Spectral Norm

We study norm-constrained linear classification under Eu clidean adversarial perturbations in a Gaussian model with a low-dimen sional informative subspace and an independent noise tail. For bounded ramp loss, we prove that a principal-space witness with risk below one half forces every near-optimal predictor to have small tail weight. A path-specific density bound yields constants without requiring positive tail variance. Under isotropic principal covariance, we establish a unique population minimizer and joint local growth. Boundary normalization then removes the common attack penalty from centered margins, giving localized finite-sample guarantees governed by principal dimension and total tail energy. Globalized growth removes the entrance condition at weaker constants; a model-aware comparison retains local guarantees. Experiments with twenty paired repetitions show decreasing excess risk and tail use with sample size, and nearly unchanged behavior when tail dimension grows at fixed total energy. Pure-noise controls and optimizer diagnostics clarify the scope and limitations of these conclusions.

We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_{q}$-norm over an $\ell_{p}$-ball of radius $R$, where $1\leq p,q\leq \infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, efficiently realizing the nearly optimal rates of (MBG+26), thereby resolving the nonsmooth end of the COLT 2015 open problem (Guz15b). In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness over an $\ell_1$-ball of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem in sublevel sets of an evolving bundle (LNN95; BBE+20): at each query we either find a point with low function value or we produce a deep cut in the current sublevel of the bundle, that we chase. The dichotomy between stability of selectors and forced movement by deep cuts bounds the number of iterations of the algorithm near optimally. For nested subsets of $R B_{p}^{d}$, we introduce a novel notion of stable center whose movement is bounded by $\widetilde{O}_{p,q}(RT^{1-1/p+(1/q-1/2)_{+}})$ in the $\ell_{q}$-norm after $T$ steps, which we show is nearly optimal in high dimensions. A Monte Carlo average of the proposed selector achieves near-optimal rates with high probability and can be implemented in polynomial time for our optimization algorithm in the real-arithmetic model.

We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p < q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include $\widetilde O(1/T)$ for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $X$ and the set of possible subgradients $H$ are convex, centrally symmetric, and admit a type of minmax theorem, advancing on a fundamental question by Sridharan [Sri12, Section 10.1.2, Q3]. As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.

Recently, a wide range of recommendation algorithms inspired by deep learning techniques have emerged as the performance leaders on several standard recommendation benchmarks. While these algorithms were built on different DL techniques (e.g., dropouts, autoencoder), they have similar performance and even similar cost functions. This paper studies whether the models' comparable performance are sheer coincidence, or they can be unified under a single framework. We find that all linear performance leaders effectively add only a nuclear-norm based regularizer, or a Frobenius-norm based regularizer. The former ones possess a (surprising) rigid structure that limits the models' predictive power but their solutions are low rank and have closed form. The latter ones are more expressive and more efficient for recommendation but their solutions are either full-rank or require executing hard-to-tune numeric procedures such as ADMM. Along this line of finding, we further propose two low-rank, closed-form solutions, derived from carefully generalizing Frobenius-norm based regularizers. The new solutions get the best of both nuclear-norm and Frobenius-norm world.

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $Θ\left( \frac{dr}{\varepsilon^2} \max\left\{1,\frac r{\sqrt t}\right\} \right)$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt t$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq ε$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $σ^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(σ^2 ε^{-3} + ε^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with ``sparse'' noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. In vector-valued linear regression with square loss $\ell_{(x,y)}(W)=|Wx-y|_2^2$, where $x\in\mathbb{R}^d$, $y\in\mathbb{R}^m$ and the learner is the empirical risk minimizer of minimal Frobenius norm, we prove that the minimal budget of weighted examples that recovers the full-data loss on every finite dataset is exactly $n^*(d,m)=(m+1)d$. We further determine two more values of the weighted selection profile $F_w(d,m,n)$: at the near-threshold budget, $F_w(d,m,(m+1)d-1)=1+\frac{1}{dm^2}$, and at the spanning budget, $F_w(d,m,d)=d+1$ for every $m$, while $F_w(d,m,n)=\infty$ for $n<d$. For the smallest open intermediate cell $(d,m)=(2,2)$ we prove $F_w(2,2,3)\in[13/8,15/8]$ and $F_w(2,2,4)\in[5/4,3/2]$, reduce the conjectured exact values $13/8$ and $5/4$ to a finite moment problem on the circle with at most seven atoms, and establish strong structural evidence for the conjecture. The upper-bound techniques (a fixed-basis conic compression lemma, a determinant-facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems) are of independent interest. As a byproduct we correct an erroneous claim circulating in a recent unrefereed preprint, exhibiting an explicit dataset with $m=2$ on which no weighted selection of $2d$ points recovers the optimal loss. All results are new only for $m\ge 2$; the scalar case $m=1$ is due to Hanneke et al.

Muon motivates designing optimizer geometry around the function of each parameter block and uses the spectral norm for hidden linear layers. For the language-model head, the spectral norm is not a faithful measure of functional change. Softmax removes shared logit shifts, whereas the spectral norm can assign arbitrarily large size to updates that change no output probability. We therefore treat the LM head and softmax as one module and derive an update geometry for their composition. Hilbert's projective distance respects this invariance as it measures the largest change in pairwise log odds. For an update $S$ with token rows $s_i^\top$, we show that the largest Hilbert distance over $\left\lVert h\right\rVert_2\leq H$ is exactly $H D(S)$, where $D(S)=\max_{i<j}\left\lVert s_i - s_j\right\rVert_2$ is the Euclidean row diameter. This diameter replaces the spectral norm in the resulting Muon-style steepest descent problem. An exact solution is possible, but its direct formulation contains one $d$-dimensional vector variable for every token pair. For a vocabulary size of approximately $50$k, this means more than one billion token pairs, making the calculation impractical at every training step. We instead impose a stronger common-ball constraint and derive projected RowNorm as an $O(Vd)$ solution. For the exact RowNorm oracle, we prove that its first-order decrease is at least $1/\sqrt{2}$ of the exact diameter-constrained optimum. With Muon on the backbone, experiments across three seeds at 190M, 380M, and 640M parameters show that RowNorm reduces mean final step diameters and empirical Hilbert RMS perturbations by factors of $45$--$60$ and $12$--$15$, respectively, with only a $0.0057$--$0.0153$ increase in mean final validation loss.

Deep graph neural networks(GNNs) suffer from oversmoothing- a progressive collapse of node representation towards a low information subspace as network depth increases because the normalized graph propagation operator is repeatedly applied directly to the hidden representations. In this work we study Householder Graph Neural Network (HouseGNN). Rather than updating the hidden state like standard GCN, HouseGNN uses the aggregated neighbourhood message solely to estimate a reflection direction; the node embedding is then updated by a Householder reflector followed by GroupSort, yielding a piecewise orthogonal layer that preserves Euclidean norm at every node and at every depth. We prove three core properties: (i) every internal layer preserves the node-wise Euclidean norm; (ii) the Householder reflector is scale scale and sign-invariant in the message; and (iii) pairwise distance between nodes can change through mismatch between node-wise orthogonal operators.

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_βu=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_β$ is a positive elliptic spectral multiplier of order $β$. Given a parameter set $Θ_n=\{θ_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(Θ_n)$ by the discrete residual on the collocation points $\{η_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(Θ_n)}\frac1m\sum_{i=1}^m\left(f(η_i^)-\mathfrak L_βv_n(η_i^)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+β$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} |u-u_{n,m}|{\mathcal H^β(\mathbb S^d)}\eqsim|f-\mathfrak L_βu{n,m}|{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} |f|{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\ |f|{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} |v_n|{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}|v_n|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).

Sharpness-Aware Minimization (SAM) aims to improve generalization by encouraging insensitivity to small, worst-case parameter perturbations. However, the notion of a "small" perturbation is inherently geometry-dependent: while existing SAM variants have explored a wide range of choices, a clear perspective on which geometries are most effective in practice remains elusive. Recent work on matrix-aware optimization, particularly the Muon optimizer, suggests that respecting the matrix structure of hidden-layer weights can lead to strong empirical performance. Motivated by this, we study matrix-aware geometry in both stages of SAM: we introduce a layerwise spectral inner perturbation for matrix-valued hidden-layer parameters and combine it with either AdamW/SGDW or Muon in the outer update. Across ImageNet-1K experiments on ViT-Small/16 and ResNet-50, we find that the combination of a spectral inner step with a Muon outer step performs consistently strongly, achieving the best validation accuracy on both models among the evaluated methods.

Reducing toxicity is often framed as a global alignment problem, yet perceptions of harmful language are subjective and context-dependent. We present the first comparative evaluation of training-free methods for aligning language generation to user-specific toxicity sensitivities across three inference-time intervention stages: pre-decoding (prompt conditioning and rewriting), in-decoding (token, logit, and representation steering), and post-decoding (candidate re-ranking). Evaluated against toxicity sensitivity targets derived from the PRISM dataset, all methods reduce alignment error by 28-47%. However, the results reveal a fundamental trade-off between alignment effectiveness, personalization, and general language quality, showing how toxicity sensitivity alignment is an inherently multi-objective problem.

We establish mean-square and concentration bounds for stochastic approximation (SA) with arbitrary norm contractive mappings, under a multiplicative noise model where the noise may scale affinely with the norm of the iterates, and the iterates are potentially unbounded. These settings arise in reinforcement learning, where operators are often contractive in the $\ell_\infty$ norm and the noise scales with the iterates. To address the arbitrary norm, earlier works replace the non-smooth squared norm with a smooth Lyapunov function constructed via the generalized Moreau envelope. For concentration analysis, these works handle multiplicative noise and unbounded iterates through a multi-stage bootstrapping argument that starts from a time-varying worst-case bound and iteratively refines it. We instead present a unified and elementary analysis that yields both bounds. Using an averaged noise sequence and corresponding auxiliary iterates, we obtain a one-step Lyapunov drift inequality for the normed error directly, without smoothing the norm or constructing an envelope. For the mean-square bound, we combine this drift inequality with an induction argument showing that the iterates remain bounded in expectation. For the concentration bound, we develop a probabilistic induction over a sequence of "good" events on which the iterates are controlled, allowing the standard Azuma-Hoeffding bound to be applied. Our approach yields the first sub-Gaussian tailed maximal (all-time) concentration bound for SA under multiplicative noise, by allowing the stepsize to depend logarithmically on the confidence level. Beyond the specific setting considered here, we discuss the generalizability of these proof techniques to other noise models and iterative algorithms.

Muon-style optimizers apply a polar map to matrix momentum, but their updates also depend on the representation of each parameter block before orthogonalization. We study this representation choice as a form of optimizer geometry and introduce {\method}, a family of Muon-style optimizers that shares the same momentum rule and Newton--Schulz backend across native, nearest-square, skinny, and vector representations. Each Frobenius-isometric representation induces a distinct polar steepest-descent geometry, in which the shorter matrix dimension determines the number of supported singular channels, the pullback scaling, and the constants in stochastic nonconvex convergence bounds. In a teacher--student model, curvature collapse and an isotropic Marchenko--Pastur spectral profile connect early-stage dissipation to the represented nuclear-to-squared-Frobenius norm ratio. Pretraining experiments on LLaMA2-130M and LLaMA2-600M, together with fixed-momentum diagnostics, show that balanced non-native representations can match the performance of the native representation, whereas reducing the shorter dimension weakens the scaling and singular-channel support, leading to behavior that increasingly resembles normalized momentum.

We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold. The latent vectors are sampled either uniformly from the sphere or from a standard Gaussian distribution. Although every edge appears with probability $p$, the edges are dependent through their shared latent vectors. For the spherical model, at the connectivity scale $np=Ω(\log n)$, we prove $\|A-\mathbb E A\|=O\left(\sqrt{np\log n}+npτ\right)$, with high probability, where $τ$ is the cap threshold. This sharpens the spectral norm bound of Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions. An analogous result holds for the Gaussian model after removing the fluctuations of the vector norms, yielding improved global synchronization guarantees for the homogeneous Kuramoto model. We then recover the latent geometry from the leading eigenspace. When $np\gg\log n$, both the latent vector and relative Gram matrix errors vanish provided $d\ll np\log(1/p)/\log n$. The required lower dimension is only $d\gg\log(1/p)$ for the spherical model and $d\gg\log^2(1/p)\log n$ for the Gaussian model, improving the recovery guarantees of Li and Schramm (2023). Finally, we prove the first exact recovery result for the Gaussian mixture block model of Li and Schramm (2023). At the optimal connectivity scale $np=Ω(\log n)$, a polynomial-time semidefinite program exactly recovers all labels in a moderate-separation regime, whereas larger separation makes exact recovery impossible because isolated vertices appear with high probability. Our proofs combine orthogonal polynomial expansions, decoupling, and matrix concentration, avoiding the trace-moment arguments used in previous work.

Which properties of a partially trained network are causally portable to a different, independently trained network? Single-trajectory interventions show necessity within one run, not portability across runs. We introduce cross-trajectory chimera interventions: given two runs from different seeds, we split each weight vector into a norm and a unit direction, recombine one run's norm with the other's direction, and continue training. On two modular-arithmetic tasks that grok, the components dissociate. Direction carries a transferable, donor-specific circuit identity: implanting a donor's direction at the recipient's norm drives the run to the donor's circuit in 40/40 cases, while an angle-matched random control yields no shift. The transfer is threshold-like, and its location is predicted by the recipient's norm, separating perfectly by norm class over all 20 pairs (joint permutation probability 1.9e-4). Norm carries only a modest, distributed delay effect and no identity signal. An adaptive bisection procedure localizes the threshold to +/-1/64. Direction indexes which solution a trajectory approaches; norm governs how susceptible that identity is to being overwritten.

Reinforcement learning (RL) post-training improves the reward alignment of flow-based generators, but often degrades perceptual quality in ways that are not captured by the reward proxy. We identify a simple structural signature of this drift: across three post-training methods (NFT, AWM, DPO), RL fine-tuning inflates the per-step velocity norm $\|v_θ\|$ by $5\%$ to $15\%$ relative to the reference. A form of norm inflation has been studied in classifier-free guidance (CFG), where rescaling the velocity back to a reference norm at inference time can mitigate the resulting artifacts. However, this inference-time correction does not transfer cleanly to RL: rescaling $v_θ$ to match $\|v_{\text{ref}}\|$ at inference time neither improves reward nor fixes the quality degradation, because the inflation is co-adapted into the model weights. Furthermore, an adjoint sensitivity analysis shows that velocity magnitude rescaling carries no coherent first-order reward signal at the batch level, indicating that suppressing norm inflation is unlikely to remove a consistently reward-carrying component. Since inference-time renormalization fails while norm suppression carries no reward cost, training-time intervention is the appropriate strategy. Together, these findings motivate NormGuard, a hinge penalty that activates only when $\|v_θ\|$ exceeds $\|v_{\text{ref}}\|$ and composes additively with any velocity-local base loss. Across two base models, three post-training methods, and two reward proxies, NormGuard consistently improves MLLM-judged image quality and forensic realism while preserving reward, with gains that amplify under few-step inference and are not explained by early stopping.

We study the stability of a classical family of metrics defined over functions' Gaussian scale-space representations, focusing on the comparison of images (functions of two variables). These metrics have precedents both in harmonic analysis, specifically the theory of Besov spaces, and in classical methods of image processing; special cases are also known to be metrically equivalent to certain Wasserstein distances. We quantify these metrics' robustness to geometric deformations, and introduce rotationally-invariant versions that are stable to changes in angle when comparing tomographic projections. We also describe computationally efficient algorithms for evaluating the metrics from finite samples, and prove their robustness to additive noise. The results are illustrated through numerical experiments.

The global minimum-variance portfolio (GMVP) is the canonical decision built from an estimated covariance matrix, yet covariance estimators are universally evaluated by matrix-norm loss, which is not the object the decision depends on. We characterise exactly how covariance-estimation error maps into GMVP suboptimality. We prove an exact regret identity and a non-asymptotic bound showing decision regret depends on the estimation error only through its action on the portfolio weights, scaled by portfolio concentration and the conditioning of the true covariance. From this we derive the decision geometry: GMVP regret is invariant to a (p-1)-dimensional projection of the p^2-dimensional error matrix, with invariance to the covariance-scale direction as an exact special case. We then apply the framework to heavy-tailed returns (tail index kappa in (2,4)), establishing the regret convergence rate implied by the centred operator-norm rate, and confirm the theory on a skew-t/t-copula simulation design with pre-registered analysis. The decision-focused advantage is a sharper constant and a concentration discount rather than a faster rate; we report an honest high-conditioning boundary of the rate prediction. The results complement recent decision-focused learning approaches by supplying the exact estimation geometry and consistency theory they lack.

Common first-order optimizers, such as Adam, implicitly treat each parameter block as an unstructured vector, which disregards the multilinear weight structure present in many modern machine learning models. Recent work has shown that exploiting matrix structure can improve optimization dynamics. A notable example is Muon, which performs steepest descent under the spectral norm constraint. We take the next step and introduce Tensorion, a tensor-aware optimizer that extends Muon's constrained optimization perspective from matrices to higher-order tensors. Tensorion is built around a linear minimization oracle (LMO) over a tensor norm ball. The norm is carefully chosen to balance two objectives: tightly bounding the tensor spectral norm, while still keeping the LMO tractable. This LMO becomes computable because it reduces to operations on adaptively selected unfolding matrices. Notably, when restricted to order-2 tensors (i.e., matrices), Tensorion recovers Muon exactly. Experiments on tensor-based computer vision problems suggest that Tensorion can offer improved convergence behavior and more stable gradient updates compared with Adam-based and existing tensor-aware baselines in the evaluated settings.

Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object. Yet every weight matrix carries two distinct quantities -- a \emph{magnitude} and a \emph{direction} -- and all optimizers stepping in the matrix as a whole couple their dynamics: the directional change from an update depends on the current magnitude, while the magnitude drifts as a byproduct of learning the direction, so neither is governed directly by the learning rate. Typical training therefore leans on surrounding recipes such as weight decay and warmup to keep learning stable at scale, though these regulate the coupling only indirectly; other recent methods instead constrain the weight to a fixed-norm sphere, but add no learnable magnitude, leaving scale control to normalization layers alone. We propose \emph{Magnitude--Direction (MD) Decoupling}, an optimizer modification that factorizes each weight into a fixed-norm direction on a hypersphere and learnable per-row and per-column magnitude gains, updated at separate learning rates, all while the model still sees a single fused weight tensor. The method is agnostic to the base optimizer and removes the need for weight decay and warmup. Across both Adam and Muon, MD Decoupling improves on well-tuned baselines, transfers the optimal LR across model width without retuning, and continues to help at scale on large Mixture-of-Experts (MoE) models. Treating magnitude and direction as separately controlled quantities thus yields more predictable training dynamics and a simple, broadly applicable improvement to modern optimizers.

Motivated by an application in machine learning optimization, this paper focuses on the challenges of sampling a matrix uniformly from the unit spectral norm ball. It is proven that all singular values of sampled matrices converge to 1 almost surely as the matrix dimensions increase. This result provides the theoretical justification for a proposed simple sampling method applicable for large dimension sizes matching matrices found in modern large language models. Experimental results demonstrate both the convergence of the singular values, as well as the exact and proposed approximate sampling methods.

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a $\log\log/\log$-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

In this work, we investigate new activation functions for achieving arbitrary-accuracy Sobolev approximation by fixed-size neural networks. We first show that any function in $W^{2,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy, measured in the $W^{1,\infty}$-norm, by a fixed-size neural network using the Elementary Universal Activation Function ($\mathrm{EUAF}$). To extend this result to $W^{s,\infty}((a,b)^d)$ for $s\in\mathbb{N}$, we introduce a smooth activation $\mathrm{DUAF}_{\infty}$ from the family of Differentiable Universal Activation Functions ($\mathrm{DUAF}_n$). We prove that any function in $W^{s,\infty}((a,b)^d)$ can be approximated with arbitrary accuracy in the $W^{s-1,\infty}$-norm by a fixed-size $\mathrm{DUAF}_{\infty}$-activated network. We further construct sigmoidal variants $\widetilde{\mathrm{DUAF}}_n$ and show that, for every $1\leq s\leq n$, fixed-size $\widetilde{\mathrm{DUAF}}_n$-activated networks still approximate any $f\in W^{s,\infty}((a,b)^d)$ with arbitrary accuracy in the $W^{s-1,\infty}$-norm. In all these results, the width and depth bounds are computed explicitly, and the proposed activations are elementary.

Modern deep learning optimization features heterogeneous parameter structures, noisy gradients, and highly nonconvex landscapes, posing significant challenges for both algorithm design and theoretical analysis. Motivated by the limitations of SGD and the success of adaptive optimizers, we propose {\it Schattor}, a family of adaptive first-order methods based on Schatten norms. Schattor unifies SGD and the recently proposed matrix-variate adaptive optimizer Muon within a single Schatten-norm-based framework. We establish dimension-free stationarity guarantees for methods in the Schattor family for stochastic matrix optimization problems via a novel matrix martingale moment bound. We also develop multi-block extensions that adaptively balance block-wise optimization progress and prove dimension-free stationarity guarantees in this more general setting.

Schatten-$\infty$ based optimizers such as Muon have shown promising empirical performance, but there remains seemingly conflicting observations regarding whether they are beneficial. We resolve this conflict by showing that the conclusion is regime dependent. Even when the objective is smooth in the Schatten-$\infty$ geometry, smaller Schatten-$p$ geometries can be optimal, specifically in the low-dimensional regime, which we show includes Chinchilla scaling. This conclusion follows from a new noise-robust acceleration result for the SODA framework for $p>2$. The same analysis explains why Muon-like methods do not require warmup, why they naturally favor large batches, and yields a batch size scaling rule for arbitrary $p$.

Non-Euclidean optimisation methods with matrix-valued updates, such as Muon and Scion, have recently shown strong empirical performance for training Transformer models, yet their theoretical advantages over Euclidean methods remain poorly understood. We address this gap in the heavy-tailed non-convex regime, where stochastic gradients have bounded $p$-th central moments, $p \in (1,2]$. We show that certain non-Euclidean methods achieve optimal sample complexity under stronger stationarity measures, while Euclidean methods incur additional dimension-dependent costs. As a consequence, for $m \times n$ matrices, Muon finds an $\varepsilon$-stationary point in nuclear norm within $\mathcal{O}\left(\min\{m, n\} \frac{Δ_1 L}{\varepsilon^2} \left(\frac σ\varepsilon \right)^{\frac p {p-1}}\right)$ samples, absorbing heavy-tailed noise without extra dimension dependence, unlike Euclidean methods. We further prove this sample complexity, including its dimension dependence, is optimal for all first-order methods under nuclear-norm stationarity. Experiments on large language models support our theory. Surprisingly, our results suggest that other Schatten geometries beyond the spectral geometry of Muon can perform competitively in certain settings.

Optimal Transport has become recently a powerful method for domain adaptation by aligning source and target distributions. We study a supervised domain adaptation problem where source and target domains are related by a rotation or a translation or a homothety in $\mathbb{R}^2$. We prove that the optimal transport map recovers the underlying map when using a $p-$norm cost with $p \geq 2$. Based on this insight, we develop a method combining $K-$means and optimal transport to estimate the underlying map, enabling adaptation of linear regression models when target data is scarce. Simulations demonstrate improved performance over baseline methods. Rather than relying on highly expressive deep learning architectures, we focus on classical machine learning models to emphasize interpretability and theoretical insight. This perspective allows us to explicitly characterize the role of optimal transport in recovering geometric transformations such as rotations, translations, and homotheties. Our contributions include a theoretical result linking optimal transport and rotations, translations and homothecies in $\mathbb{R}^2$, and a practical method for adaptation in linear regression offering both conceptual clarity and applied value in domain adaptation tasks in this space.

Bernstein--Schur kernels are products of a finite-feature kernel and a completely monotone shift-invariant kernel: nonstationary kernels falling between the shift-invariant and dot-product templates random features exploit, so neither Bochner sampling nor polynomial sketching applies to the full kernel directly. We give one random-feature construction for the whole class that randomizes both factors: it sketches the finite modulation and samples the radial factor's one-dimensional Bernstein--Widder scale before applying Gaussian random Fourier features, giving feature dimension $Dm$, free of the $O(d^2)$ size of the exact modulation feature. With the modulation kept exact (the $m\to\infty$ limit), we prove unbiasedness, an exact variance, and a matrix-Bernstein operator-norm bound controlled by the top kernel and modulation eigenvalues and an intrinsic dimension rather than the crude $N\max_{ij}$ route. Whitening this argument at the ridge makes the effective dimension $d_{\mathrm{eff}}(λ)$ the \emph{exact} intrinsic dimension of the matrix variance, so $O((1+\|P\|_{\mathrm{op}}/λ)\log(d_{\mathrm{eff}}/δ))$ radial draws preserve the kernel-ridge solution; tilting the draw by a closed-form whitened leverage improves this to the effective-dimension count $O((1+d_{\mathrm{eff}})\log(d_{\mathrm{eff}}/δ))$. Conditioning on the sketch carries every guarantee to the deployed doubly-randomized estimator up to one additive sketch term, and all hold for the whole class with the modulation Gram in place of the polynomial one. The flagship instance is the biased $yat$-kernel $k_{yat,b}(w,x)=(w^\top x+b)^2/(\|w-x\|^2+\varepsilon)$, whose family span contains the inverse-multiquadric kernel by finite differences in $b$.

Recent work has sought to understand Large Language Models (LLMs) reasoning, yet a principled, model-intrinsic signal that captures its layer-wise reasoning dynamics remains underexplored. We bridge this gap by demonstrating that the l2 norm of hidden states serves as an endogenous signal of the model's reasoning intensity. Using Sparse Autoencoders (SAEs) as a diagnostic probe, we observe that LLMs' internal reasoning is marked by a sharp increase in reasoning feature activations concentrated in late layers. Motivated by this pattern, we establish a formal link between reasoning intensity and the model's latent geometry and theoretically prove that the l2 norm of hidden states bounds the activation strength of SAE reasoning features. Empirical correlation analysis and causal interventions further validate the l2 norm as a faithful indicator, where heightened norms consistently correspond to critical reasoning steps. We then introduce three test-time scaling techniques guided by l2 norms: (i) Adaptive Layer-wise Reasoning Recursion, (ii) Endogenous Reasoning State Steering, and (iii) l2-guided Response Selection, which requires no additional training or data and is compatible with advanced inference engines. Experiments across model architectures and benchmarks show that l2-norm-based techniques significantly improve reasoning performance, offering a principled yet simple lens to perceive and control LLM latent reasoning dynamics. Our code is available at https://github.com/zjy1298/The-Tell-Tale-Norm.

The robustness of deep neural networks is crucial for safety-critical deployments, yet existing evaluation methods are often attack-dependent and lack interpretability. We propose a principled, attack-agnostic robustness metric based on the spectral norm of the Fisher Information Matrix (FIM), which quantifies the worst-case sensitivity of the model's output distribution to input perturbations. Theoretically, we establish that the FIM equals the variance of the input Jacobian and derive closed-form spectral bounds for common architectures, including VGG, ResNet, DenseNet, and Transformer, providing the first theoretical robustness ranking. To enable scalable evaluation, we develop efficient algorithms, including power iteration and Hutchinson-based estimation, that support both white-box and black-box settings. Extensive experiments across multiple datasets, including CIFAR, ImageNet, and medical images, and across multiple architectures show a strong correlation between our metric and adversarial vulnerability. Our framework serves as an interpretable diagnostic tool that complements attack-based evaluations, offering insights into architectural sensitivity and guiding the design of more robust models. Code is available at: https://github.com/franz-chang/SRP/.

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

The existing optimizers for deep neural networks (DNNs) typically rely on either the $\ell_2$ norm or the $\ell_\infty$ norm, resulting in optimizers that do not adapt well to substantial changes in curvature across parameter dimensions. Generally, the training process of DNNs often exhibits strong curvature anisotropy in the early period, whereas in the later period, the training process of DNNs tends to move toward flatter regions with weaker anisotropy. Particularly, optimizers based on the $\ell_2$-norm are usually dominated by high-curvature directions, restricting updates of optimizers along with lower curvature direction and thus leading to a slower convergence rate. While optimizers based on the $\ell_\infty$-norm are prone to oscillations in flatter regions, due to the coordinate-wise updates of the same magnitude. To address these two extreme cases generated by $\ell_2$ and $\ell_\infty$ norms, we propose a novel $\ell_p$-norm scheme with a dynamical value of $p$ and incorporate it into stochastic gradient descent (SGD) and SGD with momentum (SGDM), leading to two novel optimizers with better generalization performance: ${\ell_p}$-SGD (LPSGD) and ${\ell_p}$-SGDM (LPSGDM). Particularly, the resulting optimizers suppress the dominance of high-curvature directions in the early period by utilizing a large $p$ ($p>2$), followed by a gradual decrease of $p$ toward 2 to enable more stable and refined updates, where the latter process is motivated by the cosine annealing strategy. We establish theoretical guarantees of the resulting algorithms and analyze that both LPSGD and LPSGDM achieve an $O(T^{-1/2})$ convergence rate for the nonconvex setting. Extensive experiments are conducted on benchmark datasets, including CIFAR-10, CIFAR-100, and ImageNet-1K, with multiple DNNs such as VGG-11, ResNet-18, and ResNet-50.

We present improved bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) -- including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.

Muon-style optimizers take a matrix-valued momentum or preconditioned update $B = U \operatorname{diag}(σ_1,\ldots,σ_r) V^\top$ and replace it with its canonical partial polar factor $\operatorname{Pol}(B) = U V^\top$. This maps every nonzero singular value to one. MuCon is the clipped-Muon variant studied here: it applies singular-value clipping to the same Muon matrix, $D^{\mathrm{MuCon}}\_τ(B) = \operatorname{MClip}\_τ(B) = U \operatorname{diag}\bigl(\min\{σ\_i,τ\}\bigr) V^\top, \qquad τ> 0$. Thus, $\operatorname{MClip}\_τ$ denotes the mathematical clipping operator, while MuCon denotes the optimizer primitive that substitutes this clipped direction for Muon's polar direction. The Muon/MuCon scaling parameterization used in this work is called $\text{SpectralP}$: it is the hidden-matrix scaling recipe under which polar Muon or clipped MuCon directions are applied. The map $\operatorname{MClip}\_τ$ is the Frobenius projection onto the spectral-norm ball $\{X : \|X\|_2 \le τ\}$: it leaves singular values at or below $τ$ unchanged and modifies only the violating singular directions. This paper asks when the MuCon clipping step can be approximated without a full dense SVD. We record two exact identities, a polar/absolute-value formula and a scalar-root formulation leading to a rational Newton filter for the clipped positive-semidefinite factor, and identify the numerical obstruction common to both: singular values near the threshold make sign decisions and rational solves ill-conditioned. Matrix-function methods are therefore useful only when paired with stable polar/square-root primitives or explicit regularization near the clipping boundary.

The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than $2^{\sqrt{\log n}}$, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a $d^{1/4}$-approximation as a baseline. We give, to the best of our knowledge, the first polynomial-time approximation algorithm beating this baseline by polynomial factors. For the important special case of $q = 4$ it achieves a $d^{1/8}$-approximation. All previous algorithms required additional assumptions on $X$, or only surpassed the baseline for small values of $n$. Moreover, we construct sum-of-squares certificates for the $2 \rightarrow q$ norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its $q$-th moment.

Modern optimizers, like Muon, impose matrix-wise geometry constraints on their updates. These matrix-wise constraints can be unified under Linear Minimization Oracle (LMO) theory. However, all current methods impose fixed LMO geometries for the update rules, chosen by-design or empirically, which are not necessarily optimal according to the problem's geometry. We introduce a novel efficient datadriven criterion for dynamically choosing proxy-optimal update LMO geometries on individual Deep Neural Network layers. Derived in closed form from gradient and activation statistics using a single-step random feature regression surrogate model, our criterion navigates a design space interpolating from SGD to Muon updates. Moreover, integrating parameter-wise preconditioning allows our framework to recover SGD, Muon, Adam, and MuAdam as specific extrema. To make this adaptive approach scalable, we pair it with efficient computational strategies, achieving only a $\sim$ 3% runtime overhead on highly optimized baselines. As a proof of concept, we show that this data-driven optimizer beats or remains competitive with the performance of the best performing optimizer between Muon and AdamW across three different training scenarios. Ultimately, this work provides evidence that LMO geometry can be successfully and efficiently adapted from runtime data, opening a new pathway for optimizer design beyond static geometries.

We study the problem of reconstructing an unknown point in $\mathbb{R}^d$ from approximate linear queries. This setting arises naturally in applications ranging from low-dimensional remote sensing and signal recovery to high-dimensional data analysis and privacy-sensitive inference. Our main goal is to characterize the optimal reconstruction error as a function of the number of queries $T$, the ambient dimension $d$, and the noise parameter $δ$. We first analyze the limit $T \to \infty$ and show that the optimal reconstruction error converges to the explicit value $\sqrt{2d/(d+1)} δ$, which plays a role analogous to the Bayes optimal error in supervised learning. When the dimension is fixed, we show that the excess error above this limit decays doubly exponentially fast as $T \to \infty$, a rate that is significantly faster than those typically encountered in learning curves. When the dimension grows, we show that a number of queries on the order of $\exp(d)$ is necessary and sufficient to achieve vanishing excess error. Finally, we introduce and analyze an improper variant of the reconstruction problem. From a technical perspective, our main contribution is a generalization of Jung's theorem (1901). The classical theorem bounds the maximum possible radius of a set of diameter 1 and characterizes extremal bodies. Our generalization provides a robust variant that characterizes near-extremal bodies and is proved via geometric and dynamical arguments exploiting symmetry and Lie group actions.

Despite the global deployment of text-to-image (T2I) models, their safety frameworks are largely calibrated to a Western-centric default, creating significant vulnerabilities for the rest of the world. To embrace cultural pluralism and bring historically under-represented perspectives in T2I safety, we conduct localised community-centered red teaming studies in the Global South. Our two-fold approach prioritizes localization and participation, by focusing on secondary urban centers in these regions, and conducting community engagement and training workshops to contextualize local norms. As a result, we present PLACES, a dataset comprising over 26,000 examples of T2I model failures collected in partnership with universities in Ghana, Nigeria, and two regions of India (Karnataka and Punjab). Analysis of prompts collected reveals a wide-ranging diversity in socio-cultural and linguistic attributes, when compared to existing geography-agnostic crowdsourced red-teaming data. We observe unique adversarial patterns enabled by local cultural and linguistic nuances, and distinct clusters within region around specific themes, such as religion in India. Moreover, we uncover structural contextual gaps in existing safety frameworks by identifying novel harms showing normative dissonance (e.g., violating religious norms, ignoring local customs, and ominous symbolism). This work argues that expanding T2I safety requires moving beyond mere scale to incorporate deeply localised, participatory methodologies for data collection and contextualization. Content warning: This paper includes examples containing potentially harmful or offensive content.

A growing lesson from neural network optimization is that optimizer design should respect how the model is parametrized. The layerwise input-output structure of neural networks motivates scale-invariant optimizers, such as Muon and Scion, whose updates also support hyperparameter transfer. At the same time, stochastic gradient noise in deep learning is often far from sub-Gaussian and may exhibit heavy tails. These observations have shaped recent algorithmic principles for training neural networks, yet their joint theoretical consequences are underexplored. In particular, it remains unclear what dimension dependence is unavoidable for gradient-based methods given the problem class is defined by input-output norm and under heavy-tailed noise, and whether higher-order smoothness can accelerate training. We study these questions through nonconvex smooth stochastic optimization over $\mathbb R^{m\times n}$ equipped with general norms and under $p^\mathrm{th}$-moment heavy-tailed noise, where the goal is to achieve an $ε$-stationary point in the dual norm. Our first contribution is a dimension-dependent lower bound: when $\frac{\max\{m,n\}}{(\min\{m,n\})^2}$ is large enough, any gradient-based method requires $Ω(\min\{m, n\}ε^{-\frac{3p-2}{p-1}})$ oracles for the problem class defined by the spectral norm, which is a common input-output norm. We prove that a scale-invariant Scion method with the spectral norm can achieve the matching upper bound of $O(\min\{m, n\}ε^{-\frac{3p-2}{p-1}})$. To exploit higher-order smoothness, we propose a transported Scion method and improve the bound to $O(\min\{m, n\}ε^{-\frac{5p-3}{2p-2}})$ when the Hessian is Lipschitz. Finally, we incorporate heuristics into our transported method and evaluate it across multiple architectures and model sizes, demonstrating its flexibility and compatibility with neural network training.

Hyperparameter transfer across model architectures dramatically reduces the amount of compute necessary for tuning large language models (LLMs). The maximal update parameterization (μP) ensures transfer through principled mathematical analysis but can be challenging to derive for new model architectures. Building on the spectral feature-learning view of Yang et al. (2023a), we make two advances. First, we promote spectral norm conditions on the weights from a heuristic to the definition of feature learning, and as a consequence arrive at the Complete-P depth and weight-decay scalings without recourse to lazy-learning. Second, we consider a modified spectral norm that preserves the valid scaling law of network weights when weight matrices are not full rank. This enables (to our knowledge, the first) derivation of μP scalings for grouped-query attention (GQA). We demonstrate the efficacy of our theoretical derivations by showing learning rate transfer across the GQA repetition hyperparameter as well as experiments regarding transfer over weight decay.

Gradient clipping is a standard safeguard for training neural networks under noisy, heavy-tailed stochastic gradients; yet, most clipping rules treat all parameters as vectors and ignore the matrix structure of modern architectures. We show empirically that data outliers often amplify only a small number of leading singular values in layer-wise gradient matrices, while the rest of the spectrum remains largely unchanged. Motivated by this phenomenon, we propose spectral clipping, which stabilizes training by clamping singular values that exceed a threshold while preserving the singular directions. This framework generalizes classical gradient norm clipping and can be easily integrated into existing optimizers. We provide a convergence analysis for non-convex optimization with spectrally clipped SGD, yielding the optimal $\mathcal{O}\left(K^{\frac{2 - 2α}{3α- 2}}\right)$ rate for heavy-tailed noise. To minimize hyperparameter tuning, we introduce layer-wise adaptive thresholds based on moving averages or sliding-window quantiles of the top singular values. Finally, we develop efficient implementations that clip only the top $r$ singular values via randomized truncated SVD, avoiding full decompositions for large layers. We demonstrate competitive performance across synthetic heavy-tailed settings and neural network training tasks.

The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon's performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

Muon and related norm-constrained matrix optimizers have become central to large-scale learning problems. They are formulated as a linear maximization oracle (LMO) over an ambient matrix-norm ball in unconstrained Euclidean space. However, these do not generalize cleanly to manifold-valued parameters such as low-rank factorizations, orthogonality constraints, or symmetric positive definite (SPD) matrices. Naively restricting the Muon LMO to the tangent space (i) breaks quotient symmetries and (ii) couples the tangent-space constraint with an ambient norm bound, thereby obstructing closed-form solutions on various manifolds of interest. We resolve both issues with a single observation: every Riemannian metric canonically lifts a unitarily invariant Euclidean norm to an intrinsic norm on each tangent space, and the resulting intrinsic norm constrained LMO is symmetry preserving. Building on this, we introduce intrinsic Muon (iMuon), a unified framework that yields closed-form updates on the fixed-rank, SPD, Stiefel, and Grassmann manifolds for any unitarily invariant norm, including the spectral, Frobenius, and nuclear norms. We establish convergence guarantees for both deterministic and stochastic iMuon with rate constants that depend only on the manifold dimension. Notably, on the fixed-rank manifold this constant depends only on the rank, making the rate independent of factor conditioning and removing the runtime factor-rescaling required by prior work. Experiments on LoRA finetuning of LLMs, image classification, and subspace learning illustrate the efficacy of the proposed approach.

Muon fixes the \emph{direction} of every matrix-valued update at the polar factor of its momentum, while each layer's step \emph{magnitude} is addressed only by a static shape correction. We derive a dynamic per-layer scalar by adapting the LARS/LAMB trust-ratio principle to the orthogonalized setting, where the standard denominator candidates---the raw momentum norm or the polar-factor norm---either live in the wrong unit space or carry no update-scale information. The resulting method, \emph{OrScale}, uses the norm of the parameter-space direction actually applied and anchors each layer's ratio at one via a per-layer calibration, so that the Moonlight recipe (tuned for AdamW, shared with Muon via RMS matching) transfers with \emph{no additional sweep}; a component ablation confirms each design choice is individually load-bearing. Theoretically, OrScale retains a nuclear-norm $O(1/\sqrt{T})$ convergence rate for any clipped multiplier and achieves a strict layer-adaptive descent gain $κ_{\mathrm{eff}}>1$ under two conditions estimable from standard training diagnostics---a bound that predicts the gain should \emph{grow with architectural heterogeneity}. Experiments confirm the prediction: with every hyperparameter inherited verbatim from the Moonlight recipe, OrScale matches or beats Muon+Moonlight across dense 125M--1.1B FineWeb-Edu pre-training, and on a 16B-A3B mixture-of-experts model---where the logged trust ratios separate cleanly by layer class---the gap widens by an order of magnitude to $0.130$ nats ($3.8\%$ relative) at parity wall-clock cost.

Time-series anomaly detection is critical for ensuring safety in high-stakes applications, where robustness is a fundamental requirement rather than a mere performance metric. Addressing the vulnerability of these systems to adversarial manipulation is therefore essential. Existing defenses are largely heuristic or provide certified robustness only under $\ell_p$-norm constraints, which are incompatible with time-series data. In particular, $\ell_p$-norm fails to capture the intrinsic temporal structure in time series, causing small temporal distortions to significantly alter the $\ell_p$-norm measures. Instead, the similarity metric \emph{Dynamic Time Warping} (DTW) is more suitable and widely adopted in the time-series domain, as DTW accounts for temporal alignment and remains robust to temporal variations. To date, however, there has been no certifiable robustness result in this metric that provides guarantees. In this work, we introduce the first \emph{DTW-certified robust defense} in time-series anomaly detection by adapting the randomized smoothing paradigm. We develop this certificate by bridging the $\ell_p$-norm to DTW distance through a lower-bound transformation. Extensive experiments across various datasets and models validate the effectiveness and practicality of our theoretical approach. Results demonstrate significantly improved performance, e.g., up to 18.7% in F1-score under DTW-based adversarial attacks compared to traditional certified models.

Standard Importance Sampling (IS) collapses under label corruption because high-norm examples, prioritized for variance reduction, are often adversarial outliers. We formalize this misalignment using an $\varepsilon$-contamination model and propose Disagreement-Regularized Importance Sampling (DR-IS), a sub-sampling method based on loss rank-disagreement across independent proxy ensemble. We prove finite-sample concentration bounds showing that the empirical rank disagreement of bulk corrupted examples is bounded above, and that of boundary-clean examples bounded below, both at rate $O(\sqrt{\log(N/δ)/K})$ with probability $1-δ$; when the structural expectation gap $Δ'$ between the two groups is positive and the boundary-clean set is at least as large as the selected subset, these bounds certify strict separation and control the contamination rate of the selected subset. Empirically, DR-IS remains robust under targeted high-norm attacks that break magnitude-based methods such as the Error $L_2$-norm (EL2N) on benchmark datasets. DR-IS complements training-dynamics approaches like Area Under the Margin ranking (AUM), offering improved robustness in the loss-aligned regime alongside explicit finite-sample concentration certificates and a contamination bound limiting noise leakage from the statistical tail of corrupted points.

Normalization layers in neural operators usually compute statistics by uniformly averaging discrete grid values, making the normalization itself discretization-dependent and thereby a source of transfer error across different resolutions or meshes. To enable discretization robustness, we introduce a quadrature normalization family that replaces existing uniform averaging in normalization layers with numerical quadrature: QuadNorm and BlendQuadNorm. On endpoint-inclusive uniform grids, the proposed quadrature moments are $O(h^2)$-consistent across discretizations, meaning that their cross-resolution mismatch decays quadratically with grid spacing. A transfer-error bound then predicts how normalization-induced mismatch scales with both the resolution gap and network depth. The experiments show the same gap- and depth-scaling trends predicted by the transfer-error bound. On Darcy, QuadNorm delivers the best cross-resolution performance at every tested target resolution from $64^2$ to $256^2$; on real-data benchmarks, Transolver with QuadNorm achieves nearly resolution-invariant transfer. The largest gains appear on nonperiodic PDEs and nonspectral architectures, where native-resolution improvements also emerge. We also validate BlendQuadNorm, which stays close to LayerNorm behavior and serves as a conservative default for periodic FNO settings. These results identify normalization as a previously overlooked source of resolution dependence in neural operators.

We establish sample complexity results for stochastic optimization over the integers, especially with a view to understand the complexity with respect to the corresponding continuous optimization problem. We show that integer optimization can sometimes require strictly more samples and sometimes strictly smaller number of samples, depending on the structure of the objective and constraints. 1. For Lipschitz objectives over subsets of the $\ell_\infty$ ball, the statistical complexity of general stochastic mixed-integer, nonlinear, nonconvex optimization is exactly the same as stochastic linear optimization with just bound constraints. 2. For Lipschitz objectives over subsets of the $\ell_2$ ball, we show that integer optimization can require strictly smaller sample size compared to the continuous setting in a certain regime. To get to this result, we also establish tight sample complexity results for nonconvex continuous stochastic optimization which, to the best of our knowledge, do not appear in prior work. 3. For strongly convex, smooth objectives, integer optimization has high statistical complexity compared to the continuous setting. In particular, we show that integer optimization requires $Ω(1/ε^2)$ samples to report an $ε$-approximate solution, compared to the well-known $O(1/ε)$ sample complexity from the continuous optimization literature.

Sign-based optimization algorithms, such as SignSGD and Muon, have garnered significant attention for their remarkable performance in training large foundation models. Despite this empirical success, we still lack a theoretical understanding of when and why these sign-based methods outperform vanilla SGD. The core obstacle is that under standard smoothness and finite variance conditions, SGD is known to be minimax optimal for finding stationary points measured by $\ell_2$-norms, thereby fundamentally precluding any complexity gains for sign-based methods in standard settings. To overcome this barrier, we analyze sign-based optimizers leveraging $\ell_1$-norm stationarity, $\ell_\infty$-smoothness, and a separable noise model, which can better capture the coordinate-wise nature of signed updates. Under this distinct problem geometry, we derive matched upper and lower bounds for SignSGD and explicitly characterize the problem class in which SignSGD provably dominates SGD. Specifically, we compare the \emph{upper bound of SignSGD} with the \emph{lower bound of SGD}, illustrating that SignSGD effectively reduces the complexity by a factor of $d$ under \emph{sparse noise}, where $d$ is the problem dimension. Furthermore, we elevate this framework to the matrix domain, providing an equivalent optimal lower bound for the Muon optimizer, proving that extending the sign operator to matrices preserves this optimal scaling with dimensionality. Finally, we bridge our theoretical bounds to practice, demonstrating that the theoretical superiority of SignSGD accurately predicts its faster convergence during the pretraining of a 124M parameter GPT-2 model.

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

Kernel quadrature can exploit RKHS spectral structure and outperform Monte Carlo on smooth integrands, but optimized quadrature weights are generally signed and may be numerically unstable. We study whether spectral acceleration remains possible when the weights are constrained to be positive, i.e., simplex weights. In the exact-target fixed-pool setting, an evaluated i.i.d. candidate pool of size $N$ is already available and the task is to reweight it so as to approximate the kernel mean embedding. We show that this positive reweighting problem is governed not by the equal-weight empirical average, but by the random convex hull generated by the pool. Our main geometric result shows that the mean of a bounded $d$-dimensional random vector can be approximated by a convex combination of $N$ i.i.d. samples at accuracy $O(d/N)$ with high probability, sharper than equal-weight averaging in the fixed-dimensional regime. We transfer this $d$-dimensional convex-hull approximation to full RKHS worst-case error through an augmented Mercer-truncation argument. The resulting positive-weight KQ bounds consist of a spectral tail term and a finite-sample convex-hull term, yielding Monte-Carlo-beating rates in favorable spectral regimes, including near-$O(1/N)$ rates up to logarithmic factors under exponential spectral decay. We also provide a constructive Frank--Wolfe algorithm that operates directly on the pool atoms, maintains simplex weights, and admits an explicit optimization-error bound.

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have $$\Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p},$$ where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound $$\Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p},$$ where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

Strichartz inequalities are a cornerstone of the modern theory of dispersive PDEs, but their extremizers are known explicitly only in a handful of sharp cases. The non-convexity of the underlying functional makes the problem hard, and to our knowledge no systematic numerical attack has been attempted. We propose a simple neural-network-based pipeline that searches for extremizers as critical points of the Strichartz ratio, and apply it in three settings. First, on the Schrödinger group we recover the Gaussian extremizers of Foschi and Hundertmark--Zharnitsky in dimensions $d=1,2$ to within $10^{-3}$ relative error, with no analytical prior. Second, on $59$ further admissible pairs in $d=1$ where the answer is conjectural, the method consistently finds Gaussians, supporting the conjecture that Gaussians are the universal extremizers in the admissible range. Third, on the critical Airy--Strichartz inequality at $γ=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;α,1,0,0)$ with growing internal frequency $α$, and the discovered ratio approaches the Frank--Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\simα^{-0.9}$. We confirm the same picture with an independent Hermite-basis ansatz. We propose a precise conjecture: the supremum equals $\widetilde A_{q,r}$ and is approached, but not attained, along the breather family. The pipeline thus serves both as a validator on known cases and as a discovery tool when no extremizer exists.

We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.

The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose probability distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \geq 2$, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator, and no supremum evaluation is necessary. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics-informed neural networks (SV-PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks, the underlying principle is independent of the trial space and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher-order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second-order linear elliptic problems spanning high-frequency and multi-scale solutions, indefinite operators, variable coefficients, and non-standard domains, in which SV-PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L-BFGS steps.

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove that $\mathcal{G}$ can be uniformly approximated in $H^{t}$-norm by a neural operator with $\mathcal{O}(\varepsilon^{-d/s})$ trainable parameters. This yields an explicit complexity--error relation of the form $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}$. We then study the one-dimensional viscous Burgers solution operator $\mathcal{G}: u_{0}\mapsto u(\cdot,1)$ on a bounded $H^{1}$-ball and train FNOs with an $H^{1}$-loss. Across a sweep of model sizes, we obtain test $H^{1}$-errors down to $\mathcal{O}(10^{-7})$ and relative errors of order $10^{-3}$, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law $\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}$ with empirical exponent $α\approx 1.4$, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.

Deep residual architectures are modeled as products of near-identity Jacobians. This paper proves deterministic quotient-geometric estimates for singular spectra of Frobenius-normalized layer factors, emphasizing a normalized top-radial Cartan coordinate and fitted power-law chart. Full-rank factors are mapped from $\mathrm{GL}(d)$ to the positive cone by $A\mapsto A^\top A$, then to ordered eigenvalue data. Under Frobenius normalization, exact power-law spectra form a trace-normalized Cartan orbit. This orbit is a Gibbs family on ranks, a Fisher information line, and a Bures--Wasserstein curve with line element $d/4$ times Fisher information. The main rigidity theorem is a slack-aware margin inequality: interface radial amplitude, non-backtracking slack, and signed residual variation control displacement of the fitted Cartan coordinate. In the exact-chart zero-slack case, a depth-$L$ budget gives exponent drift of order $(\log M)/L$; generally, slack and residual increments augment the bound. We separate scalar top-radial from full-Cartan spectral control, which also needs Bures/Hellinger residual variation. We prove approximate-power-law and metric-chart versions, converse lower bounds, Fisher--KL/Bures action estimates, and near-identity expansions for normalized residual chains. Near-identity results verify transport budgets; chart quality remains measurable. Effective rank is a spectral-energy quantile, giving finite-width power-law tail bounds and robust rank-window transition estimates. Empirical static-weight exponent profiles serve as diagnostics; full verification also requires interface budgets, slacks, and residuals for the same operator chain.