A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds

Alessio Marta; Paola Causin

cs.LGAug 24, 2026

A Theory of Speciation in Generative Diffusion Models on Compact Riemannian Manifolds

Authors: Alessio Marta, Paola Causin

Organizations: Department of Mathematics, University of Milan, via Saldini 50, 20133 Milan, Italy

Abstract

Speciation in generative diffusion models denotes the emergence of distinct stable branches during denoising, through which initially undifferentiated trajectories progressively commit to different data classes. In this work we develop an intrinsic theory of speciation for diffusion models supported on compact Riemannian manifolds: the aim is to go beyond existing theoretical descriptions, which usually identify speciation with a symmetric pitchfork bifurcation and assume to work in a large-dimensional space. We characterize speciation by bifurcations of the critical points of the evolving probability density. A spectral heat-kernel representation makes explicit the role of the manifold geometry, while Poincaré-Hopf and Morse theory impose global constraints on the number and type of score equilibria and reveal topologically-imposed geometrical modes. For mixtures of heat kernels, we prove that generic speciation events have a one-dimensional critical kernel and admit an A2 fold normal form; pitchforks and simultaneous multidirectional transitions arise from nongeneric symmetric configurations. We derive geometry-dependent estimates of speciation times for bimodal mixtures and Riemannian regular simplices. We further establish structural stability of nondegenerate folds under score perturbations and show that the first-order time shift is determined solely by the component of the score error along the critical direction. The theory is illustrated on the sphere using mixtures of von Mises-Fisher distributions, where pitchfork and saddle-node bifurcations, topological modes, and hierarchical multiple speciations are observed. Finally, a chart-based intrinsic score-learning scheme based on neural networks contrasts the theoretically predicted transitions on prototypal and more complex datasets.

Figures & tables

Figure 1: Bifurcation scenarios for the 1D problem ( 3.1 ) with $t^{*}=1$ . Left: pitchfork bifurcation ( $a=0$ ). Two stable nullclines branch from the original stable equilibrium point $x=0$ , which becomes unstable after the bifurcation time ( $t_{c}=t^{*}=1$ ). Right: saddle-node “imperfect” bifurcation ( $a=1$ ). A pair of stable and unstable nullclines emerge at $x=1$ , which is not the original equilibrium point of the system; here $t_{c}=2>t^{*}$ . Solid lines denote stable equilibria, dashed lines unstable ones, and violet curves representative trajectories stemming from different initial conditions. Notice that increasing time $t$ here corresponds to noise-to-data reverse flow in the following examples.

Figure 2: Distribution $p_{t}$ as a function of the angle $\theta$ drawn at the section $\phi=\pi$ , for an initial bimodal vMF density. Read along the reverse process, the curves show the evolution from large $t$ — where $p_{t}$ is essentially the Riemannian uniform distribution $1/4\pi\approx 0.08$ — down to $t=0$ . As $t$ decreases, the two data modes emerge as maxima, while the critical point at $\theta=\pi/2$ changes nature, from a maximum to a minimum separating the two data modes. Left: modes at $\alpha=\pi/6$ ; the change of nature at $\theta=\pi/2$ occurs around $t\approx 0.8-0.9$ . Right: modes at $\alpha=\pi/3$ ; since the two centers are closer, the transition at the equator is more proximal to initial time, around $t\approx 0.2$ . In both panels $\theta=0$ and $\theta=\pi$ remain local minima (geometrical modes).

Panel of Figure 3 — Figure 3: Bimodal vMF distribution, analytic score. Sign (symbols), values (colors) and zeroes (in green) of the $\theta$ component of the score, restricted on the great circle ( $\phi=\pi$ ) connecting the centers of the bimodal distribution for $\alpha=\pi/6$ (top) and $\alpha=\pi/3$ (bottom). The speciation times $t^{*}\approx 0.89$ (top) and $t^{*}\approx 0.21$ (bottom) are in correspondence with the critical points of Figure 2 and are in good agreement with their theoretical estimates $t^{*}\approx 0.99$ and $t^{*}\approx 0.18$ , obtained via ( 3.23 ) – see Appendix A.3 for an estimate of the parameter $\sigma^{2}$ in the case of a vMF distribution. For small times $t$ , the attractors correspond to the centers of the initial bimodal distribution. In both cases, after speciation the previous attractor $\theta=\pi/2$ – the equator – becomes a repeller (dashed line). Notice that, for readability, the two panels have different time ranges and colors are mapped on different scales.

Panel of Figure 4 — Figure 4: Bimodal vMF distribution, analytic score. Study of the stability of the equilibrium points of the score for the case $\alpha=\pi/6$ , in relation to the critical points of $p_{t}$ . Top: Logarithm of the Riemannian norm of the score $S_{t}$ for different times. The centroids of the dark blue regions correspond to the real zeros of $S_{t}$ , and therefore to the critical points of $p_{t}$ . At the speciation time $t^{*}\approx 0.89$ the new zeroes of the score emerge via a pitchfork bifurcation from the original critical point, see also Figure 3 . Bottom: Logarithm of $p_{t}$ at different times. Before speciation ( $t>t^{*}\approx 0.89$ ) the critical point at $\theta=\pi/2$ is a maximum and therefore an attractor. After speciation ( $t<t^{*}\approx 0.89$ ) this point becomes a saddle, thus an hyperbolic point, while the new critical points at $\theta=\pi/2\pm\pi/3$ are maxima – stable equilibrium points of $S_{t}$ . The minima at $\theta=\pi/2$ , $\phi=0$ and $\phi=2\pi$ are geometric modes.

Figure 5: Bimodal vMF distribution, analytic score. Plot of the two eigenvalues $\lambda_{1,2}$ of $\operatorname{Hess}p_{t}$ at the critical time $t^{*}$ . Notice that at the corresponding critical point $(\theta^{*},\pi^{*})=(\pi/2,\pi)$ the eigenvalue $\lambda_{1}$ vanishes, while $\lambda_{2}$ is negative, coherently with the fact that at speciation $\ker(\operatorname{Hess}p_{t})$ is one dimensional.

Panel of Figure 6 — Figure 6: Trimodal vMF distribution, analytic score. Plot of the sign (symbols), value (colors) and zeroes (in green) of the components of the score. Top: $\theta$ -component at the section $\phi=\pi$ . Near the first speciation time $t^{*}_{1}$ , the late time branch is shifted toward $\theta=\pi/6$ , the common polar angle of $C_{1}$ and $C_{2}$ and a saddle-node bifurcation generates the attractor of $C_{3}$ at $\theta=5\pi/6$ . Bottom: $\phi$ -component at the section $\theta=\pi/6$ . At the second speciation time $t_{2}^{*}\approx 0.12$ a pitchfork bifurcation separates $C_{1}$ and $C_{2}$ . Notice that, for readability, the two panels have different time ranges and colors are mapped on different scales.

Panel of Figure 7 — Figure 7: Trimodal vMF distribution, analytic score. Behavior near the first speciation (for clarity only angles $\phi\in(\pi/2,3\pi/2)$ are shown). Top: logarithm of the Riemannian norm of $S_{t}$ . Bottom: logarithm of $p_{t}$ . At $t_{1}^{*}\approx 0.6$ , the score vanishes at the center $C_{3}$ , which is an attractor. At this time, a second attractor appears at the midpoint of the geodesics through $C_{1}$ and $C_{2}$ , which are not yet distinguished. The two attractors are separated by a saddle near $(\tfrac{7\pi}{12},\pi)$ . This corresponds to a saddle-node bifurcation: the original attractor at $\theta=\pi/6$ persists, while a saddle-node pair emerges near $(\tfrac{5\pi}{6},\pi)$ , see also Figure ( 6 ). The attractor of the $C_{1}$ - $C_{2}$ cluster is a geometric mode forced to exist by Poincarè-Hopf equality ( 3.7 ) after the $C_{3}$ speciation.

Panel of Figure 8 — Figure 8: Trimodal vMF distribution, analytic score. Behavior after (left), near (middle) and in correspondence (right) of the second speciation at $t^{*}_{2}\approx 0.12$ . For clarity only latitudes $\phi\in(\pi/2,3\pi/2)$ are shown. Top row: logarithm of the Riemannian norm of $S_{t}$ . Bottom row: logarithm of $p_{t}$ . The coarse-grained geometric mode appeared after the first speciation time at $(\pi/6,\pi)$ , representing the $C_{1}$ - $C_{2}$ cluster, is now sifted into two separate attractive basins.

Figure 9: Bimodal vMF distribution, learned score. Trajectory generated from a point sampled at about $\pi/4$ , $\phi=6/5\pi$ . First the trajectory heads to the equator, after speciation the equator becomes unstable and repels the process so that the trajectory commits to a point near the north pole. Time is parametrized by the colors of the standard viridis map.

Panel of Figure 10 — Figure 10: Bimodal vMF distribution, learned score field before (top left), and after speciation time (top right) and at the end of the reverse process (bottom). For clarity only azimuthal angles $\phi\in(2,4)$ are shown.

Panel of Figure 11 — Figure 11: Bimodal distribution, learned score. Zeros (green) and values (color) of the $\theta$ component of the learned score for. Left ( $\alpha=\pi/6$ ): the speciation time $t_{learned}^{*}\approx 0.86$ is in good agreement with the theoretical one ( $t_{theoretical}^{*}\approx 0.89$ ) – see Figure 3 . Right ( $\alpha=\pi/3$ ): the pitchfork unfolds into a $A_{2}$ bifurcation and the numerical speciation time is in good agreement with the theoretical speciation time ( $t^{\star}_{learned}\approx 0.23$ vs $t^{\star}_{theoretical}\approx 0.21$ ) In both cases, the approximation of the score given by the network breaks the perfect pitchfork, as discussed in Remark 7 . The original stable branch persists, while a secondary stable (bottom branch of the pair)–unstable (top branch of the pair) pair is created through a saddle-node.

Panel of Figure 12 — Figure 12: Trimodal vMF distribution with equal weights, learned score. Zeros (green) and values (colors) of the components of the learned score. Left: $\theta$ component at first speciation. As in the analytic model (cf. Figure 6 ), the late-time branch is steered towards the cluster at $\theta=\pi/6$ , while the center at $\theta=5/6\pi$ speciates via a saddle-node. Right: $\phi$ component at second speciation. The perfect pitchfork of the analytic solution unfolds into a saddle node due to the approximation of the score. The first speciation time is well approximated, the second (very close to the end of the reverse process) is affected by a larger error.

Panel of Figure 13 — Figure 13: Trimodal vMF distribution, learned score, equidistant centers on the equator. Left: $w_{1}=w_{2}=w_{3}=1/3$ case. Due to the symmetry, two saddle-node pairs are created at about the same speciation time. The original late-time branch at $\phi=\pi$ is preserved, and it continues to the nearest center located at $\phi=\pi$ . Notice that the speciation time emerging from the learned score for the two branches is similar but not exactly the same (grey area). Right: $w_{1}=1/2,w_{2}=w_{3}=1/4$ . The original late-time branch is steered towards the center with heaviest weight. The asymmetry of the weights leads to different speciation times for the pairs $C_{1}-C_{2}$ and $C_{2}-C_{3}$ .

Panel of Figure 14 — Figure 14: Fires on Earth dataset. Left: original dataset; Right: $500$ points generated by the model. Earth Map from [ 45 ]

Panel of Figure 15 — Figure 15: Fires on Earth dataset. Zeros (green) and values (colors) of the score components. Left ( $\theta=\pi/2$ ): $\phi$ component. The late time branch contributes to the formation of the central Africa fire cluster ( $\phi\approx 3.5$ ); The central America cluster ( $\phi\approx 1.9$ ) is generated via a bifurcation at $t_{1}^{*}\approx 0.45$ (applying ( 3.23 ) to this pairs of cluster with $\sigma=0$ yields $t_{1}^{*}\approx 0.56$ ); The saddle-node at $t^{*}_{2}\approx 0.27$ corresponds to the formation of the Indian-Southeast Asia cluster ( ( 3.23 ) yields $t_{2}^{*}\approx 0.25$ ). The repellers separating the the clusters are located in the Atlantic and Indian oceans respectively. Right ( $\phi\approx 3.9$ ): $\theta$ component. The central Africa cluster speciates from the East Europe cluster at $t^{*}_{3}\approx 0.1$ ( ( 3.23 ) yields $t_{2}^{*}\approx 0.14$ ). The repeller separating the two clusters is located in the Mediterranean sea.

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