Papers

By means of $W$-representation, we investigate the Wishart-Laguerre $\beta$-ensemble and present a compact expression for Catalan polynomials. For a special Nekrasov partition function associated with the Wishart-Laguerre $\beta$-ensemble, we show that it can be expressed as a exponential form of the special Catalan polynomials. Furthermore, we make a rule to calculate the Catalan polynomials by means of Feynman diagrams and present another expression for Catalan polynomials. We also establish the connection between certain Catalan polynomials and knot theory.

Let $L$ be a linear second-order elliptic operator satisfying a nondegeneracy condition. We establish sharp $L^q$ estimates for solutions of the adjoint problem associated with $L$ and, by duality, derive sharp $L^p$ estimates for solutions of $Lu = f$. In particular, this confirms a conjecture of Carlo Pucci.

We investigate surjective isometries between the unit spheres of facially symmetric spaces and their interaction with the underlying facial structure. It is shown that such isometries preserve the essential geometric relations associated with symmetric faces, orthogonality, and geometric tripotents. In particular, a surjective sphere isometry induces a natural correspondence between the geometric tripotents of the underlying spaces and preserves the principal structural features determined by the Peirce decomposition. These results provide a geometric framework for studying linear extension problems for sphere isometries. As an application, we establish a positive solution of Tingley's problem for finite-dimensional real neutral strongly facially symmetric spaces satisfying the joint Peirce property. We also obtain an extension result for a wider class of real neutral strongly facially symmetric spaces admitting an appropriate approximation by finite-dimensional Peirce subspaces. In these settings, every surjective isometry between unit spheres extends uniquely to a surjective real-linear isometry of the ambient spaces. The results demonstrate that the facial and tripotent structures play a decisive role in the rigidity of spherical isometries.

In this paper we introduce the model of Percolative MDLA (PMDLA) on $\mathbb{Z}^2$ which engulfs the entire occupied percolation component of each particle that reaches the aggregate. We prove a phase transition of finite time explosion, as a function of the particle density $\lambda$, where the critical value equals $\lambda_c=-\log(1-p_c^{\rm site}(\mathbb{Z}^2))$. Below $\lambda_c$, the process is nonexplosive and has a linear upper bound on its radius. For $2/3<\lambda<\lambda_c$, it also has a linear lower bound on the growth rate. Above $\lambda_c$, an infinite component is swallowed in finite time a.s. At $\lambda_c$, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.

By uncertainty principles we mean the generalized Caffarelli–Kohn–Nirenberg (CKN) inequality and the Heisenberg–Pauli–Weyl (HPW) uncertainty principle, with the latter being an endpoint case and the Hardy inequality a further limiting case. We study these \(L^p\)-type inequalities \((1<p<n)\) on \(n\)-dimensional non-trivial flat Finsler manifolds, which arise from Hilbert's fourth problem. These manifolds are forward complete but not backward complete, projectively flat, with zero flag curvature, and have infinite reversibility, making them highly asymmetric. They are the simplest non-Riemannian, non-Minkowskian solutions to Hilbert's fourth problem, providing a natural framework in which classical inequalities exhibit new behavior. For the full class, we establish a sharp threshold at the parameter value \(s=2\): the generalized CKN inequality and the HPW uncertainty principle hold for \(s>2\) but fail completely for \(s\le2\). When the generating Minkowski norm is reversible (while the Finsler metric itself remains irreversible with infinite reversibility), we determine the sharp constants for \(s>2\), which were previously unknown. These constants are approached in the limit by compactly supported radial functions. As a corollary, our results cover Berwald's metric space and resolve the open questions on its sharp constants. These findings reveal that the threshold phenomenon is structural for the entire class and is a genuinely non-Riemannian feature with no analogue in the Riemannian flat case.

We determine the $RO(C_2)$-graded $k\mathbb R$-Hurewicz image, where $k \mathbb R$ denotes the connective Atiyah's Real $K$-theory spectrum. As a consequence, we obtain four infinite families in the $RO(C_2)$-graded equivariant stable stems that are $C_2$-equivariant refinements of the classical $\mu$-families. Using a $C_2$-equivariant $v_{1,0}$-self map, we explicitly construct an equivariant Adams periodicity operator $P_{C_2}$ that captures these $v_1$-periodicity phenomena in the $C_2$-stable stems.

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$. O. Klurman, A. P. Mangerel and J. Teräväinen (2023) have shown that three primes suffice when $q$ is sufficiently smooth. We study the moduli $q=p^\gamma$ that are powers of a fixed prime $p$ and we show that the third prime can be taken subpolynomially small. For any $q_\bullet\le q$ for which $q_\bullet\ge\exp\bigl(C(\log q)^{2/3}(\log\log q)^{4/3}\bigr)$, we give an asymptotic formula for the weighted number of representations of each reduced class as a product of two primes not exceeding $q$ and a third not exceeding~$q_\bullet$. We also show that the conjecture holds for all but a proportion $\exp\bigl(-c(\log q)^{1/3}(\log\log q)^{-1/3}\bigr)$ of the reduced classes. Here $C$ and $c$ are positive constants that depend only on $p$. Both results are unconditional and they rely on a wide zero-free region with no exceptional zeros for the Dirichlet $L$-functions attached to characters modulo $p^\gamma$.We also present conditional versions of these results for arbitrary $q$ under a quasi-Riemann hypothesis.

Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{*}$-algebras generated by free semicircular systems: the isomorphism class of a $C^{*}$-algebra generated by a semicircular $n$-tuple does not depend on $n\in\{2,3,\dots,+\infty\}$. Similarly, reduced free products of $n$ copies of the Jiang–Su algebra are isomorphic for different $n$ in this range. While reduced free group $C^{*}$-algebras of different ranks are not isomorphic (by a result of Pimsner–Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of $C^{*}$- and $W^{*}$-algebras; a relative version of that property was previously used to derive free complementation results.

We construct a seven-cycle example for a periodically forced scalar equation with a continuous three-zone piecewise-linear nonlinearity. Symbolic integration and linear elimination reduce the four-flight matching conditions to two equations in the residence times. The competing exterior visits produce an additional turn of the critical response and lead to the rational parameters $(A,B,\mu)=(9,3/2,181363/100000)$. At these parameters, finite Taylor estimates give three sign changes of the return displacement. Scalar order, analyticity along transverse itineraries, and half-period reflection then establish at least seven distinct isolated periodic orbits on the phase cylinder.

In this work, we propose a curve to be a minimum of length among rectifiable, continuous curves of unit width in three-dimensional Euclidean space. We show that a previously conjectured minimizer of that problem is not of unit width, give the numerical derivation of a candidate family of unit width curves, prove that it admits a strict local minimum within that candidate family and that this minimum is of unit width and length $3.905124162243593\ldots$. The curve consists of eight straight segments, two cone geodesics, and three circular helices. Its parameters are specified by a uniquely isolated solution of the joining and stationarity equations. To establish unit width in every direction, we combine analytic estimates with interval arithmetic on a finite covering of the remaining directions. We conjecture that the curve is globally minimizing. We supplement this work with the programs used to identify the candidates, replicate the computer-assisted parts of our proof, as well as a formalization of our proof in Lean.

We study a PDE-constrained variational problem for the optimal spatial distribution of a catalytic material reacting with a substrate in a biological or materials system. With a given catalyst concentration, the substrate concentration is determined by a reaction-diffusion equation. The reaction functional for the catalyst concentrations is defined to be the total reaction output, which is the product of catalyst and substrate concentrations. Both the linear and Michaelis-Menten reaction kinetics are considered. We construct reaction-maximizing sequences through the localization of catalysts using barrier functions and show the non-existence of a global maximizer. We also derive the first and second variations of the reaction functional. By constructing a competing catalyst concentration, we show the non-existence of a local maximizer. Based on our analysis, we propose a regularized variational model and prove the existence of its global maximizer.

A Zumkeller number is a positive integer whose set of positive divisors can be partitioned into two disjoint subsets of equal sums. In this paper, we investigate the additive properties of these numbers in conjunction with prime numbers. Specifically, we study the natural density of the exceptional set \(\mathcal{E}\) consisting of all positive integers not expressible as the sum of a prime and a Zumkeller number. Let $\alpha(x) = \#\{n \le x : n \in \mathcal{E}\}$ denote its counting function. We establish a quantitative upper bound for $\alpha(x)$, demonstrating that $\alpha(x) \ll x(\log \log x)^{-A}$, where $A = \log_2\left(\frac{2\pi^2}{2\pi^2-3}\right) \approx 0.2378$, thereby proving that the natural density of this exceptional set is $0$.

Differential Privacy (DP) provides a rigorous framework for quantifying privacy guarantees. While most methods achieve DP by injecting calibrated random noise, the intrinsic randomness of certain procedures such as sampling can yield DP guarantees “for free”. In this work, we develop a unified Rényi divergence framework to characterize the inherent DP guarantees of sampling from a Bayesian posterior distribution. We show that the privacy loss incurred by One-Posterior-Sampling (OPS) is governed by posterior exponential moments of the single-record log-likelihood ratio, and establish explicit DP guarantees under uniformly bounded, sub-Gaussian, or sub-exponential tail regimes. We apply the framework to representative Bayesian models, including categorical likelihoods with arbitrary priors, Gaussian-Gaussian conjugate pair, generalized linear models from the exponential family and linear regression with Gaussian likelihood, both with Gaussian priors on the regression coefficients. Our results elucidate sample size, model structure, neighboring relations in DP (bounded or unbounded), and how prior if applicable, jointly determine the inherent DP guarantees of OPS. Furthermore, our results recover the existing DP guarantees for OPS as special cases – often with tighter privacy loss bounds – and substantially broaden the class of Bayesian models for which the inherent DP guarantees of OPS can be rigorously characterized.

Poset games form a fundamental class of combinatorial games that includes classical games such as Nim and Chomp. Determining the outcome of a poset game is known to be PSPACE-complete even for posets of height at most three. However, the exact computational complexity of Two-Level Poset Game, played on posets of height at most two, has remained an open problem for over a decade. We resolve this problem by proving that Two-Level Poset Game is PSPACE-complete via a polynomial-time reduction from Avoid True on positive CNF formulas.

We study a continuous interpolation between the type $B$ and shifted type $A$ Narayana triangles and its extension through affine moments of the continuous $q$-ultraspherical (Rogers) measure. For every fixed $0<q<1$ and $-1<\beta<1$, these affine moments form a coefficientwise Hankel-totally positive sequence in the polynomial variable $t$. The proof rests on a general continuant criterion: decreasing nonnegative off-diagonal weights and coefficientwise positivity of the leading continuants imply total positivity of the full tridiagonal matrix. A telescoping identity gives positive certificates for shifted continuants and stability under coordinatewise decreases of the weights. We determine the exact power of $t$, palindromicity, and extreme coefficients of arbitrary Hankel minors. Classical limits recover the continuous Narayana family, including its moment representation and gamma positivity. We further prove that the continuous Narayana transformations preserve strong $t$-log-convexity for every real $\theta\ge0$, and that the $d$-Hoggatt transformations have the same preservation property for every integer $d\ge1$. The preservation proofs use the Zhu–Sun criterion and a mixed-minor one-sign-change argument whose local ingredient is a hypergeometric hitting-probability inequality.

This survey reviews results on the Hausdorff dimension of shrinking target sets and recurrence sets in dynamical systems. In their basic formulations, shrinking target sets concern orbits approaching a prescribed centre, whereas recurrence sets concern orbits returning close to their initial points. A central theme is the comparison of the Hausdorff dimensions of these two classes of limsup sets. The dynamical systems considered include expanding conformal systems, irrational rotations, affine iterated function systems, and matrix transformations. Related results on Hausdorff measure are also included when they provide natural refinements of the dimension theory. Finally, we review results on badly approximating orbits and non-recurrent sets, with particular attention to winning properties and intersections with self-conformal and self-affine sets.

Given $0<r_{j}<1$ with $r_{j}\to1$, there exist a normal operator $T$ on $\ell^{2}$ with eigenvalues $\lambda_{j}$ satisfying $|\lambda_{j}|=r_{j}$ and $g\in\ell^{2}$ such that \[ \left\{ \frac{T^{n}g}{\left\Vert T^{n}g\right\Vert }\right\} _{n\geq0}\text{is a frame}\:\Longleftrightarrow\:\sum_{j\geq1}\left(1-r^{2}_{j}\right)=\infty. \] Whenever this condition holds, the frame bounds can be chosen with ratio arbitrarily close to one.

Consider the one-dimensional nonlinear stochastic heat equation $$ \frac{\partial u(t,x)}{\partial t} = \frac{\partial^2 u(t,x)}{\partial x^2} + \sigma(u(t,x))\dot W(t,x), \qquad t>0,\ x\in\mathbb R, $$ where $\dot W$ is a Gaussian noise that is white in time and fractional in space with Hurst parameter $H\in(1/4,1/2)$. Hu et al. (Ann. Probab. 45 (2017), 4561–4616) established existence and uniqueness of the solution under the assumptions that $\sigma$ is differentiable, both $\sigma$ and $\sigma'$ are Lipschitz continuous, and $\sigma(0)=0$. Hu and Wang (Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022), 379–423) subsequently removed the technical condition $\sigma(0)=0$ by developing the well-posedness theory in a suitable weighted space associated with the weight $\lambda(x)= c_H (1 + |x|^2)^{H-1}$. In this weighted framework, which allows $\sigma(0)\ne0$, we establish temporal and spatial local linearizations: small increments of the nonlinear solution are approximated by the corresponding increments of the solution to the additive equation, multiplied by the local coefficient $\sigma(u(t,x))$. The main difficulty lies in handling the nonlocal fractional-difference structure induced by the rough spatial noise. We address this difficulty using weighted heat-kernel increment estimates. These approximations yield exact Lévy-type uniform moduli of continuity in both time and space, together with Khinchin and Chung laws of the iterated logarithm.

For a positive integer $n$, we write $L(n)$ in place of the maximum number of distinct primes whose sum is $n$, assuming there is a way of writing $n$ as a sum of distinct primes, with $L(n) = 0$ otherwise. Again if $n$ admits a partition into distinct primes, then we let $K(n)$ denote the number of parts in the unique partition that has a largest product, with $K(n) = 0$ otherwise. In the latest editions of Guy's classic text on unsolved problems in number theory, the problem as to whether or not $L(n) - K(n)$ is unbounded was left open, and it appears that this has remained open. We solve this problem in the affirmative. Our proof relies, to a large extent, on a consequence of Vinogradov's theorem together with a 1948 result due to Sierpiński on the existence of primes whose gaps to the immediately preceding and following primes both exceed any prescribed positive bound.

It is known that the dynamical properties of Kleinian groups with parabolic elements are much more complex than those without parabolic elements. In this paper, we mainly study the quasi-conformal deformation properties of geometrically finite Kleinian groups with parabolic elements. We show that, for a Kleinian group $G$ with parabolic elements, there exists a subset $K$ of the limit set $\Lambda(G)$ such that the first return map $f_{K}$ is (eventually) expending. Furthermore, the expending property is invariant under quasiconformal deformations. As an application, we show that the geometrically finite Kleinian groups have Ruelle's property. Finally, we use eigenvalue algorithm to calculate the precise Hausdorff dimensions of Hecke groups, which serves as an example to support our results.

We establish nearly sharp $W^{1,\infty}$ stability estimates for the Legendre spectral Ritz projection associated with Laplacian on a square domain. Its norm grows proportionally to the square root of the polynomial degree, up to a cubic logarithmic factor. The upper bound uses an integral representation of the discrete gradient projection in terms of one-dimensional semigroup kernels and a rank-two Duhamel comparison between the mass-matrix semigroup and its infinite-dimensional counterpart. The lower bound follows from the endpoint concentration of the Legendre kernel. As an application, we derive gradient maximum-norm error estimates for the Poisson and heat equations, with spatial factors optimal up to logarithmic terms.

We present a CR-CR method for strain gradient elasticity and prove its robustness with respect to the Lame parameter $\lambda$ and the length scale $\iota$. The method introduces the strain as an independent symmetric tensor and discretizes both the strain and the displacement using CR elements. The commuting properties of the CR interpolation operator with the relevant differential operators leads the robustness with respect to $\lambda$. A stabilization term weakly imposes the relation between the strain and the symmetric gradient of the displacement. The method also applies directly to the biharmonic equation and singularly perturbed fourth-order problems. For these problems, we prove that the CR-CR method is equivalent to an RT-CR method proposed by Huang and Tang. This result extends the classical equivalence between the RT-P0 mixed method and the CR method for the Poisson equation to fourth-order problems. The equivalence shows that the stabilization term arises naturally from the RT-CR mixed formulation.

Feel-good Thompson sampling (FGTS) augments the posterior of vanilla Thompson sampling with an exploration term that favors reward functions with high predicted optimal values. This modification achieves minimax-optimal regret up to logarithmic factors in linear contextual bandits, but existing analyses of its two standard variants establish regret guarantees in expectation. In this paper, we derive the first high-probability regret bounds for both cumulative and current-step FGTS in contextual bandits with general realizable function approximation. Our bounds depend on the corresponding decoupling coefficients and the prior mass near the true reward function. In the $d$-dimensional linear setting, both variants satisfy $\mathrm{Regret}(T)=O\big(d\sqrt T\log(2T)+\sqrt T\log(1/\delta)\big)$ with probability at least $1-\delta$, where $T$ is the time horizon. This bound matches the dimension and horizon dependence of algorithms based on the upper confidence bounds, up to logarithmic factors. Relative to the expected-regret guarantees, the cost of controlling the regret with high probability is confined to additive terms with at most linear dependence on $\log(1/\delta)$. The proof ideas are partially developed by GPT-6 Astra under the authors' guidance.

In the classical art gallery problem, a guard sees a target when the segment joining them lies entirely within the polygon. We explore what changes when sight may bend along regular arcs on absolutely irreducible curves, but only those defined by primitive integer polynomials of total degree at most \(d\) and coefficient height at most \(H\). We call such curves admissible sight carriers. Arcs can dramatically improve sight. In a one-sided comb, no straight-line guard sees selected targets in two different teeth, yet a single central guard sees a chosen target in every tooth along parabolic carriers. Sight can also fail in two different ways. Geometrically, the polygon may block every sight arc of the allowed degree between two points. For instance, in an alternating comb, the number of guards needed for a row of targets spanning the comb grows linearly with the number of sides at each fixed degree, even for carriers with arbitrary real coefficients. Arithmetically, no admissible carrier may pass through the points at all. Even in the unit square, at fixed \(d\) and \(H\), the proportion of rational targets that must guard themselves tends to one as the height bound grows. An algebraic target may lie on no rational curve of degree at most \(d\), however large \(H\) is. Both failures can be overcome for finite target sets by allowing the degree and coefficient height to grow. We show that every finite set of algebraic targets in a rational polygon becomes visible from any rational interior guard, with explicit bounds. We also construct, for every \(d\), arbitrarily large Galois orbits in the unit square, all of projective height \(4^{d+1}\), whose points must each guard themselves at degree \(d\) for every \(H\). Yet at degree \(d+1\) and coefficient height one, a single guard sees the whole orbit along \(Z(Y-X^{d+1})\).

We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle $\sigma$ on $K$ and a unital right-colinear algebra map $\Gamma:A\to K$, we show that $\Gamma:A_\sigma\to K^\sigma$ is an algebra map precisely when $\sigma$ is trivial on $\operatorname{im}\Gamma\otimes\operatorname{im}\Gamma$. We describe comodule-algebra morphisms $A_\sigma\to K^\sigma$ by characters and establish injectivity criteria using the coradical and extensions of characters to convolution-invertible functionals. For coactions on Hopf quotients projecting to the coproduct, character-evaluation maps are injective, and equality of their images is determined by a unique algebra character. For the $\imath$-Hopf algebras $H^\imath_\tau$ of Chen–Lu–Pan–Ruan–Wang, with $H$ having a bijective antipode and a Hopf pairing, we obtain a Hopf bi-Galois realization $H^\imath_\tau\hookrightarrow F_\sigma(H\otimes H^{\mathrm{cop}})$. We identify algebra characters with nonzero $\tau$-twisted compatible maps $\chi$ and prove that the corresponding morphisms $\xi_{\tau,\chi}$ of Chen–Lu–Pan–Ruan–Wang are injective, yielding right coideal realizations in $(H\otimes H)^\imath$. We classify the $H^\imath_\tau$ up to comodule-algebra isomorphism and determine when these realizations have the same image.

We study the circle-relation subspace $O_n^\circ(V)$ in the defining relations of the level-$n$ Zhu algebra of a vertex operator algebra $V$. We assemble the residue relations generating $O_n^\circ(V)$ into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least $-2n-1$, modulo $O_n^\circ(V)$. This recursion is encoded in a left truncated field $Y_L$. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for $Y_L$ needed for finite-length calculations. For a finite homogeneous strong generating set $S$, we define the finite-length circle subspaces $O_n^\circ(N,S,V)$ and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the $k=0$ strip and to last vacuum mode $-1$, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for $n=1$ and $n=2$. The $Y_L$ identities are the input for the companion paper (Part II).

Digital-domain self-interference cancellation is essential in in-band full-duplex radios after passive suppression and analog cancellation. It is particularly challenging when the desired signal is strong relative to the residual self-interference. In this regime, conventional iterative digital cancellers update their coefficients from desired-signal-contaminated received samples and may suffer substantial performance degradation. This paper reformulates digital-domain self-interference cancellation as an operator-learning problem. The proposed framework learns the temporal evolution of the residual self-interference channel from historical observations and uses the learned operator together with the known local transmit signal to reconstruct the current residual self-interference. A Transformer neural operator is used to realize this framework. The resulting method is trained in a desired-signal-free setting and then applied in the presence of a desired signal. A continuity-based analysis characterizes the sensitivity of the learned operator to the resulting input perturbation. Numerical experiments in static and time-varying channels, including time-varying SIR conditions, demonstrate stable cancellation performance over a wide range of interference levels. At 30 dB SIR, the proposed method improves the cancellation performance by 5–15 dB over the considered conventional methods.

Velocity space discretization is a primary source of computational cost in deterministic methods for multiscale gas flows. A fixed velocity grid often retains redundant kinetic degrees of freedom (DOFs), particularly under near equilibrium conditions. To address this issue, this work develops a nonequilibrium-capturing formulation based on the unified gas kinetic scheme (UGKS), in which the number of kinetic DOFs adapts locally to the degree of nonequilibrium. A discrete orthogonal basis is constructed for numerical implementation, and the velocity distribution function is decomposed into an analytical local equilibrium and a basis expanded nonequilibrium component. The retained, locally adapted DOFs are evolved directly, substantially lowering computational overhead while preserving the discrete conservation of mass, momentum, and energy, as well as accurately capturing nonequilibrium states. The formulation retains the core multiscale coupling of molecular transport and collision inherent to UGKS, providing explicit and implicit schemes for unsteady and steady flows, respectively. Numerical tests across a wide range of flow speeds and rarefaction regimes (from continuum to highly rarefied) demonstrate close agreement with benchmark solutions. For unsteady flows, the explicit scheme achieves speedups of 12.64 to 58.50 over the standard UGKS. In steady state tests, only about $2\%$ of the original kinetic DOFs are required in continuum regions, accelerating each implicit iteration by a factor of 33.87. Even in the most rarefied cases, the scheme captures nonequilibrium behavior using only about $27\%$ of the original DOFs.

We study factorized gradient descent (FGD) for overparameterized asymmetric matrix sensing with the balancing regularizer $\frac18\|L^\top L-R^\top R\|_F^2$. An exact indefinite lifting of the population objective decomposes the dynamics of FGD into positive, negative, and null spectral subspaces. The first two components have linear restoring terms, whereas the null component contracts only quadratically. For independent isotropic sub-Gaussian sensing matrices and independent centered sub-Gaussian noise with scale proxy $\sigma$, we prove the local bound \[ \|L_tR_t^\top-X^\star\|_2 \lesssim\max\left\{\frac{1}{\eta t+1/\sigma_r}, \kappa\sigma\sqrt{\frac{d_1+d_2}{n}}\right\}, \] under a sufficiently small initialization error, the number of linear measurements $n\gtrsim\kappa^2(d_1+d_2)\max\{k^2,\sigma^2/\sigma_r^2\}$, and step size $\eta\lesssim\sigma_1^{-1}$. Here $\sigma_1$ and $\sigma_r$ are the largest and smallest nonzero singular values of the target $X^\star\in \mathbb{R}^{d_1\times d_2}$ respectively, and $\kappa=\sigma_1/\sigma_r$. Furthermore, a population trajectory was constructed to have tight product error $\Theta(1/t)$. These results provide a local rate characterization for ordinary gradient descent for overparameterized asymmetric matrix sensing with balancing regularizer.

A set $A$ of integers is an $h$-basis for $n$ if every integer in $[0,n] = \{0,1,2,\ldots,n\}$ belongs to the sumset $hA$, that is, can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. The classical postage stamp problem asks for the largest integer $n_h(k)$ such that $[0,n_h(k)] \subseteq hA$ for some set $A$ of nonnegative integers of size $k+1$. We consider the effect of allowing negative elements. Let $n^{\flat}_h(k)$ be the largest integer such that $[0,n^{\flat}_h(k)] \subseteq hA$ for some set $A$ of integers of size $k+1$ containing at least one negative integer. For $k=2$, we have $n^{\flat}_h(2) < n_h(2)$ if $h$ is odd or $h=4$ and $n^{\flat}_h(2) = n_h(2)$ if $h$ is even and $h \neq 4$. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.