# Deep frequency principle towards understanding why deeper learning is faster

@article{Xu2020DeepFP, title={Deep frequency principle towards understanding why deeper learning is faster}, author={Zhi-Qin John Xu and Hanxu Zhou}, journal={ArXiv}, year={2020}, volume={abs/2007.14313} }

Understanding the effect of depth in deep learning is a critical problem. In this work, we utilize the Fourier analysis to empirically provide a promising mechanism to understand why feedforward deeper learning is faster. To this end, we separate a deep neural network, trained by normal stochastic gradient descent, into two parts during analysis, i.e., a pre-condition component and a learning component, in which the output of the pre-condition one is the input of the learning one. We use a…

## 11 Citations

### Frequency Principle in Deep Learning Beyond Gradient-descent-based Training

- Computer ScienceArXiv
- 2021

Empirical studies show the universality of the F-Principle in the training process of DNNs with nongradient-descent-based training, and algorithms without gradient information, such as Powell’s method and Particle Swarm Optimization.

### A Computable Definition of the Spectral Bias

- MathematicsAAAI
- 2022

Neural networks have a bias towards low frequency functions. This spectral bias has been the subject of several previous studies, both empirical and theoretical.
Here we present a computable…

### Solving Multi-Dimensional Schr\"{o}dinger Equations Based on EPINNs

- Computer Science
- 2022

A novel numerical method that uses a neural network to solve the multi-dimensional static Schr¨odinger equation and other high-dimensional eigenvalue problems for multi-electron atoms and molecules is proposed.

### Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networks

- Computer ScienceArXiv
- 2022

An embedding principle in depth is discovered that loss landscape of an NN “contains” all critical points of the loss landscapes for shallower NNs, which serves as a solid foundation for the further study about the role of depth for DNNs.

### Empirical Phase Diagram for Three-layer Neural Networks with Infinite Width

- Computer ScienceArXiv
- 2022

The phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studyingDeep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

### Limitation of characterizing implicit regularization by data-independent functions

- Computer Science
- 2022

This work makes an attempt to mathematically define and study the implicit regularization, and proposes two dynamical mechanisms, i.e., Two-point and One-point Overlapping mechanisms, based on which they provide two recipes for producing classes of onehidden-neuron NNs that provably cannot be fully characterized by a type of or all data-independent functions.

### Overview frequency principle/spectral bias in deep learning

- Computer Science
- 2022

An overview of F-Principle is provided and some open problems for future research are proposed, which inspire the design of DNN-based algorithms in practical problems, explains experimental phenomena emerging in various scenarios, and further advances the study of deep learning from the frequency perspective.

### Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs

- Computer ScienceSSRN Electronic Journal
- 2022

A subspace decomposition based DNN (dubbed SDNN) architecture for a class of multi-scale problems by combining traditional numerical analysis ideas and MscaleDNN algorithms is constructed.

### Going Deeper in Frequency Convolutional Neural Network: A Theoretical Perspective

- Computer ScienceArXiv
- 2021

The Fourier transform theory is revisited to derive feed-forward and back-propagation frequency operations of typical network modules such as convolution, activation and pooling and extended to the Laplace transform for CNN, which can run in the real domain with more relaxed constraints.

### Towards Understanding the Condensation of Neural Networks at Initial Training

- Computer Science
- 2021

This work illustrates the formation of the condensation in multi-layer fully connected NNs and shows that the maximal number of condensed orientations in the initial training stage is twice the multiplicity of the activation function, where “multiplicity” indicates the multiple roots of activation function at origin.

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