LEARNING GRID CELLS AS VECTOR REPRESENTA- TION OF SELF-POSITION COUPLED WITH MATRIX REPRESENTATION OF SELF-MOTION
This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The model consists of the following three sub-models. (1) Vector-matrix multiplication. The movement from the current position to the next position is modeled by matrix-vector multiplication, i.e., the vector of the next position is obtained by multiplying the matrix of the motion to the vector of the current position. (2) Magnified local isometry. The angle between two nearby vectors equals the Euclidean distance between the two corresponding positions multiplied by a magnifying factor. (3) Global adjacency kernel. The inner product between two vectors measures the adjacency between the two corresponding positions, which is defined by a kernel function of the Euclidean distance between the two positions. Our representational model has explicit algebra and geometry. It can learn hexagon patterns of grid cells, and it is capable of error correction, path integral and path planning.
Updated score from 6 to 7 after the authors addressed my comments below.
Previous review:
This paper builds upon the recent work on computational models of grid cells that rely on trainable (parametric) models such as recurrent neural networks [Banino et al, 2018; Cueva & Wei, 2018]. It focuses entirely on path integration in 2D and 3D, from velocity inputs only, and it relies on two sub-networks: the motion model (an RNN) and the localization model (a feed-forward network). The avowed goal of the paper is to build a very simple and linear model for grid cells.
By linearly embedding the position x into a high-dimensional hidden vector v(x) (e.g., 96 elements), it can model motion using a linear model relying on matrix-vector multiplication: v(x + dx) = M(dx) v(x), where dx is the 2D or 3D displacement, v(.) is a vector and M(.) is a matrix. The embeddings v(.) are learnable and the paper assumes a square or cubic grid of N*N or N*N*N possible positions x (with N=40); these embeddings are also normalized to unit length and obey the kernel constraint that the dot-product between any two positions' vectors v(x) and v(y) is a Gaussian or an exponential function. The motion matrix is represented as block diagonal, where each block is a rotation of subvector v_k(x) into v_k(x + dx), where each block corresponds to a specific grid cell, and where the diagonal block is further expressed as a quadratic function of dx_1, dx_2, dx_3 elements of the displacement vector.
The strengths of the paper are that:
1) The supervision of the localization subnetwork only depends on Euclidean proximity between two positions x and y and therefore uses relative positions, not absolute ones. Similarly, the path integration supervision of the motion model uses only relative displacements.
2) The resulting rate maps of the hidden units seem perfect; the model exhibits multi-scale grid behaviour.
3) The idea of using disentangled blocks, rather than injecting noise or using dropout and a softmax bottleneck as in [Banino et al, 2018], is interesting.
4) The model accumulates little path integration error over 1000 step-long episodes.
The weakness of the paper is its simplicity:
1) The assumption that A(x, y) can be modeled by a Gaussian or exponential (Laplacian?) kernel is limiting, in particular for positions x and y that are far apart.
2) There is no discussion about what egocentric vs. allocentric referentials, and dx is assumed to be aligned with (x, y) axes (which are also the axes defining the bounding box of the area).
3) Unlike the other work on learning path integration using an RNN, the linear matrix model can only handle allocentric displacements dx_1, dx_2 (and optional dx_3 in 3D).
4) No consideration is given to non-square areas: would the network also exhibit grid-like behavior if the area was circular?
5) What happens if the quadratic parameterisation of block diagonals is dropped?
6) The paper did not use metrics accepted in the neuroscience community for computing a gridness score of the grid cells (although the grid cell nature is evident). There should however be metrics for quantifying how many units represent the different scales, offsets and orientations.
7) The authors did not consider (but mentioned) embedding locations from vision, and did not consider ambiguous position embeddings.
The experiments about path planning are unconvincing. First of all, the algorithm requires to input absolute positions of every obstacle into equation (9) - (10), which assumes that there is perfect information about the map. Secondly, the search algorithm is greedy and it is not obvious how it would handle a complex maze with cul-de-sac. Saying that "there is no need for reinforcement learning or sophisticated optimal control" is very misleading: the problem here is simplified to the extreme, and fully observed, and any comparison with deep RL algorithms that can handle partial observations is just out of place.
In summary, the authors have introduced an interesting and elegant model for grid cells that suffers from simplifications. The part on path planning should be cut and replaced with more analysis of the grid cells and an explanation of how the model would handle egocentric velocity.