These interactive 3D plots show each participant's facial-expression data projected into a participant-specific linear discriminant space using LDA (Linear Discriminant Analysis). Unlike the UMAP projections, which preserve temporal trajectory structure, LDA operates at the frame level — each frame is treated as an independent observation with no temporal context. The discriminant axes are computed to maximise linear separation between the nine emotion categories.
This makes the LDA and UMAP galleries complementary: the UMAP shows how emotional expression moves through the manifold over time; the LDA shows how discriminable a single moment of expression is from another. Convergences between the two analyses validate geometric findings across independent methods. Divergences reveal the temporal gap — the layer of dynamical structure that exists in sequences but is invisible to frame-level analysis.
Two visual versions are provided for each participant: black background (art / presentation mode) and white background (print / analysis mode). The toggle dropdown allows individual emotions or pairs to be isolated.
Axes represent the first three linear discriminant dimensions. The explained variance ratio for dimension 1 is shown for each participant — this quantifies how much of the frame-level discriminable structure is captured by a single linear axis.
Dimension 1 loading — the most important number per participant. P3's 99.96% means almost all frame-level discriminable structure lives in a single linear axis: nine emotion categories are nearly perfectly separable at the level of individual frames, without any temporal context. Compare this to the UMAP, which shows the same participant's data as an island topology — the two analyses provide convergent evidence for highly separable emotional geometry.
Flow geometry — isolate Flow (orange) and compare its position across participants. In some participants (for example P4 and P6), Flow occupies an isolated region of the manifold, whereas in others it lies adjacent to neighbouring emotional states. This variability mirrors the participant-specific organisation observed in the UMAP gallery and illustrates that even highly recognisable emotional states do not occupy a universal geometric location.
LDA vs UMAP divergences — some geometric relationships visible in UMAP are absent in LDA, and vice versa. P5 illustrates an important distinction between frame-level and temporal analyses. Although Anger and Flow are well separated in the LDA projection, they appear as neighbouring regions in the UMAP manifold because UMAP captures the continuous transitions between emotional states. LDA describes instantaneous separability; UMAP describes the geometry of emotional trajectories.
Discriminant structure — Discriminant structure — Compare the overall shape of the projections. Participant 3 is almost one-dimensional (99.96% of discriminable structure captured by a single axis), whereas Participants 4, 6 and 7 require multiple dimensions to separate the emotion categories. These differing geometries demonstrate that even frame-level emotional organisation is participant-specific.
The LDA and UMAP projections use the same 76-dimensional ARKit feature vectors but differ fundamentally in their analytical assumptions. UMAP is unsupervised and preserves the sequential structure of 60fps trajectories — it reveals how emotional expression moves through the manifold over time. LDA is supervised and treats each frame independently — it reveals what makes a single moment of expression discriminable from another.
Viewing both galleries together gives access to two independent geometric analyses at different levels of abstraction. Where they agree, the finding is robust across methods. Where they diverge, the divergence is informative — it reveals the temporal gap between frame-level structure and trajectory-level dynamics.