Fig 1.
(a) Non-metric topological graph, labeled with distance measurements. The labels are independent of each other and do not need to adhere to the triangle inequality. (b) Embedded graph from (a). To find a Euclidean embedding, the distance labels need to be adjusted to create a valid configuration, for example by stretching or compressing the edges or “wiggling” on the vertices until the difference between map and labels is minimized. As opposed to the non-metric labeled graph, changes to one label will therefore influence others. (c) Euclidean metric map. Places are directly assigned coordinates based on their position in the world. Over time, the coordinates may be refined by repeated measurements and the map will approach the Euclidean ground truth. The same can be expected from the embedded graph optimization if the labels are refined.
Fig 2.
(a) Vertex positions in the maze. Their pixel coordinates were considered the Euclidean ground truth for the model. The maze was partitioned into straight segments and corners, and one vertex was placed per corner. Two vertices, 12 and 35, were only used in a control graph without wormholes. (b) The corresponding topological graph with edges through wormholes (red dotted lines). The graph was then labeled with local distance and angle measurements based on the ground truth, except for the wormhole edges, which were manually adjusted to reflect the locally distorted topology instead. Note that the distance along the wormhole edges is shortened but not zero.
Fig 3.
(a) Layout of the wormhole maze, redrawn from Warren et al. (2017) [35]. The yellow arrows show wormhole position and magnitude. Touching one end of the arrow instantly and seamlessly teleported subjects to the other end. (b, c) Example directional estimates for object pairs from the “Route-finding and shortcuts” dataset (Experiment 1 in Warren et al. (2017) [35]) and the “Rips and folds” dataset (Experiment 2 in Warren et al. (2017) [35]). The thin arrows show the Euclidean ground truth direction between objects, the short dotted lines the corresponding subject estimates, and the thick solid line the average subject estimate. The length of the estimates has been normalized and does not reflect walked distance. In (c), the colors indicate different goals.
Fig 4.
(a, b) References for comparison. The unembedded graph (as in Fig 2) and an another embedding which was also found by the optimization method. The second embedding performed worse on the subject data and was not further used. (b) The embedded graph, i.e., the labeled graph with the vertices at coordinates that minimize the difference between map and labels. The orientation of the embeddings is arbitrary; here, they were rotated so that the edge (2, 3) is horizontal. The red dotted lines show the edges that pass through wormholes. (d) Sketch of the distorted wormhole maze according to the embedding in (c). The sketch shows how the embedding might be represented by a subject. Edges that cross each other in the embedded graph could for example be rationalized as multi-level paths, leading to a 3D representation. Alternatively, in a purely 2D map, the arms would simply intersect. Note that the edges have no coordinates in the embedding but are simply lines in the adjacency matrix.
Fig 5.
(a-c): Dataset “route-finding and shortcuts”, (d-f): Dataset “rips and folds”. (a) Shortcut predictions of the non-metric labeled graph (dotted lines) and average subject estimates (solid lines), plotted on ground truth coordinates. The gray vertices show how the graph would continue on routes through wormholes. (b) Shortcut predictions of the embedded graph, lines as in (a). The subject estimates were rotated to match the local orientation of the originating maze arm. (c) Distribution of the prediction error. The difference between the models is not significant, i.e., they predict the data equally well. (d) Example shortcut predictions (dotted lines) and subject estimates (solid lines) for three of the 24 object pairs in the “rips and folds” dataset. (e) Shortcut predictions of the embedded graph for the same object pairs as in (d). (f) Distribution of the prediction error. The difference between the models is also not significant on this dataset.