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Tolga Birdal

Publications and source records attributed to Tolga Birdal.

At least 19 recordsLinked to original sources
arXiv2026

Heat Kernel Textures: the Geodesic Gaussians That Do Not Splat

3D Gaussian Splatting has recently revolutionised novel view synthesis as well as many other 3D vision methods and applications. Drawing inspiration from this representation, we now rethink textures to overcome the main issues of UV mapping while considerably lowering their memory footprint. Heat Kernel Textures (HKTex) eliminate UV unwrapping as well as their persistent issues of wasted UV space, seams, distortions, vertex-duplication, and varying resolution. Grounded in discrete Riemannian geometry and intrinsically defined on any manifold surface discretised as a triangular mesh, HKTex uses anisotropic heat kernels as geodesic equivalents to Gaussians. Like our kernels, also the optimisation of their position and the adaptive densification strategies were redefined to operate on the surface of the object to be textureised. Our novel representation is also fully integrated with a physically based renderer and can be optimised either from existing textures or multi-view images. Our project page and code are available at circle-group.github.io/research/HeatKernelTextures.

cs.CV↗
arXiv2026

Bernstein-Vazirani Networks: Quantum Machine Learning by Interference

We introduce Bernstein-Vazirani Networks (BVNs), a non-variational quantum machine learning framework that leverages quantum interference for supervised learning, demonstrated on vision and representation learning tasks. In their standard form, BVNs follow the principle of quantum Fourier sampling: labelled data are placed in superposition and interfered in the Fourier basis to extract globally informative features. We then define generalised BVNs that enable interference in problem-adapted bases, yielding more expressive models under the same measurement budget as in the standard setting. BVNs achieve universal function approximation through (over)complete interference bases, while training of BVNs is gradient-free. Experiments on synthetic and real-world classification tasks, as well as implicit image representation, show strong generalisation capabilities and competitive performance with classical and quantum baselines.

quant-ph↗
arXiv2026

DisPOSE: Projected Polystochastic Diffusion for Self-Supervised Multi-View 3D Human Pose Estimation

Recovering 3D human poses for multiple individuals from different camera views is a fundamental bottleneck for analyzing interacting behaviors. Existing self-supervised approaches leverage synthetic catalogues of 3D poses; however, this leads to poor generalization in real-world scenarios due to distribution shifts. We therefore introduce DisPOSE, a self-supervised framework that approximates the inherently discrete multi-view person-assignment problem as a generative diffusion process over the space of polystochastic tensors. By employing differentiable Sinkhorn projections during denoising, our model learns to guide solutions toward valid and feasible assignments based on 2D image priors. The complete 3D skeletons of localized individuals are then regressed using a Hypergraph-Convolutional Decoder that explicitly models relational structures and articulated joints across multiple views. The proposed approach outperforms current state-of-the-art self-supervised methods on standard datasets and demonstrates strong performance on a newly proposed benchmark featuring highly occluded scenes from surgical operating rooms. Our diffusion-based localization demonstrates high label efficiency, retaining 99% of its performance with only 10% of the pseudo-labels. Notably, disentangling the assignment and root regression components while maintaining differentiability makes DisPOSE nearly agnostic to different camera arrangements. Project site and code: https://wngtn.github.io/DisPOSE/

cs.CV↗
arXiv2026

Latent Riemannian Flow Matching for Geometry-Grounded 3D Foundation Models

Geometric foundation models, such as the Visual Geometry Grounded Transformer (VGGT), provide strong 3D priors from unposed images. However, such models operate purely in a feed-forward, deterministic regime, \ie~they cannot generate plausible geometry beyond what the input views directly support. Generative models for 3D scenes, on the other hand, must rely on strong geometric priors to produce coherent outputs from sparse inputs. We bridge these two paradigms by performing flow matching directly in VGGT's latent space, leveraging its learned 3D priors without committing to any explicit downstream representation such as Gaussians, meshes, or video-VAE latents. This requires respecting the latent geometry: VGGT tokens occupy a product of high-dimensional hyperspheres on which standard Euclidean flow matching fails. We address this with a Riemannian Flow Matching framework defined on a product manifold of four hyperspheres, aligned with VGGT's multi-scale encoder, which keeps generated tokens on the valid data manifold required by the frozen decoding heads. On RealEstate10K, ScanNet++ and ETH3D, our method achieves strong performance against recent scene generation baselines in both per-view appearance and aggregated 3D geometry, establishing latent-space flow matching on geometric foundation models as a viable paradigm for 3D generation. The project page can be found $\href{https://lisaweijler.github.io/geometry-grounded-rfm/}{\text{here}}$.

cs.CV↗
arXiv2026

MoLingo: Motion-Language Alignment for Text-to-Human Motion Generation

We introduce MoLingo, a text-to-motion (T2M) model that generates realistic, lifelike human motion by denoising in a continuous latent space. Recent works perform latent space diffusion, either on the whole latent at once or auto-regressively over multiple latents. In this paper, we study how to make diffusion on continuous motion latents work best. We focus on two questions: (1) how to build a semantically aligned latent space so diffusion becomes more effective, and (2) how to best inject text conditioning so the motion follows the description closely. We propose a semantic-aligned motion encoder trained with frame-level text labels so that latents with similar text meaning stay close, which makes the latent space more diffusion-friendly. We also compare single-token conditioning with a multi-token cross-attention scheme and find that cross-attention gives better motion realism and text-motion alignment. With semantically aligned latents, auto-regressive generation, and cross-attention text conditioning, our model sets a new state of the art in human motion generation on standard metrics and in a user study. We will release our code and models for further research and downstream usage.

cs.CV↗
arXiv2026

SAMoR: Motion Modelling for Articulated Objects of Any Skeleton and Topology

Modeling motion for articulated objects of arbitrary skeleton topology remains difficult: existing motion generators target a fixed human skeleton, and prior adaptations either fail to share a vocabulary across rigs or discard motion detail through global pooling. Our key observation is that while joint-level motion does not correspond cleanly across species, motion of functional joint groups does: a human arm, a wolf foreleg, and a bird wing share motion structure despite differing joint counts and connectivity, a correspondence that joint names (e.g., "forearm", "wing_L1") partially expose even when topology does not. We introduce SAMoR (Skeleton-Aware Motion Representation for Articulated Objects), a cross-topology motion representation that encodes each motion segment as a small fixed number ($K=8$) of part tokens shared across arbitrary skeletons. A graph-transformer encoder consumes per-joint motion features, kinematic graph structure, and joint-name embeddings, then compresses them into part-level tokens via cross-attention pooling and residual vector quantization, yielding a discrete motion codebook shared across rigs. To keep the part queries from collapsing into redundant global representations, we introduce a topology-agnostic attention supervision loss, with joint-name dropout to reduce over-reliance on text labels. We curate a heterogeneous corpus from HumanML3D, Truebones Zoo, and animated Objaverse-XL assets, and evaluate SAMoR on held-out characters with unseen skeletons. It supports accurate reconstruction and cross-topology transfer, and enables text-conditioned generation and part-wise editing via a MaskGIT token generator. SAMoR reaches $2.75 \times 10^{-2}$ normalized MPJPE on cross-topology reconstruction, $5.8\times$ below the strongest adapted variable-$J$ tokenizer baseline, while remaining competitive with fixed-skeleton specialists on HumanML3D.

cs.CV↗
arXiv2026

Collapsed Effective Operators for Higher-order Structures

Higher-order structures are powerful relational modeling tools, yet existing spectral operators decompose the topology into separate ranks, leaving practitioners to fuse the information back to vertices through ad hoc choices. We introduce Collapsed Effective Operators, which condense higher-order degrees of freedom into a single vertex-level operator via Schur complementation of a graded Laplacian. This yields a (generally dense) operator that encodes long-range interactions mediated by topology and is applicable to arbitrary higher-order constructs. We show it preserves positive semi-definiteness with a spectral upper bound relative to the rank-0 Hodge Laplacian, effectively lowering system energy under higher-order connectivity. Empirically, our operator improves spectral clustering, signal smoothing, and enables the inclusion of topological features in neural network architectures via positional encoding. The project page can be found http://circle-group.github.io/research/CollapsedEffectiveOperators

cs.LG↗
arXiv2026

Adversarial Attacks Leverage Interference Between Features in Superposition

Why do adversarial examples exist, and why do they transfer between models? Existing explanations appeal to high-dimensional geometry, non-robust patterns in the input, and decision boundary structure, but none provides a representation-level mechanism that explains why specific perturbations succeed and why attacks transfer between models. In this paper, we show that adversarial vulnerability can stem from efficient information encoding in neural networks. Specifically, vulnerability can arise from superposition - the phenomenon where networks represent more concepts than they have dimensions, forcing non-orthogonal representation and thus interference. This interference causes perturbations targeting one representation to affect others, creating vulnerabilities determined by interference patterns. In synthetic settings with precisely controlled superposition, we establish that superposition suffices to create adversarial vulnerability. The resulting attacks are predictable: PGD-discovered perturbations align with theoretically optimal perturbations derived from the interference geometry. Models trained on similar data develop similar interference patterns, explaining attack transferability. We then show that successful attacks on image classifiers exhibit the structure predicted by our proposed mechanism. These findings reveal that adversarial vulnerability can be a byproduct of networks' representational compression, complementing existing explanations based on data properties or architectural factors.

cs.LG↗
arXiv2026

Learning Interface Breakup: A Geometry-Conditioned Latent Surrogate for Spray Formation

Designing spray nozzles requires predicting how geometry shapes transient two-phase breakup, but high-fidelity volume-of-fluid (VOF) simulations with adaptive mesh refinement (AMR) are too expensive for iterative design exploration. Standard surrogate models are also challenged by this setting because both the liquid--gas interface and the underlying adaptive discretization evolve across time and geometries. We introduce a geometry-conditioned latent surrogate trained on 797 two-phase nozzle simulations that addresses this by encoding the AMR cell-density field, rather than the full multi-channel flow state, as a compact proxy for where the solver concentrates resolution. From this representation, the model reconstructs transient density evolution and nozzle geometry, and a lightweight second stage recovers the remaining flow variables. On held-out simulations, the method accurately captures key interface dynamics while reducing inference time to 0.045 seconds per trajectory, corresponding to a speed-up of more than $6\times10^4$ relative to Basilisk CFD. These results suggest that AMR refinement structure can serve as a compact and learnable representation for geometry-conditioned surrogate modeling of transient two-phase flows.

physics.flu-dyn↗
arXiv2026

Topological Neural Operators

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

cs.LG↗
arXiv2026

COMPOSE: Hypergraph Cover Optimization for Multi-view 3D Human Pose Estimation

3D human pose estimation from sparse multi-view camera rigs is an essential task for numerous applications, including action recognition, sports analysis, and human-robot interaction. While learned methods dominate the field on benchmarks, they require large annotated datasets; training-free optimization-based methods remain promising as they circumvent 3D supervision by solving a correspondence problem across views from 2D detections. Existing combinatorial formulations rely on pairwise associations to model this correspondence problem and enforce global consistency across views only as a downstream constraint. However, reconciling locally plausible pairwise matches becomes brittle under occlusion and noisy detections, where local errors propagate globally. We propose COMPOSE, which recasts multi-view 3D human pose estimation as a weighted exact-cover optimization over a hypergraph of person hypotheses. Our formulation replaces pairwise association and post-hoc consistency enforcement with a single global combinatorial objective. To address the exponentially large candidate space, we introduce a geometric pruning strategy alongside two complementary solvers: an exact Integer Linear Programming formulation and a scalable relaxation via Belief Propagation. Without any 3D supervision, COMPOSE improves average precision by up to 31 points over the best optimization-based method and 13 points over self-supervised learned methods, demonstrating the effectiveness of higher-order combinatorial association for training-free multi-view 3D human pose estimation.

cs.CV↗
arXiv2026

Variational Inference for Lévy Process-Driven SDEs via Neural Tilting

Modelling extreme events and heavy-tailed phenomena is central to building reliable predictive systems in domains such as finance, climate science, and safety-critical AI. While Lévy processes provide a natural mathematical framework for capturing jumps and heavy tails, Bayesian inference for Lévy-driven stochastic differential equations (SDEs) remains intractable with existing methods: Monte Carlo approaches are rigorous but lack scalability, whereas neural variational inference methods are efficient but rely on Gaussian assumptions that fail to capture discontinuities. We address this tension by introducing a neural exponential tilting framework for variational inference in Lévy-driven SDEs. Our approach constructs a flexible variational family by exponentially reweighting the Lévy measure using neural networks. This parametrization preserves the jump structure of the underlying process while remaining computationally tractable. To enable efficient inference, we develop a quadratic neural parametrization that yields closed-form normalization of the tilted measure, a conditional Gaussian representation for stable processes that facilitates simulation, and symmetry-aware Monte Carlo estimators for scalable optimization. Empirically, we demonstrate that the method accurately captures jump dynamics and yields reliable posterior inference in regimes where Gaussian-based variational approaches fail, on both synthetic and real-world datasets.

cs.LG↗
arXiv2026

Root-to-Leaf Path Random Walks, Normalized Hodge Laplacians, and Cheeger Inequalities on Simplicial Complexes

We introduce root-to-leaf path random walks on double covers of graded signed graphs and analyze their behavior in a general setting. Viewing simplicial complexes within this framework, we show that these walks induce the natural normalization of the coboundary operator and of the Hodge Laplacians while preserving the basic structural features of combinatorial Hodge theory. We then derive Cheeger inequalities for the upper side of the normalized Hodge spectrum, identify the coherent structures governing these bounds, and combine the up- and down-cases into sharper estimates.

math.CO↗
arXiv2026

Generalization at the Edge of Stability

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.

cs.LG↗
arXiv2026

Parallelised Differentiable Straightest Geodesics for 3D Meshes

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes. A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics, which also allows to trace geodesics and parallel-transport vectors as a by-product. We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme. After proving our parallelisation performance and accuracy, we demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries. In particular, to showcase the versatility of our method, we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser that we apply to centroidal Voronoi tessellation. Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG.

cs.CV↗
arXiv2026

HOG-Diff: Higher-Order Guided Diffusion for Graph Generation

Graph generation is a critical yet challenging task, as empirical analyses require a deep understanding of complex, non-Euclidean structures. Diffusion models have recently made significant advances in graph generation, but these models are typically adapted from image generation frameworks and overlook inherent higher-order topology, limiting their ability to capture graph topology. In this work, we propose Higher-order Guided Diffusion (HOG-Diff), a principled framework that progressively generates plausible graphs with inherent topological structures. HOG-Diff follows a coarse-to-fine generation curriculum, guided by higher-order topology and implemented via diffusion bridges. We further prove that our model admits stronger theoretical guarantees than classical diffusion frameworks. Extensive experiments across eight graph generation benchmarks, spanning diverse domains and including large-scale settings, demonstrate the scalability of our method and its superior performance on both pairwise and higher-order topological metrics. Our project page is available \href{https://circle-group.github.io/research/hog-diff/}{here}.

cs.LG↗
arXiv2026

TopoOR: A Unified Topological Scene Representation for the Operating Room

Surgical Scene Graphs abstract the complexity of surgical operating rooms (OR) into a structure of entities and their relations, but existing paradigms suffer from strictly dyadic structural limitations. Frameworks that predominantly rely on pairwise message passing or tokenized sequences flatten the manifold geometry inherent to relational structures and lose structure in the process. We introduce TopoOR, a new paradigm that models multimodal operating rooms as a higher-order structure, innately preserving pairwise and group relationships. By lifting interactions between entities into higher-order topological cells, TopoOR natively models complex dynamics and multimodality present in the OR. This topological representation subsumes traditional scene graphs, thereby offering strictly greater expressivity. We also propose a higher-order attention mechanism that explicitly preserves manifold structure and modality-specific features throughout hierarchical relational attention. In this way, we circumvent combining 3D geometry, audio, and robot kinematics into a single joint latent representation, preserving the precise multimodal structure required for safety-critical reasoning, unlike existing methods. Extensive experiments demonstrate that our approach outperforms traditional graph and LLM-based baselines across sterility breach detection, robot phase prediction, and next-action anticipation

cs.CV↗
arXiv2026

From Data Statistics to Feature Geometry: How Correlations Shape Superposition

A central idea in mechanistic interpretability is that neural networks represent more features than they have dimensions, arranging them in superposition to form an over-complete basis. This framing has been influential, motivating dictionary learning approaches such as sparse autoencoders. However, superposition has mostly been studied in idealized settings where features are sparse and uncorrelated. In these settings, superposition is typically understood as introducing interference that must be minimized geometrically and filtered out by non-linearities such as ReLUs, yielding local structures like regular polytopes. We show that this account is incomplete for realistic data by introducing Bag-of-Words Superposition (BOWS), a controlled setting to encode binary bag-of-words representations of internet text in superposition. Using BOWS, we find that when features are correlated, interference can be constructive rather than just noise to be filtered out. This is achieved by arranging features according to their co-activation patterns, making interference between active features constructive, while still using ReLUs to avoid false positives. We show that this kind of arrangement is more prevalent in models trained with weight decay and naturally gives rise to semantic clusters and cyclical structures which have been observed in real language models yet were not explained by the standard picture of superposition. Code for this paper can be found at https://github.com/LucasPrietoAl/correlations-feature-geometry.

cs.LG↗