线性卷积网络的代数复杂度与神经簇

多项式神经网络的几何学

Geometry of Polynomial Neural Networks

本文利用代数几何研究单项式激活 PNNs 的神经流形，通过半代数集、神经簇、维数与学习次数刻画其表达能力和训练复杂度，并以实验佐证。

关于二次两神经元神经网络的欧几里得距离度

On the Euclidean Distance Degree of Quadratic Two-Neuron Neural Networks

研究二次两神经元网络的欧几里得距离度，通过代数几何方法分析其与二次切线包络的关系，并证明该度量依赖于输入维度的稳定多项式。

New Convolutional Codes Derived from Algebraic Geometry Codes

In this paper, we construct new families of convolutional codes. Such codes are obtained by means of algebraic geometry codes. Additionally, more families of convolutional codes are constructed by means of puncturing, extending, expanding and by the direct product code construction applied to algebraic geometry codes. The parameters of the new convolutional codes are better than or comparable to the ones available in literature. In particular, a family of almost near MDS codes is presented.

关于代数层的稳定欧几里得距离度

On the Euclidean Distance Degree of Shallow Polynomial Networks

本文研究了代数神经层的稳定欧几里得距离度，证明其在输入输出维度中为稳定多项式，并通过交理论和 Schubert 计算得出结果。

Geometry of Linear Convolutional Networks

We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully-connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of…

多项式卷积网络的几何与优化

On the Geometry and Optimization of Polynomial Convolutional Networks

研究使用单项式激活函数的卷积神经网络，证明其参数化映射是正则且几乎处处同构，通过代数几何方法计算神经流形的维数和度，并量化回归损失优化中临界点的数量。