This paper proposes a model for learning grid-like units for spatial awareness and navigation. In this model, the self-position of the agent is represented by a vector, and the self-motion of the agent is represented by a block-diagonal matrix. Each component of the vector is a unit (or a cell). The model consists of the following two sub-models. (1) Motion sub-model. The movement from the current position to the next position is modeled by matrix-vector multiplication, i.e., multiplying the matrix representation of the motion to the current vector representation of the position in order to obtain the vector representation of the next position. (2) Localization sub-model. The adjacency between any two positions is a monotone decreasing function of their Euclidean distance, and the adjacency is modeled by the inner product between the vector representations of the two positions. Both sub-models can be implemented by neural networks. The motion sub-model is a recurrent network with dynamic weight matrix, and the localization sub-model is a feedforward network. The model can be learned by minimizing a loss function that combines the loss functions of the two sub-models. The learned units exhibit grid-like patterns (as well as stripe patterns) in all 1D, 2D and 3D environments. The learned model can be used for path integral and path planning. Moreover, the learned representation is capable of error correction.

本篇论文提出了一个具有代表性的网格单元表示模型，该模型包含向量-矩阵乘法、放大的局部等距变换和全局邻接核三个子模型，可进行网格单元的学习、纠错、路径积分和路径规划。

自我位置的向量表示与自我运动的矩阵表示相结合的格网细胞学习