Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.
Mechanism-AI coupled dynamical systems are increasingly used to study complex biological systems. However, uncertainty associated with noisy, sparse, and partially observed data limits their reliability. This paper presents a systematic mathematical formulation for treating these uncertainties from the perspective of optimal transport and information geometry. We formulate the uncertainty quantification problem as the minimization of an energy functional on a probability density space equipped with a Riemannian metric, with the loss evaluated through a differential equation, and give a unified description of the gradient flows of the energy functional under specific metrics, such as the Wasserstein and Fisher-Rao metrics. Furthermore, we present corresponding numerical methods for solving these gradient flows using abstract adjoint sensitivity analysis. The formulation connects a range of Bayesian inference methods for dynamical systems; numerical experiments compare posterior approximation and predictive uncertainty under a common computational budget. Moreover, it also provides a geometric perspective on dynamical modeling and the study of heterogeneity evolution.
We introduce Multi-scale Closing Index (MCI), a new dissimilarity measure for evaluating the alignment of two 2D point sets. MCI is bounded between 0 and 1 and is based on multiscale mathematical morphology operations, with smaller values indicating greater structural agreement. The measure was motivated by the need to assess the quality of registration between point sets representing cell populations in biological images. To evaluate its behaviour under controlled conditions, we generated synthetic point clouds from Perlin-noise-based spatial structures, following established approaches for simulating biologically plausible spatial organization. We assessed MCI between those point clouds and different transformed versions, obtained by local perturbation, point removal, translation, rotation and addition of impulsive noise. Across these experiments, the dissimilarity MCI increased gradually and consistently with perturbation severity, while showing a clear discontinuity between clouds originating from the same underlying spatial structure and point clouds generated from different structures. We further compared MCI with commonly used point-set distances, including Hausdorff-based measures and Sliced Wasserstein distance. These experiments show the ability of MCI to capture structural agreement and quantify deviations between point sets, highlighting its potential as a complementary metric for evaluating two-dimensional point-set registration.
We provide sharp estimates on the number of vertices needed to approximate the Euclidean unit ball $B_d\subset\mathbb R^d$ with relative symmetric-difference volume error at most $\varepsilon\in(0,1)$. We show that the minimum vertex budget $N_\varepsilon(d)$ satisfies $$\log N_\varepsilon(d)\asymp d\log\left(1+\frac1{\varepsilon\sqrt d}\right).$$ In particular, $\exp(O(\sqrt d/\varepsilon))$ vertices suffice for every $\varepsilon\in(0,1)$ and all sufficiently large $d$. We also prove that achieving a fixed error $\varepsilon\in(0,1)$ requires at least $\Omega\bigl((\tfrac{1+\varepsilon}{2\sqrt\varepsilon})^{d-1}\bigr)$ facets, an exponential number in $d$. We extend our vertex approximation result to Gaussian symmetric-difference error, and give a positive answer to the question of De, Nadimpalli, and Servedio about subexponential vertex complexity for approximating Euclidean balls. We also disprove a Macbeath-type extremality principle for unrestricted vertex approximation. Whereas the ball uniformly maximizes the inscribed approximation error among all convex bodies (Macbeath, 1951), we show that approximating the hypercube $[-1,1]^d$ to relative error $\varepsilon$, with $\delta:=1-\varepsilon$, requires $N\ge\frac{\delta}{2}\exp(\Omega(\delta^2d/\log^2(2/\delta)))$ vertices whenever $d\ge16\log(2/\delta)$. Thus, when $d$ is large, the cube is harder to approximate than the ball in this sense. Finally, we determine the sharp dimensional order of the classical Laguerre-Delone triangulation number, showing that $\mathrm{ldel}_{d-1}=\Theta(d^{-1/2})$ as $d\to\infty$. This improves both the upper bound of Ludwig, Sch\"utt, and Werner and the lower bound of B\"or\"oczky by a factor of order $\sqrt d$. By classical asymptotic approximation theorems (Ludwig, 1999), this improvement applies to every convex body in $\mathbb R^d$ with $C^2$ boundary and positive Gaussian curvature.
The manifold hypothesis suggests that data lies on manifolds with smaller intrinsic dimension (ID) than their ambient dimension. However there is no empirical agreement on the estimates for ID from different estimators for realistic datasets. Thus it is important to test ID estimators (IDEs) with targeted stressors. In this work, we consider the role of anisotropy. To this end, we propose HomID, a collection of homogeneous spaces with anisotropic embedding, for benchmarking ID estimators. We observe that methods that perform well on standard benchmarks systematically degrade on HomID under identical resource allocation. We further observe that anisotropic distortion of such benchmarks also results in performance degradation. Finally, we demonstrate that controlled anisotropic distortions induce systematic shifts in the distributions on which these methods rely, providing a concrete mechanism for the resulting estimation errors in two particular IDEs. The code for the project is available at https://github.com/a-writes-code/homid
In this work, we formalize the concept of intrinsic dimension of objective landscapes introduced in Li et al. (2018), which measures the structural complexity of learning problems by training along randomly generated subspaces. We define the intrinsic dimension of a set $\Theta\subset\mathbb{R}^p$ as the minimal dimension $d_{\mathrm{int}}(\Theta)$ such that a random plane of dimension $d_{\mathrm{int}}(\Theta)$ in $\mathbb{R}^p$ intersects $\Theta$ with probability one. We show that it is a generalized notion of co-dimension that depends on the global geometry of $\Theta$. We study mathematical properties of the intrinsic dimension and develop techniques to estimate the intrinsic dimension of sets $\Theta$ that arise in learning problems. More specifically, we derive upper and/or lower bounds where $\Theta$ is the set of optimal parameters in student-teacher settings with shallow and deep ReLU networks, including finite and infinite sample cases, and with and without regularization. We discuss how our theory leads to novel types of generalization bounds, thereby showing that the intrinsic dimension is a measure of learning complexity. Our results connect techniques from seemingly different areas of machine learning research such as classification of symmetries of ReLU networks and stochastic geometry.