We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $Θ$ is represented by a reverse telescoping map $Θ(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is $x$ contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in $O(p^2)$ in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an $O(p^3)$ cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained $\mathbb{R}^{p\times p}$, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to $p=200$ in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

Being infinite dimensional, non-parametric information geometry has long faced an "intractability barrier" due to the fact that the Fisher-Rao metric is now a functional incurring difficulties in defining its inverse. This paper introduces a novel framework to resolve the intractability with an Orthogonal Decomposition of the Tangent Space ($T_fM = S \oplus S^{\perp}$), where $S$ represents an observable covariate subspace. Through the decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as ${\bf G}_f$, which is a finite-dimensional and computable representative of information extractable from the manifold's geometry. Significantly, by proving the Trace Theorem: $H_G(f) = \text{Tr}({\bf G}_f)$, we establish a rigorous foundation for the G-entropy previously introduced by us, thereby identifying it as a fundamental geometric invariant representing the total explainable statistical information captured by the probability distribution associated with a model. Furthermore, we establish a link between ${\bf G}_f$ and the second derivative (i.e. the curvature) of the KL-divergence, leading to the notion of Covariate Cramér-Rao Lower Bound(CRLB). We demonstrate that ${\bf G}_f$ is congruent to the Efficient Fisher Information Matrix, thereby providing fundamental limits of variance for semi-parametric estimators. Finally, we apply our geometric framework to the Manifold Hypothesis, lifting the latter from a heuristic assumption into a testable condition of rank-deficiency within the cFIM. By defining the Information Capture Ratio, we provide a rigorous method for estimating intrinsic dimensionality in high-dimensional data. In short, our work bridges the gap between abstract information geometry and the demand of explainable AI, by providing a tractable path for assessing the statistical coverage and the efficiency of non-parametric models.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of $x_T$ as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime $z=(x_t,t)$ that indexes the family of denoising distributions $p(x_0 | x_t)$ across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry

We observe that our approach obtains improved performance to comparable discrete diffusion and flow matching methods of Austin et al. [8], Stark et al. [68].

More specifically, we propose "Wasserstein Quantum Monte Carlo"

In this work, we study the problem of choosing the discretisation schedule for sampling from masked discrete diffusion models in terms of the information geometry of the induced probability path. Specifically, we show that the optimal schedule under the Fisher-Rao geometry recovers the popularly-used cosine schedule.

We propose causal effect estimators based on empirical Fr\'{e}chet means and operator-valued kernels, tailored to functional data spaces. These methods address the challenges of high-dimensionality, sequential ordering, and model complexity while preserving robustness to treatment misspecification. Using structural assumptions, we obtain compact representations of potential outcomes, enabling scalable estimation of causal effects over time and across covariates. We provide both theoretical, regarding the consistency of functional causal effects, as well as empirical comparison of a range of proposed causal effect estimators. Applications to binary treatment settings with functional outcomes illustrate the framework's utility in biomedical monitoring, where outcomes exhibit complex temporal dynamics. Our estimators accommodate scenarios with registered covariates and outcomes, aligning them to the Fr\'{e}chet means, as well as cases requiring higher-order representations to capture intricate covariate-outcome interactions. These advancements extend causal inference to dynamic and non-linear domains, offering new tools for understanding complex treatment effects in functional data settings.