arXiv subjects

Maja Gwozdz

Publications and source records attributed to Maja Gwozdz.

12 recordsLinked to original sources

arXiv2026

Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets

We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_μ:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad dμ(x):=Z_μ^{-1}e^{-V(x)}\,dx, \] and assume that $μ$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $ν$ be a compactly supported log-concave probability measure with support $K$, and let $\nablaΦ$ be the Brenier map from $μ$ to $ν$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}Φ\le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nablaΦ$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nablaΦ) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.

math.AP↗

arXiv2026

Sharp Continuity Moduli for Dirichlet Heat Flow in Boundary-Reservoir Transport Metrics

Let $Ω\subset\mathbb R^n$ be a bounded $C^2$ open set and let $P_t$ be the killed Dirichlet heat semigroup. We prove the sharp fixed-time power-scale modulus of $P_t$ for the Figalli--Gigli boundary-reservoir transport distances $W_{b,p}$. For every $t>0$, $P_t$ is globally Lipschitz with respect to $W_{b,1}$. For every $p>1$, and on every total-mass sublevel $\{μ:μ(Ω)\le m\}$, it is $1/p$-Hölder: \[ W_{b,p}(P_tμ,P_tν)^p \le C_{t,p,m,Ω} W_{b,p}(μ,ν). \] For $p>1$, we show that the exponent $1/p$ is optimal in the scale of power moduli. On the full finite-measure space, $P_t$ is discontinuous at the zero measure. To establish the lower bound, we rely on the amplification of the boundary layer. More precisely, a unit mass initially placed at distance $\varepsilon$ from $\partialΩ$ has input $W_{b,p}$-distance $O(\varepsilon)$ from zero, whereas after any fixed positive time, its $p$-th boundary moment is bounded below by $c\varepsilon$. As a result, in the quadratic case and in the original finite-measure $W_{b,2}$ metric, there does not exist a standard finite-$λ$ $\mathrm{EVI}_λ$ semigroup on a $W_{b,2}$-metric domain which would contain the affine constant-boundary data class and could restrict to the affine constant-boundary Dirichlet heat flow. Finally, we also describe the corresponding lower-bound obstruction for smooth uniformly elliptic perturbations in divergence form.

math.AP↗

arXiv2026

Entropic optimal transport need not select a zero-temperature limit

We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $μ\otimesμ$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(μ\otimesν)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapstoπ_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.

math.OC↗

arXiv2026

Non-rotationally invariant generalised geodesics in the disc

We study Brenier's relaxed least-action problem in the unit disc $D:=\{x\in\mathbb{R}^2:\ |x|<1\}$ at the critical time $T=π$, with the endpoint pair $i_D$ (the identity on $D$) and $-i_D$. We prove that the critical energy shell \[ S^3=\{(x,v)\in\mathbb{R}^2\times\mathbb{R}^2: |x|^2+|v|^2=1\} \] supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure $σ$ on $S^3$ and the Hopf quotient \[ Π=(N,M,L):S^3\to\mathbb S^2_{1/2}, \qquad \mathbb S^2_{1/2}:=\{(n,m,\ell)\in\mathbb{R}^3: n^2+m^2+\ell^2=1/4\}. \] Here, \[ N=\frac12(x_1^2+v_1^2-x_2^2-v_2^2),\quad M=x_1x_2+v_1v_2,\quad L=x_1v_2-x_2v_1 . \] This quotient is a first integral of the harmonic-oscillator flow. Let $τ:=Π_\#σ$, let $g$ be a bounded Borel function on $\mathbb S^2_{1/2}$ whose $L^\infty(τ)$-class is odd under $(n,m,\ell)\mapsto(n,m,-\ell)$, and let $δ\in\mathbb{R}$. If $1+δg\circΠ\ge0$ $σ$-a.e., then \[ dμ_{δ,g}=π(1+δg\circΠ)\,dσ\] has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If $δ\ne0$, this is equivalent to axisymmetry of the $L^\infty(τ)$-class of $g$, modulo $τ$-null sets. In particular, taking $g(n,m,\ell)=\ell m$ with $0<|δ|<8$ gives strictly positive non-rotationally invariant minimisers.

math.AP↗

arXiv2026

Failure of ambient closed-set large-deviation upper bounds in entropic optimal transport

Large-deviation upper bounds on compact sets do not, in general, extend to arbitrary closed sets without additional tightness. We show that this obstruction already occurs in static entropic optimal transport. More precisely, we construct a fixed-cost model with continuous cost and nonatomic marginals for which the entropic minimisers converge in total variation to an optimal plan with noncompact support, the known compact-set upper bound remains valid, but the corresponding closed-set upper bound fails on a specific closed subset of the ambient space. For a fixed closed set, we identify the exact tail criterion for passing from compact to closed sets. We show that there does not exist a full large-deviation principle (LDP) on the ambient space at speed $1/\varepsilon$ with an arbitrary lower semicontinuous rate function.

math.PR↗

arXiv2026

Instability of the ray-monotone selector for $W_1$-optimal transport

For the distance cost $c(x,y)=|x-y|$, the set $O(μ,ν)$ of $W_1$-optimal plans is generally not a singleton. Under the classical absolute-continuity hypotheses in the Euclidean case, secondary variational selection by the quadratic energy $C_2$ yields the ray-monotone $W_1$-optimal plan. We provide a counterexample to an open problem posed by Santambrogio that concerns the stability of this selector under weak convergence of the marginals. More precisely, we construct a fixed absolutely continuous source $μ$ and absolutely continuous targets $ν_n\rightharpoonupν$ such that $γ^{\mathrm{sel}}(μ,ν_n)\rightharpoonupγ^{\mathrm{hom}}$, where $γ^{\mathrm{hom}}\in O(μ,ν)$ but $γ^{\mathrm{hom}}\neqγ^{\mathrm{sel}}(μ,ν)$. We also identify the narrow Kuratowski limit of the optimal-plan sets $O(μ,ν_n)$, derive the constrained $Γ$-limit for secondary energies of the form $\int Φ(|x-y|)\,dγ$ with $Φ\in C([0,2])$, and deduce a non-commutation result for the additive perturbation $c_\varepsilon(x,y)=|x-y|+\varepsilon|x-y|^2$.

math.AP↗

arXiv2026

Uniform optimal-order Wasserstein quantisation

We address Steinerberger's Wasserstein transport problem on the cube $Q=[0,1]^d$. For every $d\ge2$, we consider a dyadic digital sequence $(x_n)\subset Q$ and prove that every prefix $\{x_1,\dots,x_N\}$ admits an exact equal-mass transport partition at the optimal scale. More precisely, for every $N\in\mathbb{N}$, there exist pairwise disjoint Borel sets $A_1,\dots,A_N\subset Q$ such that \[ λ_d(A_n)=\frac1N,\qquad A_n\subset B(x_n,6\sqrt d\,N^{-1/d})\qquad(1\le n\le N), \] and $λ_d\!\bigl(Q\setminus\bigcup_{n=1}^N A_n\bigr)=0$. In other terms, every prefix of the sequence supports an exact transport allocation of Lebesgue mass to its points with uniformly controlled radius $O(N^{-1/d})$. By an elementary partition criterion, this yields \[ W_\infty\!\left(\frac1N\sum_{n=1}^Nδ_{x_n},\,λ_d\right)\le 6\sqrt d\,N^{-1/d} \qquad(N\in\mathbb{N}). \] The bound holds for every $1\le p\le\infty$. The exponent $1/d$ is optimal, so it gives the sharp uniform prefix rate on the cube. The result settles Steinerberger's problem for all $d\ge1$ and all $1\le p\le\infty$.

math.CA↗

arXiv2026

No weakly factor-universal cellular automaton

Hochman asked whether there exists a cellular automaton $F$ such that every cellular automaton is a factor of $F$ in the dynamical sense. In particular, we do not require the factor map to commute with the spatial shifts. We show that no such cellular automaton exists. More generally, if $F$ weakly factors onto the radius-zero $q$-clock automaton $C_q^{(k)}$, then every periodic point of $F$ has period divisible by $q$. For a cellular automaton $F:A^{\mathbb Z^d}\to A^{\mathbb Z^d}$, define $φ_F:A\to A$ by $F(\underline a)=\underline{φ_F(a)}$, and let $g_F$ be the greatest common divisor of the cycle lengths of $φ_F$. We prove that if $C_q^{(k)}$ is a weak factor of $F$, then $q\mid g_F$ holds. It follows that the action of $F$ on constant configurations yields an explicit divisibility obstruction to clock weak factors.

math.DS↗

arXiv2026

Bi-Lipschitz Smoothing under Ricci and Injectivity Bounds

We prove that a complete Riemannian manifold with a positive uniform lower bound on injectivity radius and a positive uniform lower bound on Ricci curvature admits an $L^\infty$-close (bi-Lipschitz) smooth metric with two-sided Ricci curvature bounds and a uniform positive lower bound on injectivity radius. This answers Question 2 in the Morgan--Pansu list of open problems from the conference Modern Trends in Differential Geometry (São Paulo, 2018), proposed by L. Bandara. In the proof, we rely on controlled smoothing with Croke's universal local volume lower bound and the Cheeger--Gromov--Taylor injectivity radius estimate.

math.DG↗

arXiv2026

Green--Wasserstein Inequality on Compact Surfaces

Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the $\sqrt{\log n}$ factor in the two-dimensional Green--Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an $O(n^{-1/2})$ remainder for all $n$. We argue by contradiction and combine a second-moment estimate for the random Green energy of i.i.d. samples with the semi-discrete random matching asymptotics of Ambrosio--Glaudo.

math.PR↗

arXiv2022

High-performance symbolic-numerics via multiple dispatch

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

cs.CL↗

arXiv2021

Composing Modeling and Simulation with Machine Learning in Julia

In this paper we introduce JuliaSim, a high-performance programming environment designed to blend traditional modeling and simulation with machine learning. JuliaSim can build accelerated surrogates from component-based models, such as those conforming to the FMI standard, using continuous-time echo state networks (CTESN). The foundation of this environment, ModelingToolkit.jl, is an acausal modeling language which can compose the trained surrogates as components within its staged compilation process. As a complementary factor we present the JuliaSim model library, a standard library with differential-algebraic equations and pre-trained surrogates, which can be composed using the modeling system for design, optimization, and control. We demonstrate the effectiveness of the surrogate-accelerated modeling and simulation approach on HVAC dynamics by showing that the CTESN surrogates accurately capture the dynamics of a HVAC cycle at less than 4\% error while accelerating its simulation by 340x. We illustrate the use of surrogate acceleration in the design process via global optimization of simulation parameters using the embedded surrogate, yielding a speedup of two orders of magnitude to find the optimum. We showcase the surrogate deployed in a co-simulation loop, as a drop-in replacement for one of the coupled FMUs, allowing engineers to effectively explore the design space of a coupled system. Together this demonstrates a workflow for automating the integration of machine learning techniques into traditional modeling and simulation processes.

cs.CE↗