Papers

Low-Rank Adaptation (LoRA) is an effective approach for adapting large pretrained models by learning low-rank weight updates. In practice, the LoRA rank is used to control an adapter's parameter budget and representational capacity. We show that this view is incomplete: while the nominal rank determines the representational capacity, the optimizer shapes how much of that capacity is used in the induced weight-space updates. In a case study of GPT-2 adaptation with LoRA, we observe a strong rank-dependent optimizer effect. Despite using the same nominal rank, AdamW often produces per-step updates with concentrated singular spectra and low effective rank, whereas Muon uses a richer set of directions and benefits more consistently from increasing LoRA rank. These observations motivate ISO-LoRA, an optimizer that couples the LoRA factor updates through spectral descent on the induced tangent perturbation in weight space. ISO-LoRA promotes updates that distribute energy more evenly across singular directions, improving rank utilization while preserving compatibility with the LoRA parameterization. We complement this design with theoretical guarantees showing that ISO-LoRA can achieve higher effective rank than standard factor-wise optimizers through a one-step analysis under a stylized spiked-gradient model. We validate this design on language-model adaptation across 0.1B-7B-parameter models, where ISO-LoRA improves effective rank and downstream performance, with the strongest gains at moderate-to-large LoRA ranks. Our results highlight rank utilization as a key factor in LoRA optimization and suggest that optimizer design offers an important path toward stronger parameter-efficient adaptation.

We provide first examples of border rank lower bounds obtained using border apolarity beyond “weak border apolarity”. Border apolarity introduced by Buczyńska-Buczyński characterizes tensors of border rank greater than $r$ as those whose annihilator does not contain an ideal with certain properties depending on $r$ and the ambient space of the tensor. Previous applications of border apolarity depended on the algorithm developed by Conner-Harper-Landsberg. For a tensor with a large symmetry group it lists a set of ideals that satisfy some of the properties mentioned above. We give first examples where the list is non-empty but we are able to prove that no ideal on this list satisfies all necessary conditions coming from border apolarity. The new lower bound examples include tensors of interest in complexity: the first Schönhage triple sum of matrix multiplication tensors where the border rank was unknown, and the direct sum of two copies of a tensor that could potentially be used to prove the exponent of matrix multiplication is two. A third example of interest in group theory related to work of A. Leitner is also analyzed.

Stochastic gradient descent (SGD) with gradient clipping and additive noise has become a standard technique for training machine learning models, particularly in applications requiring robustness or privacy guarantees. However, clipping introduces a bias in stochastic gradients, while additive noise introduces additional variance, making the long-run behaviour of individual optimization trajectories difficult to characterize. In this work, we prove that SGD with clipping and additive Gaussian noise (SGD-CN) converges almost surely (a.s.) under smoothness and uniformly bounded stochastic-gradient noise assumptions, provided the step sizes satisfy some standard decaying conditions. Our analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, where we show that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods and suggest that, despite the bias and noise introduced by clipping and perturbation, the algorithm remains stable in both convex and nonconvex regimes.

We prove Hitchin's conjecture for $\mathfrak{sl}_n$ in every odd prime degree. More precisely, let $p=2k+1$ be prime and let $n\geq k+1$. Every nonzero primitive element $\alpha\in\Lambda^p(\mathfrak{sl}_n^*)^{\mathfrak{sl}_n}$ is nonzero on the top exterior power of the unique $p$-dimensional irreducible submodule of $\mathfrak{sl}_n$ for the adjoint action of a principal $\mathfrak{sl}_2$-subalgebra. The conjecture also holds for types $B_\ell$ and $C_\ell$, with $\ell\geq2$, in every prime degree $p=4r-1$ with $1\leq r\leq\ell$.

We develop a martingale-transform framework for Dunkl harmonic analysis and apply it to prove $L^p$ estimates for the Dunkl Riesz transforms using the martingale decomposition of the Dunkl process. A fundamental difference from the classical Brownian setting is that the martingale representing the Dunkl Riesz transform of a function is not, in general, differentially subordinated to the Poisson martingale of the function itself. Our main argument bypasses this obstruction by applying Burkholder's Bellman function directly. The possible positive defect of the continuous Itô drift is compensated by the negative contribution produced by the reflection jumps. This yields $L^p$ estimates for single Dunkl Riesz transforms for $1<p<\infty$, and vector-valued estimates for $p\geq2$ with a spectral dependence on the root system. For $G$-invariant functions, differential subordination can be recovered and the resulting estimates are independent of the root system.

We study limited memory steepest descent (LMSD) with exact algebraic rank compression for strictly convex quadratic optimization. At each cycle, the method restricts the gradient history to its column space and applies the reciprocals of all Ritz values of the compressed projection in the next sweep. If the block-start gradient has at most $p$ active distinct eigenvalues, the delayed sweep terminates finitely. For nonterminating trajectories, a determinant and Cauchy–Binet formula for the complete-sweep polynomial yields global convergence without a full-rank history or a run-wise normalized-conditioning bound. Consecutive sweep endpoints define a continuous positively homogeneous map on a closed cone of compatible states. Compactness then gives an $R$-linear endpoint estimate, uniform over compatible initial states, with constants depending only on $H$ and $p$; the estimate extends to inner gradients, iterate errors, and objective gaps. Every fixed positive spectral weight $W=\omega(H)$ is linearly conjugate to the standard method. The weighted methods therefore share its decay factor, while the Euclidean-norm prefactor is increased by at most $\sqrt{\kappa_2(W)}$. This includes harmonic-Ritz LMSD, fixed power weights, and the delayed BB1 and BB2 recurrences.

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.

We describe in explicit detail the rich relationship between division algebras, split composition algebras, triality, Clifford algebras, spinors, triality Lie algebras, generalized reflections, exceptional magic square Lie algebras, triality eigenspaces, 3-symmetric spaces, Vinberg theta algebras, and Exceptional Unification in particle physics.

We prove finiteness and Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach–Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro-étale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues–Fontaine curve, which in turn reduce to Poincaré duality for perfect complexes over period rings. We give a simple proof of the latter duality via the same diagrammatic argument as in our previous work for finite coefficients. Along the way, we establish optimal $v$-descent for perfect complexes over certain period rings on affinoid perfectoid spaces.

We study limit theorems for random dynamical systems. We recast the random dynamics via the skew-product of the two-point motion (the Varadhan trick). Using this approach we prove several statistical properties for the system, including quenched central limit theorems with rates, quenched local limit theorem, large deviations, and decay of correlations. Then, in applications we show how to verify the general conditions using spectral properties of appropriate operators. A key application is the study of effectively expanding on average random diffeomorphisms (in any dimension, which are allowed to be dissipative). In particular, this is the first instance of proving local limit theorems in such a broad setting, without imposing restrictions on the dynamics. We provide several new examples.

We prove that the critical value $2\pi^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere $\mathbb{S}^{3}$. Specifically, there exists $\varepsilon_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2\pi^2+\varepsilon_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.

We give an $R$-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global $R$-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan's theorem yields pointwise stability on the entire cone. Finally, the transformation $g\mapsto W^{1/2}g$ conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.

Detecting whether a transmitter is actively communicating with a receiver is a fundamental problem in molecular communications. A spherical receiver may measure not only the number and arrival times of absorbed molecules, but also their absorption locations on the receiver surface. These locations contain a directional signature that is lost in count-only detection. In this letter, we develop a molecular signal detector based on a Legendre polynomial expansion of the receiver-surface hit density and extend it to a Legendre-based Viterbi sequence detector. By exploiting the surface-level directional signature of molecular arrivals, the proposed detectors provide geometry-aware, efficient, and better-performing alternatives to count-only detection.

We investigate the operator system structure of finite Hankel matrices and its connection with complex polynomials on the real line. Using polynomial matrix spectral factorization, we establish a unital complete order isomorphism between the operator system of (n + 1) x (n + 1) Hankel matrices and the dual of the operator system of polynomials of degree at most 2n. We also show that the operator system of complex polynomials is minimal, while the Hankel operator system is maximal. As an application, we characterize positivity of block Hankel matrices with entries in a C*-algebra in terms of completely positive maps on the polynomial operator system.

The Barzilai–Borwein (BB) method is $R$-linearly convergent on strongly convex quadratics, although neither the objective value nor the gradient norm is generally monotone. We construct an explicit current-point Lyapunov function for the quadratic dynamics after the warm-up step, assuming that both endpoints of the initial active spectrum survive this step. If $a<b$ are these endpoints and $P_a,P_b$ are the corresponding spectral projectors, then \[ \overline{f}(x) :=\lVert P_a\nabla f(x)\rVert^{\frac{2b}{a+b}} \lVert P_b\nabla f(x)\rVert^{\frac{2a}{a+b}} \] satisfies, for BB1, BB2, and every fixed positive weighted delayed Rayleigh rule, \[ \overline{f}(x_{k+1}) =\left(\frac{b-a}{b+a}\right)^2 e^{-2D_{a,b}(\tau_k)}\overline{f}(x_k), \qquad D_{a,b}(\tau_k)\ge0, \] where $\tau_k=1/\alpha_k$ is the reciprocal step. The function is obtained by denormalizing the endpoint coboundary certificate in the sharp-rate analysis of Yang and Yuan. We prove that the endpoint exponents are the unique minimax choice among endpoint monomials and illustrate the Lyapunov law on a nonmonotone BB1 trajectory. Using this law, we also give a proof of $R$-linear convergence of the method in finite dimensions.

This work considers multi-agent coordination with arbitrary information networks among the agents using a game-theoretic approach. A system designer aims to assign local utility functions to the agents to guide their actions toward a desired system objective. The performance of the assigned local utilities is measured by the well known pure price of anarchy (pPoA) metric that equals the ratio of the system objective at the worst pure Nash equilibrium of the corresponding game to the optimal system objective. Our aim is to derive the utility functions which optimize the pPoA-based performance guarantees for any given information network and system objective. We develop a linear program that derives the optimal pPoA for any arbitrary information network and arbitrary system objective. Our work is the first to solve optimal utility design for arbitrary networks; our techniques generalize previous approaches which considered only the full-information setting. For supermodular objective functions, we prove that counterintuitively, a fully communication-denied utility design is optimal irrespective of the original information network. For submodular system objectives, an exhaustive numerical analysis suggests that the optimal utility design is robust to communication failures even for this case. When the system objective is weighted maximum coverage, the marginal contribution utility design provably optimizes the pPoA for a wide variety of information networks of interest.

Let $Y$ be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in $Y^{\mathrm{an}}$ associated with formal loops $\mathrm{Spec}\, \mathbb{C}(\!(t)\!)\to Y$. When $Y$ is very affine, every formal loop determines an integral point of $\mathrm{Trop}(Y)$. For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when $Y$ is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type $A$, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.

We consider channels with additive colored Gaussian noise and noiseless feedback. Kim's seminal work derived a stationary variational characterization of feedback capacity and further asserted that the capacity-achieving stationary input need not contain a feedback-independent Gaussian component. These results led to the construction of a simple coding scheme, based on the Schalkwijk–Kailath (SK) refinement principle, which was shown to be capacity-achieving. A recent note identified a gap in the proof of the feedback-independent component-removal assertion, thereby leaving the optimality of the SK scheme and subsequent results that rely on it incomplete. In this paper, we prove the component-removal assertion for channels with stationary Gaussian noise that has a rational power spectral density. Our proof uses a perturbation analysis of a convex optimization formulation of feedback capacity and, indeed, shows that every optimizer assigns zero power to the feedback-independent component. Using this stronger property, we construct from any optimizer an explicit SK coding scheme that achieves every rate below feedback capacity with doubly-exponentially decaying maximal error probability.

We introduce a general machinery to study thermodynamic formalism out of equilibrium: The thermodynamics of a topological Markov shift (denoted by $\Sigma^-$) where the potential is given by a random walk on a compact metric space $X$ (and the randomness is driven by a Gibbs process). We introduce the semi-Ruelle operator, which acts on $C(\Sigma^-\times X)$. We construct conformal measures and harmonic functions for the semi-Ruelle operator. We present a few applications: (1) We provide a new proof to the POE variational principle (POE stands for the pressure out of equilibrium which is associated with the process), and we show that maximizing measures in the POE variational principle admit positive entropy out of equilibrium, and satisfy semi-Gibbs estimates. (2) In the setting where the random walk on the fiber $X$ is given by $C^{1+}$ diffeomorphisms (which are allowed to be very dissipative), and it satisfies the open condition of effective expansion on average, we show that the averaged semi-Ruelle operator is quasi-compact when acting on a Sobolev function space. An application includes proving a spectral gap when assuming volume decay of correlations, and proving bounds on the dimension of stationary measure in terms of similarity dimension.

Channel simulation has recently emerged as a useful component in machine learning systems where samples from a prescribed probability distribution are to be compressed. Yet, general channel simulation algorithms often suffer from high computational costs, random stopping times or, in the worst case, can require generating an infinite number of shared random samples. We introduce a scheme for both exact and approximate simulation of discrete-to-continuous channels which conversely uses a fixed number of random samples, and therefore has a runtime independent of the channel and the input. Unlike existing channel simulation schemes which generate a sequence of independent samples from a proposal distribution, our approach generates one sample, or alternatively a fixed number of samples, from each potential target distribution. We then apply a latent permutation to the samples before performing sample selection using an exponential race. Our scheme provides a flexible tradeoff between the number of generated samples and the compression rate. Using polar and multilevel coding, we scale our approach to handle long blocklengths in $O(n \log n)$ time in order to benefit from reduced per-symbol overhead. We conclude by demonstrating applications to variable-rate compression with stochastic VQ-VAEs and communication-efficient differentially private distributed mean estimation via exact simulation of the Gaussian mechanism.

We study the mixing time of a simple random walk on the small-world network defined by adding edges to $\mathbb{Z}_n^d$ as follows: for each pair $\{x,y\}$ we add an edge with probability $Z_n/\|x-y\|^{d}$ with $Z_n$ chosen so that the average number of added edges to every vertex is $1$. When $d\geq 3$, we show that with high probability the mixing time is of order $\log n$ and that the random walk does not exhibit cutoff.

We study an entanglement transition at logarithmic depth in random hypercube linear optical networks. Starting from an initial Gaussian state with all modes squeezed, we prove that random hypercube linear optical networks on $n$ modes generate ensemble-averaged subsystem entanglement within a constant factor of its maximum in all subsystems at circuit depth $\Theta(\log n)$. Below depth $\log_2n$, there is an extensive subsystem with no entanglement. The entanglement generation in logarithmic depth can be compared to recent progress towards $O(\log n)$ depth average-case sampling hardness for Gaussian boson sampling in a hypercube network.

Moment Lagrangian $L_\mu$ is a Lagrangian in $T^*G^- \times Y^- \times Y$ associated to a Hamiltonian $G$-space $Y$ with a moment map $\mu$. In this paper, we prove that $L_\mu$ is tautologically unobstructed under mild assumptions on $Y$. As a key ingredient in the proof, we constructed a new symplectic groupoid structure on $T^*G^- \times Y^- \times Y$ over $G\times Y$ for which $L_\mu$ is simultaneously the unit and the fixed locus of the inversion, which might be of independent interest.

Let $\Omega\subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and let $\psi\in SH_m(\Omega)$ be a fixed negative $m$-subharmonic function. In this paper we introduce a relative finite energy class $\mathcal E_{m,\psi}(\Omega),$ which may be viewed as a Hessian analogue of the relative energy classes appearing in the pluripotential theory of complex Monge–Ampère equations. We develop a systematic pluripotential theory in this setting. More precisely, we introduce a relative Hessian capacity associated with the prescribed singularity type $\psi$, construct relative mixed Hessian products, and establish their fundamental properties. We prove a monotone convergence theorem and a Bedford–Taylor type continuity theorem for Hessian measures in the class $\mathcal E_{m,\psi}(\Omega)$. A central result of the paper is a relative comparison principle, which yields uniqueness of solutions to complex Hessian equations with prescribed singularities. As an application, we establish an existence and uniqueness theorem for the equation $(dd^c u)^m\wedge\beta^{n-m} =\mu $ for a large class of positive Radon measures that do not charge $m$-polar sets. The results obtained here provide a relative finite energy framework for complex Hessian equations and extend several fundamental aspects of Cegrell's theory to the setting of prescribed singularity types.

We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.

We consider the spectrum of the Orr-Sommerfeld operator, which is obtained from the linearized Navier-Stokes (LNS) operator near a laminar flow in an infinite two-dimensional channel in the large Reynolds number limit. For a rather general class of laminar flows, which behave locally near the boundary like a Couette flow and are bounded from below (above) by a linear increasing (decreasing) function, we show that there exists an eigenvalue of the LNS operator which coincides, to leading order, with the eigenvalue of LNS near Couette flow, which was formally obtained by Wasow in 1953.

Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of $P_p(R^d)$ are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of $L^p$ -valued martingales.

We establish an a priori bound for the dynamical parabolic $\Phi_3^4$ model with harmonic potential. This bound yields the global well-posedness of the equation and, via the Krylov-Bogoliubov method, the existence of an invariant measure, shown to be non-Gaussian. The argument builds on the strategy developed by Mourrat and Weber for the periodic $\Phi_3^4$ model, with substantial modifications to handle the non-compact geometry of $\mathbb{R}^3$ and the spectral framework imposed by the harmonic oscillator. We further prove that this measure is unique in the small-coupling regime.

We propose to learn the unknown dynamics in the proximity of a small body directly from tracking data, representing them as a feed-forward neural network embedded in the system Hamiltonian. The equations of motion form a Neural Hamiltonian Ordinary Differential Equation, whose variational equations provide exact training gradients: estimation uses position and velocity arcs at realistic noise levels, without acceleration or potential labels, and a continual learning approach warm-starts the network as new data are acquired. The known part of the Hamiltonian carries whatever is available, from the central term and spin state to the constant-density model of the imaged shape. We assess the method against a normalized spherical harmonics expansion estimated from identical arcs through the same machinery, on scenarios built on the shapes of Itokawa, 67P, Bennu and Eros. The network remains usable inside the Brillouin sphere: it plans ballistic descents at Itokawa to \SI{4.6}{m} median touchdown error from tracking alone, against 5.1–\SI{48.9}{m} for harmonics of degree 4–12, and to \SI{0.9}{m} with the imaged shape as prior, a configuration that also recovers localised density anomalies invisible to any harmonics degree. The two representations are complementary, and we discuss their combined use across the phases of a small-body mission.

This paper investigates Yamabe-type soliton structures on the Sultana-Dyer (SD) black-hole spacetime. By employing the explicit metric and curvature quantities associated with the SD spacetime, we demonstrate that the non-Killing vector field $\partial_t$ induces an $h$-almost Yamabe soliton for explicitly determined functions $h$ and $\lambda$. We also establish the condition under which this structure reduces to an almost Yamabe soliton and examine why the coordinate vector fields $\partial_r$ and $\partial_\theta$ do not satisfy the corresponding Yamabe flow conditions. Furthermore, we show that the energy-momentum tensor, consisting of a timelike dust component and a null radiation fluid, is compatible with the Riemann, conformal (Weyl), concircular, conharmonic, and projective curvature structures. This compatibility reveals a strong geometric correspondence between the matter distribution and the causal structure of the spacetime. However, when the time-dependent scalar-curvature profile is subject to the constraints associated with late-stage cosmic expansion, the null, weak, and dominant energy conditions are violated. These findings provide a geometric characterization of the time-dependent Sultana-Dyer spacetime within the framework of generalized Yamabe solitons.