Nested relationship of the Klein hierarchy of geometries.

Stratify geometrical invariants in ascending order of stability: Euclidean geometry, affine geometry, and projective geometry. The pairs of shapes in each circle differ in corresponding geometries.

(A) Examples of stimulus arrays designed to measure the perceptual learning effects of the geometries at different levels of structural stability in an odd quadrant task. They represent discriminations based on (a) a difference in collinearity, a kind of projective property, (b) a difference in parallelism, a kind of affine property, and (c) a difference in angle orientation, a kind of Euclidean property. (B) Procedure of Experiment 1.

The configural superiority effect, adapted from (Pomerantz et al., 1977; Pomerantz and Portillo, 2011).

In the odd-quadrant discrimination task, observers were required to locate the odd stimulus in an array of four figures as quickly as possible. The left panel shows the base display where the odd quadrant differs from the rest in line slope. The center panel shows the context display with four identical quadrants. The composite display is shown in the right panel, which is simply the superposition of the base and context displays. Mean correct reaction times (RTs) and error rates are shown. Note that the reaction times and accuracies for the composite display were better than the base display.

Results of Experiment 1, RTs of each discrimination tasks measured at Pre-test and Post-test were compared by one-tailed, paired t-test.

(A) Results from the group trained on the collinearity task (n=15). Performances of the collinearity task were improved after training (p = 0.0008). (B) Results from the group trained on the parallelism task (n=15). Performances of the collinearity (p = 0.008) and parallelism (p < 0.0001) task were improved after training. (C) Results from the group trained on the orientation task (n=14). Performances of the collinearity (p = 0.0007), parallelism (p = 0.002) and orientation task (p = 0.0002) were improved after training. (***p < 0.001, **p < 0.01, *p < 0.05). Error bars denote 1 SEM across subjects.

Accuracies for the three discrimination task measured at Pre-test and Post-test.

Accuracy was defined as the average percentage correct per block. At Pre-test, the accuracies of the collinearity task were significantly higher than that in the parallelism (p < 0.0001) and orientation task (p = 0.0001). At Post-test, the accuracies of the collinearity task were still significantly higher than the other two tasks (p < 0.0001 for the parallelism task, p = 0.0001 for the orientation task). Statistical significance was calculated by paired t-test with FDR correction. (***p < 0.001, **p < 0.01, *p < 0.05). Error bars denote 1 SEM across subjects.

The learning indexes of the three geometrical invariants in Experiment 1.

Error bars denote 1 SEM across subjects.

Examples of the layout of a stimulus frame with two stimuli presented in diagonally opposite quadrants.

The top row demonstrates trials with a “target” (surrounded by orange dashed box), and the bottom row demonstrates the catch trials without “target”. The blue dashed lines represent the “base” orientation for each stimulus, and θ is the angle separation of the discrimination task. (A) Stimulus examples of the collinearity (colli.) task, the upper example shows a “target” (a line with one bend along its length) located at the lower right quadrant. Two straight lines were presented on a catch trial. (B) Stimulus examples of the parallelism (para.) task, the upper example shows a “target” (a pair of unparallel lines) located at the lower right quadrant. Two pairs of parallel lines were presented on a catch trial. (C) Stimulus examples of the orientation (ori.) task, the upper example shows a “target” (the more clockwise line) located at the upper right quadrant. Two lines with identical orientation were presented on a catch trial.

Examples of stimuli in Experiment 2.

Sample stimuli in the collinearity (left), parallelism (middle) and orientation (right) discrimination task. The blue dashed lines represent the “base” orientation for each stimulus, and θ is the angle separation of the discrimination task.

Schematic descriptions of a trial in Experiment 2.

A trial included three intervals: a 500-ms pre-stimulus epoch, a 150-ms stimulus epoch, and the response epoch. In the stimulus epoch, two stimuli were presented at two diagonally opposite quadrants. Subjects maintained fixation in a green fixation point until the stimulus disappear, and indicated if there was a “target” by pressing the corresponding key (“J” for Yes, “F” for No). They should further report the position of the “target” if they pressed the J key. Negative feedback tone was presented at the end of a trial if incorrect response was given.

Results of Experiment 2, Thresholds of each discrimination task measured at Pre-test and Post-test were compared by one-tailed, paired t-test.

(A) Results from the group trained on the collinearity task (n = 15). Performances of the collinearity task were improved after training (p = 0.026). (B) Results from the group trained on the parallelism task (n = 15). Performances of the collinearity (p = 0.007) and parallelism (p = 0.003) task were improved after training. (C) Results from the group trained on the orientation task (n = 15). Performances of the collinearity (p = 0.008), parallelism (p = 0.036) and orientation task (p < 0.0001) were improved after training. (***p < 0.001, **p < 0.01, *p < 0.05). Error bars denote 1 SEM across subjects.

The learning indexes of the three geometrical invariants in Experiment 2.

Error bars denote 1 SEM across subjects.

Bootstrap estimated learning curves across the three training tasks (n=7 for each task, the training lasted for 5 daily sessions, with each session consisting of 8 blocks).

(A) Threshold data across blocks were fitted with a power function (y = axb, where y is the block threshold) to estimate learning rate for each task. Subjects were randomly resampled with replacement 1,000 times, and parameters were fitted to each resample. Smooth curves were plotted based on the mean of the estimated parameters for each task. (B) Learning curves and performance across blocks for three training groups. Open circles indicate mean threshold per block.

Results of Experiment 3.

(A) Thresholds of each discrimination task measured at the first trained block, the last trained block, and the transfer test. (B) The specificity indexes of the three geometrical invariants. Two-tailed, two-sample t-test were conducted to compare the SIs between each pair of the three tasks. Significant difference was only found between collinearity and parallelism task (p = 0.020). Error bars denote 1 SEM across subjects.

Performance of the model when trained under different discrimination tasks.

(A) Accuracy trajectories against training iterations from the models trained on collinearity (left), parallelism (middle), and orientation task (right), with the error bar representing 1 SEM. t95 is the iteration where the fully plastic network reached 95% accuracy, depicted by green dashed lines. The numbers located at the end of each curve are the final accuracies of the last iteration. (B) The learning speed which was indexed by t95 of the three tasks. The learning speed of the collinearity task was faster than the parallelism (p = 0.018) and orientation task (p < 0.0001). The learning speed of the parallelism task was faster than the orientation task (p < 0.0001). Statistical significance was calculated by paired t-test with FDR correction. (***p < 0.001, **p < 0.01, *p < 0.05). Error bars denote 1 SEM across subjects. (C) Final mean accuracies when the network was trained and tested on all combinations of tasks

Stimulus examples in Experiment 3.

Examples of the pairs of stimulus images for the three discrimination tasks in Experiment 3. The examples here are selected from the stimulus condition with the following parameters: angle separation (10° for colli. & para. and 20° for ori.), distance for para. (40 pixels), location of gap for colli. (the front one-third).

Layer change under different training tasks.

(A) Layer change trajectories during learning. (B) Iteration at which the rate of change peaked (PSI) in layers 1-5. (C) Final layer change in layers 1-5. The error bar representing 1 SEM.