Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure. Despite recent progress in one-dimensional settings and general upper bounds, fundamental questions about the localization rate of QOT optimizers around the Monge coupling have remained open. In this work, we establish a general lower bound showing that the support of the QOT optimizer cannot concentrate around the Monge graph faster than order $\varepsilon^{\frac{1}{d+2}}$ in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in \citet{wiesel2025sparsity}. We also show that the QOT value gap controls the mean-squared deviation $\mathbb E_{π_\varepsilon}\|y-T(x)\|^2$ by the scale of $\varepsilon^{\frac{2}{d+2}}$. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order $\varepsilon^{\frac{1}{d+2}}$ by reducing the problem to self-transport and applying recent self-transport sparsity results. Finally, we validate our theoretical bound with a synthetic experiment in high-dimensional settings.

Policy learning in modern operations environments faces a fundamental tension between limited operational data and the large, often continuous, state and action spaces over which good decisions must be identified and deployed. We study value-based policy learning in stochastic optimal control: a greedy policy induced by an estimate of the optimal action-value function $Q^*$ is deployed, and its performance is measured by regret. The empirical success of this approach calls for statistical insight into the structures that enable fast regret convergence. We show that, in continuous action spaces, fast policy learning is induced by three geometric structures: a growth exponent $p$, which quantifies how quickly $Q^*$ separates suboptimal actions from its maximizers; a margin-mass exponent $m$, which controls how much deployment mass lies on states with weak growth; and an action-wise regularity exponent $q$, which measures the smoothness of the $Q^*$-estimation error across actions. Given a $n^{-1/2}$-accurate estimator of $Q^*$, we show that the minimax-optimal policy regret convergence rate is \[ \widetildeΘ\left( n^{-\min\left\{\frac{p}{2(p-q)},\frac{m+1}{2m}\right\}} \right), \] up to a logarithmic factor at the boundary between the two regimes. The exponent $q$ is crucial: $q>0$ yields faster-than-$n^{-1/2}$ regret. This regime is natural in operations applications. In particular, we verify $q>0$ under mild regularity conditions in dynamic inventory control and service allocation examples, while the mechanism underlying this fast rate regime extends beyond these settings.

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.

Generative models are powerful tools for sampling from a learned distribution $\mathcal{P}(Y \mid X)$, and inverse-design methods invert this map to find an input $x$ that produces a desired point output $y^*$. However, many design goals are naturally distributional rather than pointwise, incorporating the inherent uncertainty of $Y$ and targeting a specific form for it, a task not addressed by standard inverse design. To address this issue we introduce Conditional Distribution Matching (CDM), a new inverse-design problem class in generative modeling: given a joint distribution $\mathcal{P}(X, Y)$ and a target distribution $\mathcal{G}(Y)$, find an input $x^*$ whose induced conditional distribution $\mathcal{P}(Y \mid X = x^*)$ matches $\mathcal{G}$. We formally define two variants: Conditional Distribution Matching Sampling (CDMS) and Conditional Distribution Matching Optimization (CDMO). To solve these problems, we propose MLGD-F (Matching-Loss Guided Diffusion with a Fast inner sampler), a plug-and-play inference-time algorithm that combines a pretrained score-based diffusion model with a pretrained fast conditional sampler, requiring no additional training or fine-tuning. By leveraging single-step conditional sampling, MLGD-F enables tractable gradient computation, making the estimation of $\mathcal{P}(Y \mid X)$ both memory-efficient and computationally lightweight. We validate MLGD-F on synthetic benchmarks, structured image transformations, and generative editing optimization, demonstrating reliable recovery of inputs whose conditional distributions match diverse user-specified targets, including discrete mixtures and continuous low-rank supports.

Should we care whether AI systems have representations of the world that are similar to those of humans? We provide an information-theoretic analysis that suggests that there should be a U-shaped relationship between the degree of representational alignment with humans and performance on few-shot learning tasks. We confirm this prediction empirically, finding such a relationship in an analysis of the performance of 491 computer vision models. We also show that highly-aligned models are more robust to both natural adversarial attacks and domain shifts. Our results suggest that human alignment is often a sufficient, but not necessary, condition for models to make effective use of limited data, be robust, and generalize well.

What can linearized neural networks actually say about generalization? As mentioned in the main text, all our models are trained using the same scheme which was selected without any hyperparameter tuning, besides ensuring a good performance on CIFAR2 for the neural networks. Namely, we train using stochastic gradient descent (SGD) to optimize a binary crossentropy loss, with a decaying learning rate starting at 0.05 and momentum set to 0.9. Furthermore, we use a batch size of 128and train for a 100epochs. This is enough to obtain close-to-zero training losses for the neural networks, and converge to a stable test accuracy in the case of the linearized models1.

In what follows, we give some details of content omitted in the paper due to space limit. The supplements are organized as follows. We give some proof of Lemma 1, 2, Proposition 1, Lemma 3, 4, and Theorem 2 in Section A.1 -A.6, respectively. We provide some training details in Section A.13 as well as experiment details and results in Section A.14. We compare polynomial CBFs with NCBF in A.15, compare NCBFs with different activation functions in A.16. A.1 Proof of Lemma 1 We prove by induction on L. If L =1, then x 2 X(S) if the pre-activation input to the (1,j) neuron is nonnegative for all j 2 S1 and nonpositive for all j/2 S1. We have that the pre-activation input is equal to WT1jx+r1j, establishing the result for L =1 .

This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSyncyields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSyncto the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync.

We analyze here briefly some basic notions of the geometry of the sphere that we use in our algorithm and convergence analysis. We refer the reader to [1, p. 73-76] for a more comprehensive presentation. Tangent Space: The tangent space of the r-dimensional sphere Sr at a point p is an r-dimensional vector space, which generalizes the notion of tangent plane in two dimensions. We denote it by TpSr and a vector v belongs in it, if and only if, it can be written as α(0), where α: ( ε,ε) Sr (for some ε > 0) is a smooth curve with α(0) = p. The tangent space at pcan be given also in an explicit way, as the set of all vectors in Rr+1 orthogonal to p with respect to the usual inner product.

The goal of machine learning is to find models that minimize prediction error on data that has not yet been seen. Its operational paradigm assumes access to a dataset $S$ and articulates a scheme for evaluating how well a given model performs on an arbitrary sample. The sample can be $S$ (in which case we speak of ``in-sample'' performance) or some entirely new $S'$ (in which case we speak of ``out-of-sample'' performance). Traditional analysis of generalization assumes that both in- and out-of-sample data are i.i.d.\ draws from an infinite population. However, these probabilistic assumptions cannot be verified even in principle. This paper presents an alternative view of generalization through the lens of sensitivity analysis of solutions of optimization problems to perturbations in the problem data. Under this framework, generalization bounds are obtained by purely deterministic means and take the form of variational principles that relate in-sample and out-of-sample evaluations through an error term that quantifies how close out-of-sample data are to in-sample data. Statistical assumptions can then be used \textit{ex post} to characterize the situations when this error term is small (either on average or with high probability).