What Can the Shape of Eye Movements Tell Us?

Imagine recording someone's gaze as they watch a video. You obtain a long sequence of coordinates: left, right, up, down, a pause, another jump. A familiar approach is to measure fixations and saccades, then calculate statistics of those events. But a sequence has a structure that an average cannot describe. Two people can visit the same positions in very different orders.

In Persistent Patterns in Eye Movements: A Topological Approach to Emotion Recognition, my coauthors and I explored a simple question: can the geometry of eye-movement dynamics contain useful information about emotional state? Our approach connects dynamical systems, topology, and ordinary machine learning. The interesting part is how we get from a time series to a shape.

1. The order of points matters

A gaze trace is a sequence g_t = (x_t, y_t), not merely an unordered collection of points. To see the difference, consider two paths that visit exactly the same positions. In one, the eye travels smoothly around them. In the other, the eye jumps between them. Their ordinary point clouds are identical. Their dynamics are not.

Here is the mathematical trick. Rather than represent a signal by its value at one instant, take a short collection of its values at separated instants:

This construction is called a delay embedding. It turns a one-dimensional signal into a trajectory in a higher-dimensional space. We apply it separately to horizontal and vertical gaze coordinates. In the paper, each produces a three-dimensional point cloud; we also study the original two-dimensional gaze cloud. The idea is inspired by Takens' theorem, though our empirical choice of parameters is not a proof of the theorem's conditions for this dataset.

A useful intuition is that the delayed coordinates let the past and future of the signal appear next to one another. Repeated patterns can then show up as geometric structure, even when they are difficult to recognize from summary statistics.

2. Let the point cloud grow

We now have points in a space. What is their shape? At a very small scale they are separate islands. As we increase a distance threshold, nearby points connect. Eventually a ring may appear, and at a still larger scale the ring fills in. The connected components and holes that survive over a range of scales are precisely the kinds of features studied by persistent homology.

There is a subtle point here. A loop is not important simply because it appears. What matters is how long it survives while we increase ε. We summarize the lifetime of a topological feature by its birth and death scales, (b_i, d_i). Its persistence is d_i − b_i. This gives us a way to separate short-lived geometric accidents from longer-lived structure without choosing one arbitrary scale in advance.

3. From shapes to numbers

A machine-learning model needs a fixed-length vector. We calculate persistent homology in dimensions 0 and 1 for the raw gaze cloud and the two delay-embedded clouds, then summarize their persistence diagrams. Examples include the number of recorded birth-death pairs, the mean and maximum of their birth, death, and persistence values, and entropy-type summaries. These do not tell us the meaning of a loop. They quantify its geometry across scales.

Finally, a random forest learns which combinations of these features help distinguish the four quadrants of the circumplex model of affect: positive or negative valence, each with high or low arousal. These are broad experimental labels, not a claim that every feeling has a unique topological signature.

4. What did we find?

We tested the method on the VREED dataset, in which 34 participants watched immersive VR videos while their eye movements and other measurements were recorded. Our analysis used left-eye gaze trajectories. The paper reports the following four-class classification accuracies:

The improvement in this experiment is encouraging: temporal shape can complement the familiar statistics of eye movements. But it is not evidence that a camera can reliably read a stranger's emotions. The dataset is relatively small, the paper used sample-level randomized splits, and it does not report a participant-held-out evaluation. Measurements from the same participant may therefore appear in different splits. Generalization to unseen people remains an open question.

5. Why I find this interesting

For me, the attraction of this problem is not the classifier. It is the change of language. A sequence of eye positions becomes a trajectory in a reconstructed state space. A trajectory becomes a family of geometric objects. Those objects become topological measurements. Only at the end do we ask whether the measurements predict anything.

There is also a useful restraint in this viewpoint. Topology does not explain emotion, and two different processes can have similar topology. What it offers is a particular family of geometric descriptors of dynamic data. The next meaningful tests would include participant-independent evaluation, sensitivity to the delay and sampling choices, and comparisons across more datasets.

And this is the broader question worth keeping: what other time series might reveal something when we stop looking only at their individual values and start studying the shape of their evolution?