Daily Papers

We study how a radially oscillating thin shell outside a black hole (BH) affects gravitational wave ringdown and derive matching conditions for polar and axial perturbations from the linearized Israel junction conditions on the moving worldtube. Periodic shell motion produces Floquet sidebands in the scattered wave. For a distant shell, the motion has little effect on the prompt ringdown and only weakly modifies the late time echoes. When the shell is closer to the BH, a replica of a long lived shell mode enters the incident frequency band, increasing the late time echo amplitude by roughly two orders of magnitude. We also identify an exceptional point where two polar modes coalesce and find the contribution of its second order pole in the late time waveform.

Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem may be high-dimensional and/or time-varying. Most problems in these applications are nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing Omega(d) function valuations per iteration (with d denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on 2m (for any 1 leq m leq d) function evaluations per iteration can not only find epsilon-second order stationary points polynomially fast, but do so using only Oleft(d{mepsilon^{2}psi}right) function evaluations, where psi geq Omegaleft(epsilonright) is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

Coordinate networks are widely used in computer vision due to their ability to represent signals as compressed, continuous entities. However, training these networks with first-order optimizers can be slow, hindering their use in real-time applications. Recent works have opted for shallow voxel-based representations to achieve faster training, but this sacrifices memory efficiency. This work proposes a solution that leverages second-order optimization methods to significantly reduce training times for coordinate networks while maintaining their compressibility. Experiments demonstrate the effectiveness of this approach on various signal modalities, such as audio, images, videos, shape reconstruction, and neural radiance fields.

The Redfield equation with time-dependent coefficients (or TCL2 equation) gives one of the simplest perturbative descriptions of open quantum system dynamics derived from a microscopic theory. Its status as an approximation is nevertheless subtle: while perturbation theory fixes the reduced dynamics order-by-order in the system-environment coupling, solving the Redfield equation yields a particular resummation that need not preserve complete positivity. Here we compare Redfield evolution with an exponentiated (Magnus) resummation of the same second-order perturbative map, which is completely positive and trace preserving (CPTP) for a broad class of physical conditions. We also construct a simple CPTP resolvent resummation, illustrating the non-uniqueness of resummations of the same perturbative map. The Redfield and Magnus prescriptions agree to second order but differ through time-ordering corrections at higher orders. We exhibit a weak-coupling criterion which guarantees that Redfield evolution, even when non-Markovian, remains close to a CPTP map. We illustrate the criterion in two examples, focusing on finite-dimensional (qubit) systems. Our results clarify when, and why, Redfield evolution can remain reliable even when it is highly non-Lindbladian.

Subspace representation is a fundamental technique in various fields of machine learning. Analyzing a geometrical relationship among multiple subspaces is essential for understanding subspace series' temporal and/or spatial dynamics. This paper proposes the second-order difference subspace, a higher-order extension of the first-order difference subspace between two subspaces that can analyze the geometrical difference between them. As a preliminary for that, we extend the definition of the first-order difference subspace to the more general setting that two subspaces with different dimensions have an intersection. We then define the second-order difference subspace by combining the concept of first-order difference subspace and principal component subspace (Karcher mean) between two subspaces, motivated by the second-order central difference method. We can understand that the first/second-order difference subspaces correspond to the velocity and acceleration of subspace dynamics from the viewpoint of a geodesic on a Grassmann manifold. We demonstrate the validity and naturalness of our second-order difference subspace by showing numerical results on two applications: temporal shape analysis of a 3D object and time series analysis of a biometric signal.

Studies show that the model-independent, fully non-perturbative covariant cosmographic approach is suitable for analyzing the local Universe (zlesssim 0.1). However, accurately characterizing large and inhomogeneous mass distributions requires the fourth-order term in the redshift expansion of the covariant luminosity distance d_L(z,n). We calculate the covariant snap parameter S and its spherical harmonic multipole moments using the matter expansion tensor and the evolution equations for lightray bundles. The fourth-order term adds 36 degrees of freedom, since the highest independent multipole of the snap is the 32-pole (dotriacontapole) (ell=5). Including this term helps to de-bias estimations of the covariant deceleration parameter. Given that observations suggest axially symmetric anisotropies in the Hubble diagram for z lesssim 0.1 and theory shows that only a subset of multipoles contributes to the signal, we demonstrate that only 12 degrees of freedom are needed for a model-independent description of the local universe. We use an analytical axisymmetric model of the local Universe, with data that matches the Zwicky Transient Facility survey, in order to provide a numerical example of the amplitude of the snap multipoles and to forecast precision.

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

We propose PRISM, an optimizer that enhances first-order spectral descent methods like Muon with partial second-order information. It constructs an efficient, low-rank quasi-second-order preconditioner via innovation-augmented polar decomposition. This mechanism enables PRISM to perform anisotropic spectral shaping, which adaptively suppresses updates in high-variance subspaces while preserving update strength in signal-dominated directions. Crucially, this is achieved with minimal computational overhead and zero additional memory compared to first-order baselines. PRISM demonstrates a practical strategy for integrating curvature-adaptive properties into the spectral optimization paradigm.

Second-order optimization has been developed to accelerate the training of deep neural networks and it is being applied to increasingly larger-scale models. In this study, towards training on further larger scales, we identify a specific parameterization for second-order optimization that promotes feature learning in a stable manner even if the network width increases significantly. Inspired by a maximal update parameterization, we consider a one-step update of the gradient and reveal the appropriate scales of hyperparameters including random initialization, learning rates, and damping terms. Our approach covers two major second-order optimization algorithms, K-FAC and Shampoo, and we demonstrate that our parameterization achieves higher generalization performance in feature learning. In particular, it enables us to transfer the hyperparameters across models with different widths.

This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law A_k(a) = log(1/a) + log(k-2) + B_0 + O(a), with B_0 = log q - 2log 2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B_0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a) sim (2a)^3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ^*(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ_1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.

Linear transformation of the inputs alters the training performance of feed-forward networks that are otherwise equivalent. However, most linear transforms are viewed as a pre-processing operation separate from the actual training. Starting from equivalent networks, it is shown that pre-processing inputs using linear transformation are equivalent to multiplying the negative gradient matrix with an autocorrelation matrix per training iteration. Second order method is proposed to find the autocorrelation matrix that maximizes learning in a given iteration. When the autocorrelation matrix is diagonal, the method optimizes input gains. This optimal input gain (OIG) approach is used to improve two first-order two-stage training algorithms, namely back-propagation (BP) and hidden weight optimization (HWO), which alternately update the input weights and solve linear equations for output weights. Results show that the proposed OIG approach greatly enhances the performance of the first-order algorithms, often allowing them to rival the popular Levenberg-Marquardt approach with far less computation. It is shown that HWO is equivalent to BP with Whitening transformation applied to the inputs. HWO effectively combines Whitening transformation with learning. Thus, OIG improved HWO could be a significant building block to more complex deep learning architectures.

The eight-component relativistic wave equation for spin-1{2} particles of Robson and Staudte (FV1{2}), a Feshbach--Villars linearization of the Feynman--Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the n=1--3 levels of hydrogenic ions from H to U^{91+} in the Schrödinger, Dirac and FV1{2} theories, evaluating all spectral sums, continuum included, by the Dalgarno--Lewis method. The FV1{2} and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin--field coupling the first-order roots become complex, and without the negative-norm states the (Zα)^2 coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the 2s_{1/2}--2p_{1/2} degeneracy needs its own spin--gradient term, and shifts placed by hand on the κ-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular (Zα)^{-2} second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the (Zα)^0 coefficients analytically and relates the π^2 terms of the (Zα)^2 coefficients to the singular ones. The Dirac weak/strong-field crossover for n=2 of hydrogen lies near 2.8~kV cm^{-1}, and the perturbative window set by the fine structure grows as Z^5.

We study a kinematical consequence of the Hopf-algebraic momentum composition law in κ-Minkowski spacetime. The same curved momentum space can be described in different coordinates. In the bicrossproduct basis the ordered-plane-wave labels are the translation-generator eigenvalues, so the relevant map is one-to-one. In the classical basis, instead, the translation eigenvalues P_μ are nonlinearly related to the ordered-plane-wave labels p_μ. This relation can fail to be globally one-to-one in a high-momentum region. When a given classical-basis four-momentum admits more than one real auxiliary preimage, the branch-sensitive quantity P_+equiv P_0+P_4=κe^{p_0/κ} enters the coproduct and resolves the branches in two-particle states. Imposing the vanishing total-momentum constraint therefore gives branch-dependent κ-deformed back-to-back momentum correlations. In a single-branch regime this is just a deformed correlated product, while in a multibranch regime a state specified only by P_μ can be expanded into distinct auxiliary branches. If P_μ are taken as the directly meaningful momenta, the physical content is the resulting deformed correlation pattern. If the auxiliary variables p_μ are assigned operational meaning, the same constrained state can be interpreted as a superposition over different auxiliary branches. We also compare this structure with standard regular self-adjoint nonrelativistic minimal-length models and find no analogous smooth local two-real-branch inversion on their physical domains.

By introducing a dynamical version of the second fundamental form, we generalize a recent result of Filip-Fisher-Lowe to the setting of magnetic systems. Namely, we show that a real-analytic negatively s-curved magnetic system on a closed real-analytic manifold has only finitely many closed totally s-magnetic hypersurfaces, unless the magnetic 2-form is trivial and the underlying metric is hyperbolic.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

Necessary conditions for high-order optimality in smooth nonlinear constrained optimization are explored and their inherent intricacy discussed. A two-phase minimization algorithm is proposed which can achieve approximate first-, second- and third-order criticality and its evaluation complexity is analyzed as a function of the choice (among existing methods) of an inner algorithm for solving subproblems in each of the two phases. The relation between high-order criticality and penalization techniques is finally considered, showing that standard algorithmic approaches will fail if approximate constrained high-order critical points are sought.

Layered viscoelastic interiors control the frequency dependence of planetary tidal dissipation, but repeated layered Love-number calculations are expensive in long-term orbital and spin integrations. Equivalent homogeneous rheologies provide a cheaper representation of the same response, but their effective parameters do not by themselves identify the interior layers they represent. We develop a local interpretation of such parameters by differentiating the layered-to-homogeneous construction about a fixed-geometry baseline, perturbing only selected layer rigidities and viscosities. We use the pole-residue representation of the degree-two Love number to construct a physical-to-effective transfer matrix. We then use its singular values, resolution matrix, and mode-dominance matrix to determine which physical perturbations are visible in the Love-number response and how they appear in effective-rheology coordinates. Applied to a five-layer lunar model, the full degree-two response is sensitive to seven of eight solid-layer rheological parameters. The single near-null direction is the inner-core rigidity, consistent with shielding by the fluid outer core and with the weak dependence of Love numbers on core elastic structure. Over the LLR frequency interval, fewer physical directions are visible. To first order, the elastic spring, isolated dashpot, and two Voigt elements reproduce the nine-pole response over this interval. Finite perturbations confirm the tangent prediction to within a few percent, with the validity radius limited by a nearly degenerate pair of long-timescale modes.

Second-order training methods have better convergence properties than gradient descent but are rarely used in practice for large-scale training due to their computational overhead. This can be viewed as a hardware limitation (imposed by digital computers). Here we show that natural gradient descent (NGD), a second-order method, can have a similar computational complexity per iteration to a first-order method, when employing appropriate hardware. We present a new hybrid digital-analog algorithm for training neural networks that is equivalent to NGD in a certain parameter regime but avoids prohibitively costly linear system solves. Our algorithm exploits the thermodynamic properties of an analog system at equilibrium, and hence requires an analog thermodynamic computer. The training occurs in a hybrid digital-analog loop, where the gradient and Fisher information matrix (or any other positive semi-definite curvature matrix) are calculated at given time intervals while the analog dynamics take place. We numerically demonstrate the superiority of this approach over state-of-the-art digital first- and second-order training methods on classification tasks and language model fine-tuning tasks.

Approximate second-order optimization methods often exhibit poorer generalization compared to first-order approaches. In this work, we look into this issue through the lens of the loss landscape and find that existing second-order methods tend to converge to sharper minima compared to SGD. In response, we propose Sassha, a novel second-order method designed to enhance generalization by explicitly reducing sharpness of the solution, while stabilizing the computation of approximate Hessians along the optimization trajectory. In fact, this sharpness minimization scheme is crafted also to accommodate lazy Hessian updates, so as to secure efficiency besides flatness. To validate its effectiveness, we conduct a wide range of standard deep learning experiments where Sassha demonstrates its outstanding generalization performance that is comparable to, and mostly better than, other methods. We provide a comprehensive set of analyses including convergence, robustness, stability, efficiency, and cost.

Zeroth-order optimization addresses problems where gradient information is inaccessible or impractical to compute. While most existing methods rely on first-order approximations, incorporating second-order (curvature) information can, in principle, significantly accelerate convergence. However, the high cost of function evaluations required to estimate Hessian matrices often limits practical applicability. We present the subspace-based approximate Hessian (ZO-SAH) method, a zeroth-order optimization algorithm that mitigates these costs by focusing on randomly selected two-dimensional subspaces. Within each subspace, ZO-SAH estimates the Hessian by fitting a quadratic polynomial to the objective function and extracting its second-order coefficients. To further reduce function-query costs, ZO-SAH employs a periodic subspace-switching strategy that reuses function evaluations across optimization steps. Experiments on eight benchmark datasets, including logistic regression and deep neural network training tasks, demonstrate that ZO-SAH achieves significantly faster convergence than existing zeroth-order methods.

We study the transverse-field axial next-nearest-neighbor Ising (ANNNI) model on the square lattice and on a bilayer. At the classical multiphase point κ=J_2/J_0=1/2, every configuration of straight domain walls that separates domains of at least two sites is a ground state. The walls are rigid, so perturbation theory in the transverse field g=Γ/J_0 can be carried out for each configuration separately, and we take it to order g^8. Quantum fluctuations select langle 3rangle (uparrowuparrowuparrowdownarrowdownarrowdownarrow) at order g^2, because spins next to a wall gain the most energy from virtual flips. The structures langle 23rangle, langle 223rangle and langle 2223rangle then open between langle 3rangle and the antiphase langle 2rangle at orders g^4, g^6 and g^8, in the order of the Fisher-Selke sequence of the classical three-dimensional model. A second layer suppresses each successive phase more strongly, beyond the stiffening of the local fields. We test the two widest phases with stochastic series expansion quantum Monte Carlo, comparing the free energies of the competing ordered states by thermodynamic integration, since annealing freezes in the modulation that forms at the ordering transition. The simulations reproduce the langle 3rangle wedge to about 1%, and in a pre-registered test langle 23rangle is stable at six of seven points, with the seventh resolved by post-hoc runs.

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables z_i, we derive a system of partial differential equations w.r.t.\ new variables x_j, which parameterize the differentiable constraints z_i=y_i(x_j). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.