Papers

Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0<p\leq 1$. Specifically, \[ B^s_{p,q}(\Omega)\hookrightarrow \mathcal L_1(\mathbb{D}) \] when $s\geq k+d/p$ for $0<q\leq 1$, and when $s>k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(\Omega), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.

We study robust estimation of simple random tensors of arbitrary order $q\in\mathbb{N}$ under finite-moment assumptions and adversarial contamination. We propose the first robust estimator achieving near-optimal dimension-free statistical rates in this setting. The estimator attains the near-optimal corruption rate whenever $p\ge2q$ moments are finite and continues to provide nontrivial guarantees throughout the weak-moment regime $q\le p\le2q$. Being based on directional trimmed means and minimax aggregation, our estimator is adaptive to $p$ and upper bounds on hypercontractive constants without resorting to interval-intersection procedures. Our analysis extends the trimmed-mean framework underlying recent advances in robust mean and covariance estimation to arbitrary tensor order. In particular, we establish concentration inequalities for higher-order counting and truncated empirical multi-vector product processes. We believe these inequalities could be of independent interest beyond the present application, including algorithmic robust estimation.

We study polygonal general helices in Euclidean three-space using the turning and signed torsion angles of the discrete Frenet frame. For helices whose axis is not orthogonal to the edge tangents, we prove that the global constant-angle condition is equivalent to the existence of a conserved Frenet-frame vector. On the generic branch, elimination of the auxiliary coefficient yields an intrinsic finite-step compatibility relation involving three consecutive turning angles and two consecutive torsion angles. Complementary phase and linear-subspace formulations cover the antipodal-binormal case. These characterizations reconstruct the helical axis and the helix angle and yield a sharp bound for each turning angle. We also give a spherical formulation through the tangent indicatrix: its vertices lie on a plane section of the unit sphere, which is a small circle in the non-orthogonal case, whereas the orthogonal case is exactly planar. A nonconstant Frenet-data example illustrates the criterion. Finally, for uniform chordal sampling of a smooth curve, the discrete Lancret-type quotient and the reconstructed helical direction converge with second-order accuracy.

Let \( \mathcal{S}_{g,n} \subset \mathcal{M}_g \) be the locus of curves of genus \( g \geq 2 \) admitting a model \( y^n = h(x) \) with \( h \) separable; such curves $C$ have a cyclic group \( C_n \leq \operatorname{Aut}(C) \) of order \( n \) with \( C/C_n \cong \mathbb{P}^1 \). % We give an algorithm which, given an absolutely irreducible plane model \( F(x,y) = 0 \) of a curve \( C \) over a field \( k_0 \) of characteristic zero, decides for which \( n \) the curve lies in \( \mathcal{S}_{g,n} \) and returns a model \( y^n = h(x) \) together with the birational transformation to it.

Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.

Preimage entropy measures the complexity generated by the inverse-image structure of a non-invertible dynamical system. For a continuous map $f:X\to X$ on a compact metric space, Hurley's pointwise topological preimage entropies $h_m(f)$ and $h_p(f)$ are natural invariants measuring the complexity of preimage sets. The question of whether they admit unconditional variational principles in terms of suitable measure-theoretic counterparts remains open. In this paper we resolve it by using empirical metric preimage entropies $h^*_{m,\mu}(f)$ and $h^*_{p,\mu}(f)$, defined by restricting preimage fibers to orbit segments whose empirical measures are close to a prescribed invariant measure $\mu$. We prove the variational principles $$ h_m(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{m,\mu}(f), \qquad h_p(f)=\sup_{\mu\in\mathcal M_f(X)}h^*_{p,\mu}(f) $$ for every continuous map on a compact metric space. We also show that, in general, the set of all invariant measures in these formulas cannot be replaced by the set of ergodic invariant measures. We then compare the empirical entropies with the pointwise metric preimage entropy $h_{m,\mu}(f)$. For every ergodic invariant measure $\mu$, we prove $h^*_{p,\mu}(f)\ge h_{m,\mu}(f)$. Moreover, if $f$ has uniform separation of preimages, then for every ergodic invariant measure $\mu$, $$ h^*_{m,\mu}(f)=h^*_{p,\mu}(f)=h_{m,\mu}(f). $$ Examples show that $h_{m,\mu}(f)$ is not comparable with the empirical quantities in general. We further introduce a resolving-partition property, weaker than uniform separation of preimages, under which the variational principle for $h_m(f)$ and $h_{m,\mu}(f)$ holds. Finally, we establish corresponding variational principles for preimage pressure and give an example showing that uniform separation of preimages does not imply forward expansiveness.

In this paper, we study the evolution of nearly radial concentrated vortices for the incompressible Euler equation in a bounded planar domain. We prove that if a single vortex is initially concentrated near a strict local minimum point of the Robin function of the domain and is close, up to translation, to its symmetric decreasing rearrangement, then both its shape and location remain uniformly controlled for all time. We also establish an analogous result for a pair of opposite-sign vortices near a strict local minimum point of the corresponding Kirchhoff–Routh function. No symmetry is imposed on the domain or the initial data, and the initial data need not be close to any steady state. To prove these results, we develop a new Lyapunov mechanism for the evolution of vorticity governed by the Euler equation starting from such initial data, providing quantitative control of both vortex shape and location. Specifically, we combine kinetic-energy conservation and vorticity equimeasurability with several fixed-time estimates to obtain a conditional estimate, and then use a set-valued first-exit argument to propagate it globally in time.

Bilevel optimization provides a natural modeling language for hierarchical decision problems. However, applying existing numerical solvers usually requires substantial manual analysis and reformulation. In this paper, we introduce disciplined bilevel programming (DBLP), a symbolic framework that allows users to specify and solve optimistic bilevel problems in a high-level, human-readable way that is close to the mathematical formulation. For problems with a disciplined nonlinear upper problem and a convex lower problem satisfying the disciplined parameterized programming rules, DBLP automatically canonicalizes the lower problem into conic form and constructs an equivalent single-level reformulation using the conic Karush-Kuhn-Tucker conditions. We relax the resulting complementarity constraint and use a gap continuation procedure to approximately solve a sequence of smooth nonlinear problems. We implement DBLP in the open-source Python package BLVPY, an extension of CVXPY for bilevel programming. We demonstrate the modeling and solution capabilities of BLVPY on a range of bilevel optimization problems from several application domains. The proposed framework and implementation allow users to specify and solve bilevel optimization problems within a few lines of code, without prior expertise in bilevel modeling and numerical optimization.

We prove that for every integers $d \ge 1$ and $s\ge2$ there exists an integer $D$, that depends only on $d$ and $s$, such that for every graph $P$ with maximum degree at most $ d$, there is a graph $H$ with maximum degree at most $D$ in which every $s$-coloring of $V(H)$ yields a monochromatic subdivision of $P$.

We characterize the Borel sets with an infinite version of Karchmer and Wigderson's game linking finite circuit complexity to communication complexity. To this end, we formulate an infinite version of communication complexity and prove that a given subset of the Cantor space is Borel if and only if a certain infinite communication game is solvable. We utilize this connection to provide new elementary and purely combinatorial proofs of some classical results in descriptive set theory, including the analytic separation theorem and the equivalence of monotone and positive Borel sets. Another consequence is a characterization of Borel separability via the winner of a certain "cut-and-choose" game, which we use to obtain new combinatorial proofs that neither the set of ill-founded trees nor any infinite parity function is Borel.

We study the initial boundary value problem of the two-dimensional compressible nematic liquid crystal flow with the shear viscosity $\mu$ being a positive constant and bulk viscosity $\lambda$ being a power function of density with the power exponent $\beta$. Under the condition $\beta>1$, we establish the global existence and large time behavior of the radially symmetric strong solutions to this Vaigant–Kazhikhov type model of simplified compressible Ericksen-Leslie system of Dirichlet boundary conditions for the velocity and Neumann boundary ones for the director with arbitrary large data and vacuum. This work improves the results of Zhong and Zhou (Math. Ann. 390, 2024; J. Math. Pures Appl., 212, 2026) for general 2D domains by removing the geometric angel condition on the director and relaxing the constrain on $\beta$ from $\beta>4/3$ to $\beta>1$. The key ingredient is that the rigidity mechanism arising from the radial symmetry of director prevents concentration phenomena in the transported harmonic heat flow.

We study entropy and periodic-orbit growth for flows on metric spaces. First, we prove that the finite topological entropy of a flow on a compact metric space is completely carried by compact subsets of its regular set. Next, we establish a Bowen–Walters inequality for geometrically separating flows on possibly noncompact metric spaces, under uniform control at time zero and dynamical isolation at infinity. As applications, we obtain the Bowen–Walters inequality for singular suspension flows over expansive homeomorphisms, multisingular-hyperbolic sets—extending the upper-bound part of \cite{pyyz}—and asymptotically sectional-hyperbolic attractors such as Rovella's \cite{r}.

Let $U$ be Haar-uniform on $\mathbb S^{p-1}$, let $a_1,\ldots,a_k$ be arbitrary nonzero vectors, and let $w_1,\ldots,w_k$ be simplex weights. Define \[ g(U)=\sum_{j=1}^k w_j\frac{a_j}{a_j^\top U}, \qquad N(U)=\frac{g(U)}{\|g(U)\|}. \] We prove the universal incidence law \[ \{U^\top N(U)\}^2\sim\operatorname{Beta}\!\left(\frac12,\frac{p-1}{2}\right), \] independently of the number, arrangement, rank, or overcompleteness of the directions and of the weights. Thus a deterministic, generally non-Haar function of $U$ has the same squared-cosine law as an independent Haar direction. One proof combines a Herglotz–Cauchy boundary principle, a Haar-random two-plane with one common phase, and an exact Beta–Cauchy tangent-projection equivalence. A second proof specializes the positive-semidefinite Pillai–Meng identity. The planar structure leads to converses: plane-conditional Cauchy laws recover positivity, while for signed measures an exact phase-cancellation deficit equals twice the hidden negative mass. This yields local-to-global rigidity under a phase-norming condition strictly weaker than injectivity and an unconditional exclusion of negative atoms. The law extends to probability measures under almost-sure reciprocal integrability. We characterize this condition by an exact Wiener–Dini belt series, prove finite Shannon entropy to be the sharp universal criterion for countable weights, and give an entropy–geometry extension for clustered measures. Every compact carrier of zero one-dimensional Hausdorff measure is admissible, whereas a nonzero rectifiable arc component forces divergence on a set of positive Haar measure. In orthogonal coordinates, the theorem also gives a weight-free scaled $F$ law for Pearson divergence from a fixed simplex vector to a $\operatorname{Dirichlet}(1/2,\ldots,1/2)$ vector.

Chen, Liu, Lu, and Zhang recently constructed a pretriangulated category with invertible suspension in which Verdier's octahedral axiom fails. We introduce a general enlargement construction for right pretriangulated categories and show that it preserves axioms (RTR1)-(RTR3), while failure of (RTR4) is detected by the forgetful functor. Applied to their type $A_5$ example, the construction yields a right pretriangulated category that is not right triangulated. In this example, the suspension is faithful but not essentially surjective.

We review the current state of the study of the SRB entropy functional over open components of families of uniformly hyperbolic systems, including Anosov systems, hyperbolic attractors, expanding maps, and Markov transformations. We address problems concerning the functional's infimum, supremum, and local extrema; the gradient flow induced by the functional; and the functional's behavior near the boundary of open components.

We establish the cardinal inequality \(|X|\leq 2^{t(X)H\psi(X)}\) for every Hausdorff SDL space \(X\), where \(t(X)\) and \(H\psi(X)\) denote the tightness and the Hausdorff pseudocharacter of \(X\), respectively. Since both invariants are bounded by \(\chi(X)\), this yields \(|X|\leq 2^{\chi(X)}\). As a consequence, every first-countable Hausdorff strongly cellular–Lindelöf space has cardinality at most the continuum. These results answer Questions~2.1 and~2.2 of Bella and Spadaro. An intermediate result is a uniform bounded-decomposition property for SDL spaces; in particular, their strict quasi–Lindelöf number satisfies \(\sqL(X)\leq t(X)\).

The magnetized Vlasov–Poisson equation is a fundamental kinetic model for collisionless plasmas. We establish its quasi-neutral limit in two dimensions in the presence of a spatially inhomogeneous and time-dependent external magnetic field. In the same setting, we also establish the combined mean-field and quasi-neutral limit of the associated repulsive Coulomb particle system, thereby extending the results of \cite{han2021newton}. Both limits lead to the incompressible Euler equations with Lorentz forcing. A key structural feature is the pointwise skew-symmetry of the Lorentz operator, which produces exact cancellations in the modulated-energy estimates and is crucial for controlling the limiting dynamics. We further derive a closed vorticity formulation in which the sum of the fluid vorticity and the magnetic field is transported by the flow, coupled to an evolution equation for the spatial mean velocity. Finally, we establish global well-posedness of the resulting limiting system in the Lipschitz class.

Powell asked whether the smoothness assumptions underlying classical superlinear convergence force a fixed power law between adjacent iterates of exact-line-search variable-metric methods. We answer this question negatively for BFGS: within the smooth strongly convex setting, the smallest possible adjacent-iterate Q-order is one, and this boundary is attained by a single nonterminating run. In every finite dimension at least two, and for any prescribed radius and Hessian tolerance, we construct an infinitely differentiable, globally strongly convex objective that equals the standard quadratic outside the corresponding ball and whose Hessian remains within the prescribed tolerance of the identity in operator norm. The objective has its unique minimizer at the origin and identity Hessian there. Exact-line-search BFGS, initialized with the identity matrix and started inside that ball, converges Q-superlinearly, yet no fixed power greater than one controls all sufficiently late adjacent errors.

Let $X$ be a Jacobian elliptic surface with finite Mordell-Weil group and exactly one reducible fiber. We prove that if $\chi(\mathcal O_X)\ge \rho(X)$, then the Mori cone of $X$ is simplicial. As an application, assuming additionally $q(X)=0$, we show that $X$ satisfies the bounded cohomology property (BCP): there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal O_X(C))$ for every curve $C$ on $X$. We also establish a necessary and sufficient condition for the BCP to hold on minimal smooth projective surfaces $Y$ with $\kappa(Y)\ge 1$, $q(Y)=0$, and rational polyhedral Mori cones.

Erdős and Graham asked whether every finite coloring of the plane must contain a monochromatic rectangle of any prescribed area. Kovač gave a negative answer by constructing a $25$-coloring with no monochromatic rectangle of area $1$. We reduce the number of colors to $21$ by replacing the square cells in his construction with regular hexagons.

We study an exterior Bernoulli-type free boundary problem associated with the Riemannian $p$-Laplacian on a compact Riemannian manifold. Following the shape-optimization strategy of Ly and Seck, we formulate the corresponding volume-constrained shape optimization problem, derive its first shape derivative, obtain the overdetermined Bernoulli condition through a Lagrange multiplier, and establish a conditional monotonicity result under a Riemannian contraction hypothesis. The final successive-approximation argument will require additional geometric and boundary-flux stability assumptions.

We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $\chi_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $\chi_\mathrm{stable}(G)$. This shows that $\chi_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.

We introduce the notion of Hodge–Tate splitting for schemes in positive characteristic. For a smooth variety $X$ over a perfect field $k$ of characteristic $p > 0$, we say that $X$ is Hodge–Tate split if the natural morphism $\mathcal{O}_X \to F_*\Omega^\bullet_{X/k}$ induced by the absolute Frobenius admits a splitting in $\mathcal{D}_{\mathrm{qcoh}}(X)$. We prove that this condition is equivalent to the existence of a decomposition $F_*\Omega^\bullet_{X/k} \simeq \bigoplus_{i=0}^{\dim X}\Omega^i_{X/k}[-i]$ of its de Rham complex. Consequently, for smooth projective varieties, Hodge–Tate splitting implies Akizuki–Nakano vanishing and the $E_1$-degeneration of the Hodge-to-de Rham spectral sequence. Furthermore, we establish criteria and permanence properties for Hodge–Tate splitting and use them to construct many new examples of varieties whose de Rham complexes decompose. These include blow-ups of quasi-$F$-split varieties along strata of simple normal crossings divisors, complete intersections in toric varieties, and linearly reductive quotients of Hodge–Tate split varieties. Among these examples, we obtain smooth projective varieties whose Hodge-to-de Rham spectral sequences degenerate at $E_1$, whereas their Hochschild–Kostant–Rosenberg spectral sequences do not degenerate. As an application in mixed characteristic, we prove an Akizuki–Nakano-type vanishing theorem for smooth projective globally $+$-regular varieties over the Witt ring of a perfect field.

In the application of incomplete Cholesky preconditioning to the incomplete Cholesky-conjugate gradient (ICCG) method, forward and backward substitutions exhibit sequential dependencies that constitute a major bottleneck for parallelization in multicore environments. To alleviate this bottleneck, the algebraic block multi-coloring (ABMC) method achieves both parallelism and data locality through block-wise coloring. However, ABMC requires the number of blocks to be specified as an input parameter in advance. This study evaluates the Leiden method as an alternative blocking approach for parallel preconditioning in the ICCG method. As a community detection technique that maximizes a quality function for graph partitioning, the Leiden method automatically generates blocks that reflect the matrix structure without requiring the number of blocks a priori. We partition the adjacency graphs of sparse matrices using the Leiden method and utilize the resulting blocks for parallel preconditioning. We implement the Leiden method using modularity and the constant Potts model as quality functions and compare its performance with that of the ABMC method in terms of the number of iterations, execution time, and L2 cache efficiency across eight symmetric positive definite matrices. The experimental results demonstrate that the Leiden method with the constant Potts model achieves performance comparable to that of the ABMC method configured with an optimized number of blocks.

Kaneko's val function is defined as the normalized cycle integral of the elliptic modular $j$-function along closed geodesics on the modular surface. We prove that, when primitive hyperbolic conjugacy classes are ordered by geodesic length, its values concentrate at the single point 720. More generally, an analogous concentration result holds for every weakly holomorphic modular function $f$ of weight 0, with the concentration point given by Atkin's inner product $(f, 1)_{\mathrm{At}}$. The proof combines an equidistribution theorem following Pollicott with the ergodicity of a continued-fraction suspension flow and uses the decomposition formula of Bengoechea-Imamoḡlu to construct a bounded continuous observable.

Let $X$ be proper CAT$(-1)$ and let $\Gamma\le\Isom(X)$ be finitely generated and discrete. The sharp structural theorem states that, if $\Gamma$ is accessible over finite subgroups and $\delta_X(\Gamma)<1$, then $\Gamma$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and $\partial\Gamma\to\Lambda_\Gamma$ collapses exactly their conjugate two point boundaries. The result is hereditary, and below $1/2$ every finitely generated subgroup is convex-cobounded (\cref{thm:accessible-main}). Hence non-virtually-free accessible groups have $\delta_X(\Gamma)\ge1$ (\cref{cor:accessible-gap}). Consequences cover finitely presented groups, groups with uniformly bounded finite-subgroup orders, characteristic-zero linear groups, and Kleinian groups, also infinite parabolic-free Kleinian groups have finite-index classical Schottky subgroups (\cref{cor:accessibility-extension,cor:linear-groups,cor:kleinian-classical}). Hence we cover substantial larger class than \cite{LiuWang2023},\cite{Hou2001}, also see \cref{rem:strictness-sharpness}. Finally, we also state consequences for finite JSJ representatives and hierarchies (\cref{thm:JSJ,thm:hierarchy,cor:hierarchy-dimension}).

We prove that the Higgs–Demailly system on a compact Riemann surface admits, for suitable fixed parameters, a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. Here H-ampleness is understood in the modified sense of Biswas–Misra–Ray. In particular, the argument gives an analytic proof of the equivalence between this H-ampleness and Griffiths-positive Hitchin–Simpson curvature. The proof extends the Demailly–Pingali–Murakami approach using a priori estimates and Leray–Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.

We introduce an operator-valued maximal f-divergence for completely positive (CP) maps between von Neumann algebras, associated with an operator convex function f on (0,+infinity). The construction takes values in the extended lower-semibounded self-adjoint part of the codomain von Neumann algebra. We prove independence of the common CP upper bound used in its computation, joint subadditivity, monotonicity under unital precomposition and normal postcomposition, a martingale convergence theorem for normal CP maps, and joint lower semicontinuity in the point-sigma-weak topology. For normal positive functionals, our construction recovers Hiai's maximal f-divergence. As a consequence, we establish its joint weak lower semicontinuity for arbitrary von Neumann algebras, answering a question left open by Hiai. For eta(t)=t log t, we obtain an operator-valued Belavkin–Staszewski (BS) relative entropy. Moreover, for normal channels with sigma-finite codomain, we prove that the BS channel divergence of Hollands and Ranallo coincides with the extended norm of our operator-valued divergence. Finally, we give finite-dimensional examples and obtain an explicit formula for a finite-index conditional expectation. In the BS case, this formula reduces to the logarithm of the Jones–Kosaki index.

A bottleneck in the theory of kernel methods in machine learning is the storage requirement. To ameliorate this, a standard trick is to replace the kernel with an explicit feature map. Perhaps, the most well known example is the Gaussian kernel which can be expressed in terms of the Fourier features. Analytically, the kernel $K$ can be expressed in terms of an integral expression that involves a possibly asymmetric kernel $G$ representing the feature map. Numerically, one needs to approximate this integral by a suitable numerical integration scheme, typically Monte Carlo. In this paper, we demonstrate that the eigenvalues of $K$ are approximated much better by the eigenvalues of the kernel obtained by discretizing the integral using suitable quadrature formulas instead. We illustrate this fact in the case of the Gaussian kernel and neural tangent kernels on the unit sphere of a four-dimensional Euclidean space corresponding to the sigmoid and ReLU activation functions.

Existing coded-computing designs do not explicitly exploit the intrinsic structure of the input data. In communication systems, statistical structure and redundancy are often removed through source coding (or compression) before channel coding is applied. This principle, however, does not transfer directly to coded computation. In many computational tasks, particularly in machine learning, the structure of the data is precisely what the computation seeks to exploit to infer outputs or learn meaningful patterns. Consequently, coded-computing schemes should preserve and leverage this structure in their code design, rather than ignoring or eliminating it through source coding. This observation motivates a different perspective on code construction. In many channel-coding schemes, such as Reed-Solomon codes, coded symbols are generated by evaluating a low-dimensional algebraic representation at selected points. In contrast, many high-dimensional datasets naturally concentrate near low-dimensional manifolds. In this paper, we exploit this intrinsic geometry by designing coded samples that follow the natural manifold of the data, rather than imposing an artificial low-dimensional structure unrelated to the data distribution. Inspired by graph-based manifold learning, we propose a manifold-aware encoding strategy for general coded computing (GCC). Experiments on neural network inference and high-dimensional polynomial evaluation demonstrate that the proposed strategy consistently and significantly reduces the mean squared recovery error under straggling compared with standard GCC.