This paper studies the identifiability and stability of drifting fields within the framework of Generative Modeling via Drifting. The motivating question is whether a zero-drift equilibrium identifies the target distribution, and whether an approximate zero drift implies weak distributional convergence. Since the original drifting model employs the Laplace kernel by default, we first analyze why standard Gaussian score-based arguments fail to apply. This analysis motivates the introduction of companion-elliptic kernel families, which are characterized by a companion potential satisfying an elliptic closure relation. We show that this class naturally contains the Laplace kernel and consists precisely of Gaussian and Matérn kernels with smoothness parameter $ν\ge 1/2$. Within this class, we establish field identifiability for arbitrary Borel probability measures on $\mathbb{R}^d$: if the drifting field vanishes identically, then the two measures must coincide. As for stability, we demonstrate that field convergence alone does not guarantee weak convergence, since mass may escape to infinity while remaining invisible to the field. Although tightness of the sequence directly removes this obstruction and restores weak stability, we prove that, even without tightness, every $C_0$-vague cluster point lies exactly on the defect ray $\{cp:0\le c\le1\}$. Consequently, a single scalar $C_0$-observable suffices to detect the missing mass and recover weak convergence.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point.For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

We examine the continuous-time counterpart of mirror descent, namely mirror flow, on classification problems which are linearly separable. Such problems are minimised'at infinity' and have many possible solutions; we study which solution is preferred by the algorithm depending on the mirror potential. For exponential tailed losses and under mild assumptions on the potential, we show that the iterates converge in direction towards a $\phi_\infty$-maximum margin classifier. The function $\phi_\infty$ is the horizon function of the mirror potential and characterises its shape'at infinity'. When the potential is separable, a simple formula allows to compute this function. We analyse several examples of potentials and provide numerical experiments highlighting our results.

When the potential is separable, a simple formula allows to compute this function.

In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers.

Neural Information Processing Systems http://nips.cc/

This kernel is central to describe the generalization features ofANNs.