Erkan Turan

Erkan Turan, Maks Ovsjanikov

Generative Modeling via Drifting has recently achieved state-of-the-art one-step image generation through a kernel-based drift operator, yet the success is largely empirical and its theoretical foundations remain poorly understood. In this paper, we make the following observation: \emph{under a Gaussian kernel, the drift operator is exactly a score difference on smoothed distributions}. This insight allows us to answer all three key questions left open in the original work: (1) whether a vanishing drift guarantees equality of distributions ($V_{p,q}=0\Rightarrow p=q$), (2) how to choose between kernels, and (3) why the stop-gradient operator is indispensable for stable training. Our observations position drifting within the well-studied score-matching family and enable a rich theoretical perspective. By linearizing the McKean-Vlasov dynamics and probing them in Fourier space, we reveal frequency-dependent convergence timescales comparable to \emph{Landau damping} in plasma kinetic theory: the Gaussian kernel suffers an exponential high-frequency bottleneck, explaining the empirical preference for the Laplacian kernel. We also propose an exponential bandwidth annealing schedule $σ(t)=σ_0 e^{-rt}$ that reduces convergence time from $\exp(O(K_{\max}^2))$ to $O(\log K_{\max})$. Finally, by formalizing drifting as a Wasserstein gradient flow of the smoothed KL divergence, we prove that the stop-gradient operator is derived directly from the frozen-field discretization mandated by the JKO scheme, and removing it severs training from any gradient-flow guarantee. This variational perspective further provides a general template for constructing novel drift operators, demonstrated with a Sinkhorn divergence drift.

Erkan Turan, Gaspard Abel, Maysam Behmanesh et al.

Graph Neural Networks (GNNs) learn node representations through iterative network-based message-passing. While powerful, deep GNNs suffer from oversmoothing, where node features converge to a homogeneous, non-informative state. We re-frame this problem of representational collapse from a \emph{bifurcation theory} perspective, characterizing oversmoothing as convergence to a stable ``homogeneous fixed point.'' Our central contribution is the theoretical discovery that this undesired stability can be broken by replacing standard monotone activations (e.g., ReLU) with a class of functions. Using Lyapunov-Schmidt reduction, we analytically prove that this substitution induces a bifurcation that destabilizes the homogeneous state and creates a new pair of stable, non-homogeneous \emph{patterns} that provably resist oversmoothing. Our theory predicts a precise, nontrivial scaling law for the amplitude of these emergent patterns, which we quantitatively validate in experiments. Finally, we demonstrate the practical utility of our theory by deriving a closed-form, bifurcation-aware initialization and showing its utility in real benchmark experiments.

Erkan Turan, Aristotelis Siozopoulos, Louis Martinez et al.

Continuous Normalizing Flows (CNFs) enable elegant generative modeling but remain bottlenecked by slow sampling: producing a single sample requires solving a nonlinear ODE with hundreds of function evaluations. Recent approaches such as Rectified Flow and OT-CFM accelerate sampling by straightening trajectories, yet the learned dynamics remain nonlinear black boxes, limiting both efficiency and interpretability. We propose a fundamentally different perspective: globally linearizing flow dynamics via Koopman theory. By lifting Conditional Flow Matching (CFM) into a higher-dimensional Koopman space, we represent its evolution with a single linear operator. This yields two key benefits. First, sampling becomes one-step and parallelizable, computed in closed form via the matrix exponential. Second, the Koopman operator provides a spectral blueprint of generation, enabling novel interpretability through its eigenvalues and modes. We derive a practical, simulation-free training objective that enforces infinitesimal consistency with the teacher's dynamics and show that this alignment preserves fidelity along the full generative path, distinguishing our method from boundary-only distillation. Empirically, our approach achieves competitive sample quality with dramatic speedups, while uniquely enabling spectral analysis of generative flows.

Maysam Behmanesh, Erkan Turan, Maks Ovsjanikov

Graph alignment, the problem of identifying corresponding nodes across multiple graphs, is fundamental to numerous applications. Most existing unsupervised methods embed node features into latent representations to enable cross-graph comparison without ground-truth correspondences. However, these methods suffer from two critical limitations: the degradation of node distinctiveness due to oversmoothing in GNN-based embeddings, and the misalignment of latent spaces across graphs caused by structural noise, feature heterogeneity, and training instability, ultimately leading to unreliable node correspondences. We propose a novel graph alignment framework that simultaneously enhances node distinctiveness and enforces geometric consistency across latent spaces. Our approach introduces a dual-pass encoder that combines low-pass and high-pass spectral filters to generate embeddings that are both structure-aware and highly discriminative. To address latent space misalignment, we incorporate a geometry-aware functional map module that learns bijective and isometric transformations between graph embeddings, ensuring consistent geometric relationships across different representations. Extensive experiments on graph benchmarks demonstrate that our method consistently outperforms existing unsupervised alignment baselines, exhibiting superior robustness to structural inconsistencies and challenging alignment scenarios. Additionally, comprehensive evaluation on vision-language benchmarks using diverse pretrained models shows that our framework effectively generalizes beyond graph domains, enabling unsupervised alignment of vision and language representations.