changcun Huang

This paper aims to understand the training solution, which is obtained by the back-propagation algorithm, of two-layer neural networks whose hidden layer is composed of the units with smooth activation functions, including the usual sigmoid type most commonly used before the advent of ReLUs. The mechanism contains four main principles: construction of Taylor series expansions, strict partial order of knots, smooth-spline implementation and smooth-continuity restriction. The universal approximation for arbitrary input dimensionality is proved and experimental verification is given, through which the mystery of ``black box'' of the solution space is largely revealed. The new proofs employed also enrich approximation theory.

Changcun Huang

A neural network with one hidden layer or a two-layer network (regardless of the input layer) is the simplest feedforward neural network, whose mechanism may be the basis of more general network architectures. However, even to this type of simple architecture, it is also a ``black box''; that is, it remains unclear how to interpret the mechanism of its solutions obtained by the back-propagation algorithm and how to control the training process through a deterministic way. This paper systematically studies the first problem by constructing universal function-approximation solutions. It is shown that, both theoretically and experimentally, the training solution for the one-dimensional input could be completely understood, and that for a higher-dimensional input can also be well interpreted to some extent. Those results pave the way for thoroughly revealing the black box of two-layer ReLU networks and advance the understanding of deep ReLU networks.

Changcun Huang

This paper aims to interpret the mechanism of feedforward ReLU networks by exploring their solutions for piecewise linear functions, through the deduction from basic rules. The constructed solution should be universal enough to explain some network architectures of engineering; in order for that, several ways are provided to enhance the solution universality. Some of the consequences of our theories include: Under affine-geometry background, the solutions of both three-layer networks and deep-layer networks are given, particularly for those architectures applied in practice, such as multilayer feedforward neural networks and decoders; We give clear and intuitive interpretations of each component of network architectures; The parameter-sharing mechanism for multi-outputs is investigated; We provide an explanation of overparameterization solutions in terms of affine transforms; Under our framework, an advantage of deep layers compared to shallower ones is natural to be obtained. Some intermediate results are the basic knowledge for the modeling or understanding of neural networks, such as the classification of data embedded in a higher-dimensional space, the generalization of affine transforms, the probabilistic model of matrix ranks, and the concept of distinguishable data sets.

Changcun Huang

This paper presents a basic property of region dividing of ReLU (rectified linear unit) deep learning when new layers are successively added, by which two new perspectives of interpreting deep learning are given. The first is related to decision trees and forests; we construct a deep learning structure equivalent to a forest in classification abilities, which means that certain kinds of ReLU deep learning can be considered as forests. The second perspective is that Haar wavelet represented functions can be approximated by ReLU deep learning with arbitrary precision; and then a general conclusion of function approximation abilities of ReLU deep learning is given. Finally, generalize some of the conclusions of ReLU deep learning to the case of sigmoid-unit deep learning.