Papers

We study the recovery of forward and reverse quadratic optimal-transport maps from unpaired samples in high dimensions. We introduce a bidirectional neural framework in which the learned maps induce forward and backward displacement interpolants, while the training objective combines bidirectional quadratic action, discriminator-restricted Jensen-Shannon endpoint objectives, and two-sided cycle consistency. The construction requires neither precomputed sample pairings nor an explicit convex-potential parameterization. For absolutely continuous probability measures supported on a compact convex set, and under the stated generator-approximation, discriminator-richness, and minimizer-attainment conditions, we prove a population recovery theorem: for every prescribed accuracy, the sum of the corresponding \(L^2\) errors between any global minimizer and the forward and reverse quadratic Brenier maps is below that accuracy, provided the discriminator level is sufficiently large and the annealing action weight becomes sufficiently small. Moreover, the cycle loss is bounded above by \(\lambda W_2^2(\mu_0,\mu_1)\). Complementary results quantify approximate invertibility and show that exact endpoint Jensen-Shannon divergence and cycle consistency control missing target mass and many-to-one map collapse, respectively. Experiments on Swiss roll, MNIST, CelebA, single-cell perturbation data, and chest X-ray images evaluate endpoint fidelity, transport cost, inverse consistency, and the geometry of the induced interpolations.

We prove a quantitative polynomial Roth theorem for corners in \(\mathbb F_p^2\) for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer $d$, there are constants $p_0$ and $C$ (depending only on $p$) so that for every $ p>p_0$, if polynomials \(\phi_1,\phi_2\in \mathbb \mathbb{F}_p [y]\) are of degree $\leq d$ vanishing at $0$ and are not linearly dependent, then every \(A\subset\mathbb F_p^2\) with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner $$ (x_1,x_2),\qquad (x_1+\phi_1(y),x_2),\qquad (x_1,x_2+\phi_2(y)) $$ for some \(y\in\mathbb F_p^\times\). This improves the estimate $p^{2-1/16}$ of Han–Lacey–Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han–Lacey–Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map \(\widetilde F:W\to\mathbb A^3\) from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin–Schreier sheaf and Katz–Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An \(\ell^2\) matrix estimate adapted to such sets completes the resonant case.

Let $T_{m,n}$ be the tadpole graph obtained by joining a cycle $C_m$ to a path $P_n$ by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs $(m,n)$ for which the Artinian algebra defined by the edge ideal of $T_{m,n}$ together with the squares of all variables has the weak Lefschetz property.

In 1929, Banach and Kuratowski proved under CH a combinatorial theorem, which implies that there is not a non-vanishing $\sigma$-additive finite measure on $\mathbb{R}$ which is defined for every set of reals. In 2003 Bartoszyński and Halbeisen proved that Banach and Kuratowski theorem is equivalent to the existence of a K-Lusin set of the cardinality continuum an the existence of such sets is independent of $ZFC + \neg CH$. On the other hand in 1965 Efimov introduced the strong sequences method which used to prove some well-known theorems in dyadic spaces. The aim of this paper is to show that all this notions can be generalized, i.e. considered in the generalized Baire space and to show some equivalences among them. Moreover, some applications in the direction of calibres, boundedness and partitions relations are also shown.

This paper introduces adaptive moving mesh methods for the numerical simulation of axisymmetric mean curvature flow, addressing both isotropic and anisotropic cases. The methods are developed within the framework of the mesh equidistribution principle, where a carefully designed tangential velocity is employed to dynamically redistribute mesh points during the evolution. To accurately capture the key geometric features of the evolving interfaces, we select monitor functions based on the curvature $\kappa$, its arc-length derivative $\kappa_s$, and the squared curvature $\kappa^2$. These monitor functions can be flexibly tailored to suit different problem settings and play a vital role in determining the resulting mesh quality and numerical accuracy. Spatial discretization is performed using central finite differences, while temporal integration is handled with first- and second-order time-stepping schemes, including the BDFk ($k=1,2$) and Crank-Nicolson methods. Additionally, a Lagrange multiplier approach is incorporated into the adaptive system to enforce the underlying geometric constraint, resulting in energy-stable numerical schemes. Numerical experiments confirm the convergence and energy stability of the proposed methods. More importantly, the results clearly show that the proposed methods offer significant advantages in complex geometric evolutions: by utilizing appropriately designed monitor functions, the adaptive methods achieve dynamic redistribution of mesh points, efficiently capturing localized geometric features, significantly improving numerical accuracy, and effectively preventing mesh degeneration, particularly in anisotropic cases.

We introduce cvxgenrust, an open-source tool for generating custom Rust code that solves families of parameterized convex optimization problems modeled in CVXPY. cvxgenrust canonicalizes a problem family, extracts affine maps to Clarabel cone-program data, and generates a specialized Rust crate that updates parameters and calls Clarabel natively at runtime. The generated solver can also be exposed to Python and registered as a custom CVXPY solver. Our code generator supports a wide range of convex optimization problems up to semidefinite programs and exponential-cone problems. Numerical experiments show reduced runtime relative to direct CVXPY solves and performance comparable to CVXPYgen on shared problem classes.

We study intrinsic quantum uncertainty associated with a single observable from the viewpoint of skew information. By relaxing ordinary additivity while retaining the other fundamental uncertainty requirements, we introduce a class of Hiai–Petz nonadditive skew informations that contains Hansen's metric-adjusted skew informations as the boundary case $\theta=1$. Our main quantitative result is a sharp single-observable uncertainty relation for the full Hiai–Petz class. For a fixed state, the optimal coefficient depends only on its largest and smallest eigenvalues and remains sharp even after the maximal classical contribution to the variance is retained. The general theorem yields, as special cases, sharp bounds for metric-adjusted skew informations, the SLD quantum Fisher information, and the power-commutator family. We further show that $K_s(\rho,A)=\frac12\|[\rho^s,A]\|_{\mathrm{HS}}^2$, $1/2\le s<1$, is realized within the Hiai–Petz framework through Stolarsky operator means, thereby establishing its convexity for $1/2<s<1$. More generally, for each fixed operator monotone function $f$, the Hiai–Petz spectral kernel interpolates geometrically between the metric-adjusted skew information $I^f$ and the common endpoint $K_1$. The resulting family obeys an exact power-trace-weighted composition law, replacing ordinary additivity in the interior $0<\theta<1$.

Developing an asymptotic theory for restricted maximum likelihood (REML) estimators in generalized linear mixed models (GLMMs) has remained an open problem for decades. In this paper, we establish the asymptotic distributions of the REML and maximum likelihood (ML) estimators for GLMMs when both the number of clusters and the cluster sizes tend to infinity. Under very mild conditions, requiring only finite-moment assumptions on the random effects rather than normality and imposing no restriction on the relative rates at which the number of clusters and the cluster sizes diverge, both estimators are proved to be asymptotically normal with an explicit block-diagonal covariance structure, and the bias-correction property of REML is established. Simulation studies support the asymptotic results, and a genetic data analysis illustrates the methodology.

The Swing filter extends a feasible line through its previous recording and restarts from a fitted value at the last accepted sample. We show that this inherited endpoint causes a singular loss of compression on rough input. For uniformly sampled Brownian motion at fixed tolerance, every fixed number of recording times converges to the first continuous feasibility boundary, with alternating boundary errors. The longest post-first segment tends to zero in probability. The segment-count ratio against a continuous approximation through true sampled knots diverges almost surely whenever that boundary precedes the observation horizon. For unit Gaussian increments, constrained least squares and no maximum lag, the first mean segment length has quadratic order in the tolerance $E$, the second has linear order, and every fixed mean from the third onward has order $E\log E$. We explain the logarithm through the conditional survival tail and prove a geometric bound on the subsequent approach to the stationary coefficient in the large-tolerance limit. The stationary mean satisfies $C_E\sim\gamma_{\mathrm{Swing}}E\log E$. An explicit one-dimensional kernel specifies the single coefficient $\gamma_{\mathrm{Swing}}\approx1.8120703$ and admits a $3/4$ Wasserstein contraction and convergent evaluation brackets. The decimal is not a certified error interval. A half-budget slope corridor, true endpoint chords and one-sample bridges repair the degeneration while preserving the original error bound, continuous output and constant working memory. Including bridge costs, the long-stream mean span per output segment is asymptotic to $21\zeta(3)E^2/(8\pi^2)$. Its explicit coefficient follows from the known Brownian anchored lifetime. Thus a change to the recording rule restores the quadratic scale.

In this paper, we introduce a variance-based inaccuracy measure for order statistics based on the concept of varextropy. The proposed measure provides a framework for quantifying the variability of the inaccuracy function between an order statistic and its parent distribution. We establish a transformation property of the proposed measure under strictly increasing differentiable transformations. We establish theoretical upper and lower bounds for this measure. We further develop a dynamic residual version of the proposed inaccuracy measure and establish characterization results for the underlying distribution. Furthermore, we derive bounds for the dynamic residual inaccuracy measures based on extropy and varextropy. Finally, a nonparametric estimator of the proposed measure is developed and applied to real data to assess the goodness-of-fit of candidate distributions and facilitate model selection. 1.

For \(n\ge2\), let \(A(n)=\pi(n)-\omega(n)\) and \(B(n)=\phi(n)-\pi(n)+\omega(n)\) denote the numbers of prime and nonprime totatives of \(n\), respectively, with \(1\) included in \(B(n)\). We construct an explicit injectively parametrized family \(\mathcal C_n\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-\gamma}-o(1))\log A(n)/\log\log A(n)\), realizing the sharp classical leading constant \(e^{-\gamma}\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \((\log\log A(n))^{-1/4}\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \(e^{-\gamma}\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \(N_k\) and \(M_k\) be the least integers such that \(\phi(n)>k\pi(n)\) for every \(n\ge N_k\) and \(B(n)>kA(n)\) for every \(n\ge M_k\), respectively. We prove \(M_k\le N_{k+1}\) for every \(k\ge1\), determine \(N_1,\ldots,N_6\) and \(M_1,\ldots,M_5\) exactly, and obtain \(M_k=N_{k+1}\) for \(1\le k\le5\).

In the paper, we classify all polynomial maps of the form $H=(u(x,y),\allowbreak v(x,y,z),h(x,y,z))$ in the case that $JH$ is nilpotent and $\deg_zv\geq 2\deg_zh$. Then we give the structure of $H=(u(x,y),v(x,y,z),h(x,y,z))$ if $JH$ is nilpotent and $\deg_zh\leq 3$.

The deformed Noumi-Sano operator $H_{n, m}^d(x, y)$ of order $d$ is a simultaneous extension of the Macdonald operator and the Noumi-Sano operator. In this paper, we construct an integral transform associated with eigenvalue problem of $H_{n, m}^d(x, y)$. Using the integral transform, we obtain an integral representation of eigenfunctions for $H_{n, m}^d(x, y)$.

Let $X_w$ be a Type A local complete intersection Schubert variety. We prove Contact Rigidity: the existence of two smooth singular components forces some pair of singular components to contain a common Schubert subvariety of codimension one in each. If the singular locus is a single smooth component $X_z$, the rational comparison kernel is $IC_z$, and $P_{u,w}(q)=1+q^{(\ell(w)-\ell(z)-1)/2}$ for every $u\leq z$. The first proof combines pattern avoidance with a computer-assisted finite overlap classification, rectangle inheritance, and extremal repairs. The second establishes the hypotheses of Woo's theorem and uses Euler characteristics and Bruhat triangularity to identify the entire perverse kernel.

We study rare nonresonant regions in a canonical one-dimensional disordered quantum spin chain by directly implementing the iterated Schrieffer-Wolff construction underlying the recent work of De Roeck, Giacomin, Huveneers and Prosniak on subdiffusive transport. For finite systems, we compute the probability that a disorder realization remains nonresonant through successive scales of the flow, while separately monitoring operator proliferation and the spatial localization of dressed local observables. The survival probability decays at most exponentially with system size over the numerically accessible regime. Resolving this decay scale by scale yields failure rates associated with successive Schrieffer-Wolff steps. The first rate is obtained analytically in excellent agreement with numerics, while higher-scale rates decrease rapidly throughout the regime studied. These results support the summability mechanism required for exponentially rare but parametrically long nonresonant regions, which provide the insulating bottlenecks responsible for subdiffusive transport. At the smallest couplings studied, the survival probability under our conditions exceeds the rigorous lower bound by six to seven orders of magnitude, demonstrating how conservative the constants required by the proof are while confirming that its underlying physical mechanism is quantitatively visible at accessible scales. More broadly, our results show how direct numerical implementations can provide an independent and physically transparent test of technically demanding constructive proofs, a methodology that may become increasingly useful as machine-assisted proofs become more common.

In the one-dimensional log-modified nonlinear Schrödinger equation, there are finitely many normalised standing-waves, up to space translation and phase multiplication. In the defocusing case, there is exactly one normalised standing-wave having a prescribed mass and least energy. Both results are obtained by exploiting properties of the mass function: the analyticity in the focusing case, and the monotonicity in the defocusing case.

We study a discrete memoryless two-way wiretap channel under strong one-sided secrecy, where User 1's message is confidential and User 2's message is not required to be secret. We develop non-adaptive and key-assisted adaptive inner bounds from a common resolvability estimate for selected index components. For the adaptive construction, we prove a multiround leakage theorem that accounts both for information revealed about a key before its later use and for the possibility that an encoder uses an incorrectly decoded key. The proof includes an ideal-to-implemented coupling, deterministic codebook selection, and a growing-round argument that removes the initialization-rate loss. We eliminate the internal component rates exactly and characterize the adaptive gain at each fixed auxiliary distribution. For an open two-parameter family of binary channels, we also prove a converse for arbitrary non-adaptive stochastic codes and obtain an operational adaptive-versus-non-adaptive separation. For one representative channel, the complete non-adaptive strong one-sided secrecy capacity region is identified exactly.

Let $\nu(G)$ be the maximum number of edge-disjoint triangles in a graph $G$ and $\tau(G)$ the minimum number of edges meeting every triangle. Tuza conjectured that $\tau(G)\le 2\nu(G)$. We prove that $\tau(G)\le (165/59)\nu(G)$. The constant $165/59\approx 2.797$ improves the bound $66/23\approx 2.870$ that Haxell proved in 1999. The key observation is that, for a suitable red-blue coloring, the families left over in Haxell's construction contain every triangle with exactly one red edge. Such a family $\mathcal{F}$ admits an exchange that forces certain red edges to lie in a single triangle once the blue edges of a maximum packing are deleted, which gives $\tau(\mathcal{F})\le (8/3)\nu(\mathcal{F})$.

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

We introduce a notion of viscosity solution for Hamilton–Jacobi–Bellman (HJB) equations with distributional drift, based on paracontrolled test functions and related through a Zvonkin transformation to classical viscosity theory. The equations considered are of the form \[ \left(\partial_t+\frac12\Delta+b\cdot\nabla\right)h(t,x) =-H(t,x,h(t,x),\nabla h(t,x)), \] where $b$ is singular in the sense of \cite{paradistrib} and has regularity $\mathcal C^{-\alpha}$ for $\alpha\in(1/2,2/3)$. Using doubling-of-variables arguments, we derive a priori gradient estimates that also cover Hamiltonians with slightly superquadratic growth in $\nabla h$. We also obtain probabilistic representations through singular control problems for convex $H$ and weak singular forward–backward SDEs for possibly nonconvex Hamiltonians with at most quadratic growth.

Let $(X,\tau)$ be metrizable and let $a\in X$. We give a constructive account of metrizable topologies $\sigma\subseteq\tau$ that agree with $\tau$ on $X\setminus\{a\}$. Applying Hausdorff's classical metric collapse construction, for every noncompact $(X,\tau)$ and every compatible metric $d$ we obtain a strict coarsening with a metric $p\le d$ that agrees with $d$ on a common neighborhood of each point other than $a$. A prescribed countably infinite closed discrete set $\{x_n:n\in\N\}\subseteq X\setminus\{a\}$ can be made to satisfy $p(a,x_n)\le\lambda_n$ for any positive null sequence $(\lambda_n)$. The resulting metric is greatest among the metrics dominated by $d$ that satisfy these bounds, and is complete whenever $d$ is complete. We exhibit its realization as a classical metric quotient. We also represent all localized metrizable coarsenings by continuous scalar gauges using a standard cone metric. Inclusion is expressed by the cofinal comparison of sublevel sets familiar from extension-trace theory, while pointwise maximum and minimum realize finite joins and meets. A closed-discrete criterion detects strictness. Standard preservation results for Borel structure, complete metrizability, and Polishness, together with function-space and local-field examples, complete the account.

We propose and analyze a non-iterative domain decomposition time integrator for the linear acoustic wave equation. The method is based on an overlapping decomposition in space and uses a localized implicit prediction to provide artificial boundary data at the interfaces. We use conforming finite elements of arbitrary order to discretize in space and allow spatially varying material coefficients. We prove an error bound relative to the global Crank–Nicolson approximation that decays exponentially with the size of both the subdomain and prediction overlaps. Compared to earlier work, this new method avoids a CFL-type step size restriction and allows the use of finite elements of arbitrary order. The analysis combines a discrete Saint-Venant principle with new decay and truncation estimates for the localized prediction. Numerical experiments support the analysis and demonstrate stable behavior for all tested time step sizes, accuracy comparable to that of the global Crank–Nicolson approximation, and the potential for efficient parallel execution.

We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation $\Delta^2 u=u^{-7}$ in $\mathbb R^3\setminus\{0\}$, which extend continuously to the origin with $u(0)=0$. First, we prove that if the associated conformal metric $g=u^{-4}|dx|^2$ has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile $u(x)\sim(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$ as $x\to 0$; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition $u(x)=o(|x|)$ at infinity forces $u(x)\equiv(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.

Solving inverse problems with differentiable physics simulators holds the potential to revolutionize scientific discovery and engineering design, as it enjoys both the strict physical correctness from rigorous numerical physics simulators, and the high efficiency and effectiveness from automatic differentiation and gradient-based optimization. However, currently, this paradigm faces performance issues in optimization. In this work, we target benchmarking the performance of different optimizers to solve various inverse problems. We construct 12 differentiable physics simulators spanning physics domains including discrete mechanics, continuous mechanics, atomistic simulations, rendering, and semi-empirical physics models. Based on these simulators, we design corresponding inverse problems that can be categorized into parameter identification, inverse design, and optimal control. Finally, we conduct extensive experiments to compare the performance of different optimizers, including regular first-order methods, approximate second-order methods, as well as global optimizers, on these inverse problems, and analyze the results to provide insights on how to choose and design optimizers for differentiable programming. We hope such benchmarks can inspire the development of more effective optimizers, and further promote the applications of differentiable programming in various scientific and engineering domains.

We show that if an n-vertex 5-connected graph has at least 4n-10 edges, then for any choice of five of its vertices, we can contract disjoint connected subgraphs containing these vertices to obtain $K_5$ as a minor. The bound on the number of edges is the best possible.

In this paper we introduce relative mean sensitivity both in topological and measure-theoretical settings under group actions. It turns out that relative positive entropy is relative mean $n$-sensitive between minimal systems; relative positive $\mu$-entropy for an ergodic measure $\mu$ is relative mean $\mu$-$n$-sensitive.

We study an inexact interior-point method for nonsmooth, possibly nonconvex optimization in a Hilbert space with inequality constraints ordered by a cone in a Banach lattice, with particular emphasis on infinite-dimensional state-constrained optimal control. The objective function is given by the sum of a smooth, possibly nonconvex term and a convex, possibly nonsmooth term with a computable proximal mapping. The constraints are formulated by means of an order cone in a Banach lattice. This setting covers finite-dimensional nonsmooth nonlinear problems with componentwise constraints as well as infinite-dimensional PDE-constrained optimization problems with pointwise state constraints. The method is based on barrier-regularized subproblems, which are solved inexactly by a proximal-gradient method. We consider logarithmic and power-type barriers and derive the differentiability and curvature estimates needed for the convergence and complexity analysis. Under suitable constraint qualifications and compactness assumptions, we establish approximate KKT conditions for the original problem and convergence of the inexact interior-point sequence. For logarithmic and power-type barriers, we derive complementarity estimates and outer iteration bounds; the corresponding power-barrier rates and total inner-outer complexity bounds are stated under explicit barrier-path and uniform smoothness assumptions. For convex problems, we obtain stronger convergence results. We apply the framework to state-constrained semilinear elliptic optimal control and sparse dictionary learning with nonlinear side constraints. Numerical experiments illustrate the proposed method.

We investigate solitary states in coupled oscillator networks, where a single oscillator detaches from a synchronized cluster and evolves at a distinct mean frequency. We focus on resonant solitary states, in which the mean frequency of the detached oscillator is shaped by resonance with a mode of the cluster. Here, we introduce a normal-form and symmetry-based approach to identify such states. We reduce the high-dimensional network dynamics to a two-dimensional second-order system that couples a rotator representing the solitary oscillator to a harmonic oscillator representing the dominant Laplacian mode of the cluster. We use normal-form transformations to introduce symmetry into this system. Using this symmetry, we reformulate the frequency selection problem as the search for relative equilibria of the normal form symmetry. We derive explicit algebraic conditions for the existence of these relative equilibria, and hence for resonant solitary states and their effective frequencies.

Let $Q$ be a commutative unital quantale and $\QRel$ the ordered category of sets and $Q$-relations. We develop a categorical foundation for formal and property-oriented concept lattices of $Q$-relations via the diagonal category $\DQRel$ and back-diagonal category $\BQRel$. Our main results are: (1) the formal concept lattice construction is induced by a representable functor on $\BQRel$; (2) the property-oriented concept lattice construction corresponds to a representable functor on $\DQRel$; and (3) for regular $Q$-relations, the property-oriented concept lattice of their pointwise tensor product is naturally isomorphic to the tensor product of the respective property-oriented concept lattices.

Operator learning is formulated on function spaces, but training data are typically available only through finite-dimensional representations. In encoder–decoder architectures, a matrix-valued kernel on the encoded space induces an operator-valued kernel on the original function spaces, and the corresponding reproducing kernel Hilbert spaces are isometrically isomorphic. As the input and output resolutions increase, the induced kernels converge to a limiting kernel, in the sense of operator-norm convergence of their associated integral operators, allowing regularity assumptions to be stated independently of the encoding resolution. For regularized stochastic gradient descent, we establish upper bounds for decreasing and fixed step sizes, separating the encoding and regularization terms from optimization terms of order \(t^{-\theta}\) and \(T^{-\theta'}\), respectively, for any \(\theta,\theta'\in(0,1)\). We further prove lower bounds showing that these encoding-induced terms are generally unavoidable. The analysis is further extended to encoder–decoder neural networks through the limiting neural tangent kernel (NTK), yielding error bounds with an additional finite-width term and polynomial parameter and sample complexity guarantees when the encoding errors decay algebraically. The framework covers matrix-valued kernels constructed from radial and dot product kernels, NTKs arising from wide encoder–decoder neural networks, and encoder–decoder pairs based on Fourier, Legendre polynomial, wavelet, PCA, or pointwise sampling representations.