Daily Papers

Batch Normalization (BN) has become an out-of-box technique to improve deep network training. However, its effectiveness is limited for micro-batch training, i.e., each GPU typically has only 1-2 images for training, which is inevitable for many computer vision tasks, e.g., object detection and semantic segmentation, constrained by memory consumption. To address this issue, we propose Weight Standardization (WS) and Batch-Channel Normalization (BCN) to bring two success factors of BN into micro-batch training: 1) the smoothing effects on the loss landscape and 2) the ability to avoid harmful elimination singularities along the training trajectory. WS standardizes the weights in convolutional layers to smooth the loss landscape by reducing the Lipschitz constants of the loss and the gradients; BCN combines batch and channel normalizations and leverages estimated statistics of the activations in convolutional layers to keep networks away from elimination singularities. We validate WS and BCN on comprehensive computer vision tasks, including image classification, object detection, instance segmentation, video recognition and semantic segmentation. All experimental results consistently show that WS and BCN improve micro-batch training significantly. Moreover, using WS and BCN with micro-batch training is even able to match or outperform the performances of BN with large-batch training.

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

The dynamics of Einstein-conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman-Robertson--Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (`Small Bang'), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (`Big Bang'). Thus, in the present EccH model there is a new kind of physical spacetime singularity (`Small Bang'). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout-Englert-Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.

In this paper, we consider a critical Grushin-type problem with double potentials. By applying the reduction argument and local Pohozaev identities, we construct a new family of solutions to this problem, which are concentrated at points lying on the top and the bottom circles of a cylinder.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

In the present paper we study the singularity structure of the exterior field of neutron stars with the aid of the four-parameter exact solution of the Einstein-Maxwell equations. The complete analysis of this problem in the generic case becomes possible due to the implementation of the novel analytical approach to the resolution of the singularity condition, and it shows the absence of the ring singularities off the symmetry axis in the positive mass case, as well as the possibility of the removal of the ring singularity by a strong magnetic field in the negative mass case. The solution takes an extraordinarily simple form in the equatorial plane, very similar to that of the Kerr solution, which makes it most suitable for astrophysical applications as the simplest model of a rotating magnetized deformed mass. It also provides a nontrivial example confirming a recent claim that the varphi component of the electromagnetic four-potential has features inconsistent with the intrinsic properties of the electrovac metric, while the magnetic field is represented correctly by the t component of the dual electromagnetic four-potential.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

In the era of proliferation of large language and image generation models, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the setting of high-dimensional regression and obtain analytic formulae which quantitatively outline this phenomenon in a broad range of regimes. In the special case of polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally projecting a convex set onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge-Ampere equation.

We introduce a classification conjecture for kappa-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of Z_2^2times O_3-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A,B,C,D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.

We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert Space followed by an imposition of - what effectively amounts to - a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach-Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlargers the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of R^n, under compactly supported variations. The critical point solves a fourth order nonlinear equation in double divergence form. We show that for smooth convex functionals, a W^{2,infty} critical point with bounded Hessian is smooth provided that its Hessian has a small bounded mean oscillation (BMO). We deduce that the interior singular set of a critical point has Hausdorff dimension at most n-p_0, for some p_0 in (2,3). We state some applications of our results to variational problems in Lagrangian geometry. Finally, we use the Hamiltonian stationary equation to demonstrate the importance of our assumption on the a priori regularity of the critical point.

In this note we establish a 1-to-1 correspondence between the class of generalized quadrangles with ovoids and the class of balanced incomplete block designs that posses a non-triangular local resolution system and have the appropriate parameters. We present a non-triangular local resolution system for a difference family BIBD construction of Sprott.

A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index r equipped with two different mathbb P^{r-1}-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the DK-conjecture holds for a class of simple K-equivalent maps arising from bundles of roofs. As an example, we propose a pair of eight-dimensional Calabi--Yau varieties fibered in dual Calabi--Yau threefolds, related by a GLSM phase transition, and we prove derived equivalence with the methods above.

We introduce the specialization map in Scholzes theory of diamonds. We consider v-sheaves that behave like formal schemes and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a Zariski site, and an etale site. When the kimberlite comes from a formal scheme, our sites recover the classical ones. We prove that unramified p-adic Beilinson--Drinfeld Grassmannians are kimberlites with finiteness and normality properties.

For any given dimension d, all reflexive d-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of (d+1)-tuples of integers (weights), or combinations of k-tuples of weights with k<d+1. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge numbers.

We construct a gauged linear sigma model with two non-birational K\"alher phases which we prove to be derived equivalent, L-equivalent, deformation equivalent and Hodge equivalent. This provides a new counterexample to the birational Torelli problem which admits a simple GLSM interpretation.

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

This paper studies the problem of recovering cameras from a set of fundamental matrices. A set of fundamental matrices is said to be compatible if a set of cameras exists for which they are the fundamental matrices. We focus on the complete graph, where fundamental matrices for each pair of cameras are given. Previous work has established necessary and sufficient conditions for compatibility as rank and eigenvalue conditions on the n-view fundamental matrix obtained by concatenating the individual fundamental matrices. In this work, we show that the eigenvalue condition is redundant. We provide explicit homogeneous polynomials that describe necessary and sufficient conditions for compatibility in terms of the fundamental matrices and their epipoles. In this direction, we find that quadruple-wise compatibility is enough to ensure global compatibility for any number of cameras. We demonstrate that for four cameras, compatibility is generically described by triple-wise conditions and one additional equation involving all fundamental matrices.

This note describes the development of an exact solver for Minimal Directed Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered largely by aggressively trying to reduce the DFVS problem to a Minimal Cover problem, and applying reduction rules adapted from Vertex Cover literature. The resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The resulting solver performed the second-best in the competition, although a bug at submission time disqualified it. As an additional note, we describe a new vertex cover reduction generalizing the Desk reduction rule.

We examine the geometry of neural network training using the Jacobian of trained network parameters with respect to their initial values. Our analysis reveals low-dimensional structure in the training process which is dependent on the input data but largely independent of the labels. We find that the singular value spectrum of the Jacobian matrix consists of three distinctive regions: a "chaotic" region of values orders of magnitude greater than one, a large "bulk" region of values extremely close to one, and a "stable" region of values less than one. Along each bulk direction, the left and right singular vectors are nearly identical, indicating that perturbations to the initialization are carried through training almost unchanged. These perturbations have virtually no effect on the network's output in-distribution, yet do have an effect far out-of-distribution. While the Jacobian applies only locally around a single initialization, we find substantial overlap in bulk subspaces for different random seeds. Our code is available at https://github.com/EleutherAI/training-jacobian

We construct S-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations X/S with relative Picard rank 1 and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of X/S.

Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis functions are used to approximate both the state and the position of the Lagrangian computational domain. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global basis. The proposed approach is successfully demonstrated for the reduction of several simple but representative problems.