Spectral Regularization

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6 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Jul 13Week of Sep 28

Latest papers 31

Oct 5, 2026cs.LG

ORCA: The Annealed Spectral Conditioning Optimizer for Faster, Better LLM Training

Modern LLM optimizers such as Muon often produce weight matrices with higher effective rank than Adam, yet further spectral control has delivered only modest gains. We identify a tension behind this result: concentrated spectra can suppress gradient directions in coupled weight matrices and slow optimization, while constraints maintained throughout training can limit task-specific adaptation and raise the attainable loss floor. We introduce ORCA (Orthogonal Regularization, Cooled After), a minimal optimizer intervention that applies strong but temporary soft orthogonality regularization early in training, then removes it. This allows the weights to benefit from a broader spectrum early on and adapt freely afterward. Across LLaMA, Qwen3, and fine-grained mixture-of-experts models ranging from 130M to 8B parameters, ORCA achieves lower final validation loss than Muon. Its loss reduction relative to Muon matches or exceeds Muon's reduction relative to Adam. Ablations support the early-shaping, later-release design. Further, ORCA requires no architectural changes and adds minimal overhead.

Yuanshi Liu, Boyuan Jiang, Liang Hou +4

School of Intelligence Science and Technology, Peking University · Kling Team

Spectral Regularization Deep Learning Optimization LLM Training

Sep 30, 2026stat.ML

Principal Component Regression Dominates all Monotone Spectral Filters for Linear Regression

We compare the instance-wise, finite-sample risks of monotone spectral filters for linear regression, a broad class of estimators including principal component regression (PCR), gradient descent (GD), and ridge regression. We show that PCR dominates all monotone spectral filters: compared to any such filter, the risk of optimally tuned PCR is no bigger by a constant factor for all problems. Furthermore, the dominance is strong if the filter is separated from step functions (e.g., GD and ridge): there exist problem instances for which the risk of PCR is smaller by a polynomial factor in sample size dependence. Our comparison results show that PCR is optimal and thus admissible among monotone filters, significantly extending Wu et al. (2026)'s result that GD strongly dominates ridge. From a technical perspective, we establish new upper and lower bounds for general spectral filters, which are instance-wise sharp when specialized to ridge or GD, recovering or improving the best-known bounds.

Juno Kim, Hengyu Fu, Peter Bartlett +2

University of California, Berkeley · Google DeepMind

Spectral Regularization Linear Regression Statistical Learning Theory

Sep 28, 2026cs.LG

$λ$ -JEPA Spectral Anti-Collapse Regularization for Self-Supervised Learning

Joint-embedding self-supervised learning typically combines an invariance objective across augmented views with additional mechanisms to prevent representational collapse. These objectives are often applied after a projection head, while downstream tasks use the backbone representation before the projector. We find that this mismatch does not necessarily prevent dimensional collapse in the backbone, which can retain low effective rank and potentially limit downstream transfer. To address this, we introduce SACReg, a spectral anti-collapse regularizer motivated by an analysis of

λ

-balance, which captures the relative scale of weight matrices across layers. In a two-layer linear network, we show that (i)

λ

-balance prevents collapse, and (ii) our regularizer applied to the backbone induces

λ

-balance. In the nonlinear case, this regularizer leads to anti-collapse as well and, in realistic architectures on ImageNet100, it empirically increases the representations' ranks. We apply SACReg to JEPA and propose

λ

-JEPA, which improves over LeJEPA and VISReg on ImageNet-1k classification and in average linear-probe transfer performance across eight downstream image datasets. On video self-supervised learning,

λ

-JEPA improves over LeVJEPA and V-JEPA 2 on the Something-Something-v2 and Kinetics-400 benchmarks. Code is available at https://github.com/berkerdemirel/lambda-jepa.

Berker Demirel, Clémentine Dominé, Valentino Maiorca +3

Institute of Science and Technology Austria (ISTA) Am Campus 1, 3400 Klosterneuburg, Austria

Spectral Regularization Joint-Embedding Predictive Architecture Video Representation Learning Self-Supervised Visual Representation Learning

Sep 28, 2026cs.SD

Prox-Friendly Log-Magnitude Prior on Complex-Valued Signal

The logarithmic transform is essential in audio signal processing since human auditory perception is approximately logarithmic with respect to magnitude. However, directly incorporating prior knowledge about signals (e.g., harmonic structure) in the log-magnitude domain into optimization problems solved by standard proximal splitting algorithms remains challenging. To address this issue, this paper proposes a novel regularizer termed EPILOG (Exponential Penalty for Imposing priors on LOG-magnitude). EPILOG indirectly imposes prior knowledge on the log-magnitude of a complex-valued signal through regularization of an auxiliary variable that is shown to be linked with the log-magnitude. Furthermore, we derive its variable-wise proximity operators and develop a proximal splitting algorithm using these operators. Experiments on speech dereverberation demonstrate the effectiveness of the proposed regularizer, particularly in promoting cepstral-domain sparsity.

Kazuki Matsumoto, Keidai Arai, Kohei Yatabe

Tokyo University of Agriculture and Technology (TUAT), Tokyo, Japan

Spectral Regularization Speech Enhancement

Sep 23, 2026cs.LG

Graph Learning with Spectral Connectivity Priors for Scarce Data

Learning a sparse graph from scarce data is practically important but challenging. Motivated by the desirable combination of local sparsity and strong global connectivity exhibited by expander-like graphs, we propose spectral connectivity-regularized graph learning (SCoGL), a framework that incorporates a family of Laplacian spectral priors to explicitly promote global connectivity. Specifically, SCoGL augments a combinatorial-Laplacian-constrained graphical lasso (GLASSO) objective over a target adjacency matrix

\mathbf{W}

with a general connectivity prior computed from Laplacian eigenvalues. We derive gradients for several representative connectivity priors and develop a projected gradient descent (PGD) algorithm with Armijo backtracking to efficiently optimize

\mathbf{W}

. Experiments show that the proposed SCoGL variants improve graph recovery and enhance downstream tasks such as graph signal denoising when signal observations are scarce.

Mingxiao Liu, Bahar Oveisgharan, Bingyan Zou +3

Tsinghua University, China · York University, Canada

Graph Signal Processing Graph Structure Learning Spectral Regularization

Sep 9, 2026stat.ML

Why Learning Rediscovers the Closed-Form Diagonal Regularizer

We identify a diagonal saturation principle in modal inverse problems: when truncation noise is isotropic, the Bayes-optimal Tikhonov shape is a closed-form power law Gamma_k proportional to lambda_k^|s| set by the prior alone, independent of the domain. Berry's random-wave conjecture decorrelates the truncation noise across modes, and Weyl's eigenvalue counting law supplies enough modes for the conclusion to survive empirical Berry violations. Together they predict an approximately flat loss landscape across the per-mode family, leaving narrow scope for a diagonal regularizer to robustly beat the closed form. On FEM-simulated acoustic rooms, the closed form is near-optimal relative to per-room oracle tuning across observation windows, and three diagonal architectures trained on the same data match its reconstruction error within 1 pp despite learning qualitatively different spectra. The framework extends to heat diffusion via a known exponential Green's function correction with no new free parameters. Saturation is restricted to the diagonal family: Learned Iterative Ridge crosses the boundary by exploiting cross-mode coupling, locating where learning starts to help.

Jeahn Han, Pyojin Kim

GIST

Spectral Regularization PDE Inverse Problems Inverse Problems

Sep 7, 2026cs.LG

Think Wider: Mitigating Latent Rank Collapse in Implicit Chain-of-Thought Reasoning

Chain-of-thought (CoT) reasoning improves the reasoning ability of large language models by introducing intermediate computation, but explicit rationales increase decoding length, latency, and context cost. Implicit CoT offers a more efficient alternative by moving intermediate reasoning into continuous latent states. However, latent reasoning can be unstable: successive latent states may become overly similar and collapse toward a shared dominant direction, reducing the diversity of the reasoning trajectory. In this work, we identify

\textit{latent rank collapse}

and propose

\textbf{WIDER}

, a lightweight spectral regularizer for implicit CoT. During training, WIDER estimates the shared direction of each latent trajectory and penalizes projections onto this direction, encouraging latent states to span a broader representational subspace. The method is plug-and-play and leaves the backbone model, latent schedule, and inference-time decoding procedure unchanged. We further formulate this collapse as a geometric bottleneck in implicit reasoning, casting its mitigation as a training-time regularization problem rather than an inference-time decoding change. Extensive experiments show that WIDER improves matched implicit CoT baselines, while mechanistic analyses reveal higher effective rank, lower dominant-direction energy, and reduced redundancy among latent steps. These results highlight latent subspace utilization as an important factor for efficient continuous reasoning, providing a geometric perspective for analyzing and improving implicit CoT. Code is available at https://github.com/whitesweater/WIDER.

Yuwen Hao, Menglin Yang

The Hong Kong University of Science and Technology (Guangzhou)

Spectral Regularization Continuous Latent Reasoning Representation Geometry in Language Models Latent CoT Reasoning

Sep 2, 2026cs.LG

Spectral Initialization and Scheduled Graph Smoothness for Uncertain Knowledge Graph Completion

Uncertain knowledge graphs (UKGs) extend knowledge graphs by assigning each triple a continuous confidence score. Since most possible triples lack observed confidences, recent methods rely on semi-supervised learning to generate pseudo-labels. These methods initialize entity embeddings without using the confidence-weighted graph, discarding its global community and hub structure. We introduce QUEST, which adds no trainable parameters to the standard confidence-distribution learning pipeline. First, QUEST initializes entity embeddings using the smallest non-trivial eigenvectors of the confidence-weighted graph Laplacian, incorporating community and hub structure before training. Second, QUEST applies an unbiased mini-batch Dirichlet energy regularizer to enforce early-stage structural consistency. On two UKG datasets, QUEST improves confidence prediction and link prediction on six of eight metric-dataset pairs over prior methods and matches the previous best on the remaining two, while removing the instability spike observed on dense graphs. These results indicate that spectral structural priors combined with a graph Dirichlet energy regularizer improve accuracy, training stability, and checkpoint reliability in UKG completion.

Md Abrar Jahin, Taufikur Rahman Fuad, Jay Pujara +1

University of Southern California · Islamic University of Technology

KG Completion Link Prediction Graph Embedding Spectral Regularization

Jul 27, 2026stat.ME

Spectral Truncation in Synthetic Control

Synthetic control (SC) matches a treated unit's pre-treatment trajectory to a weighted combination of donor units. We study Spectral SC, which instead matches the treated unit in coordinates defined by the leading temporal singular vectors of the donor panel, and a hybrid estimator that places separately tunable weight on retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints. We prove that the family reduces exactly to raw-path SC at full rank, that exact balance on

K

retained dimensions with

N_0

donors is underdetermined whenever

N_0>K+1

, with an affine solution set of dimension

N_0-K-1

, and that spectral imbalance maps to treatment-effect bias through a finite-sample best-linear-predictor decomposition. We evaluate the estimators across eleven data-generating regimes, using

400

replications per regime and donor-only placebo validation to select regularization and the mixing weight. Truncated Spectral SC has significantly higher RMSE than tuned raw-path SC in every regime, with paired differences equal to

4

to

11

Monte Carlo standard errors. The hybrid estimator selects raw-path matching in most replications and is statistically indistinguishable from tuned SC in most regimes. The result is highly sensitive to preprocessing. With raw inputs, the performance gap is large; after removing unit and time fixed effects before spectral decomposition, as suggested by the assumptions behind our bound, the gap nearly disappears and placebo validation begins to favor truncation. We interpret these findings diagnostically rather than as evidence that Spectral SC should replace raw-path SC. Basis-estimation noise, balancing underdetermination, and fixed-effects contamination determine when spectral matching can help.

Mojtaba Eslami

University of Calgary

Spectral Regularization Causal Effect Estimation Spectral Methods

Jul 24, 2026cs.LG

Spectral-Aware Analytic Class-Incremental Learning for Long-Tailed Distributions

Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches. Recent ACL methods are based on Recursive Least Squares (RLS) and have achieved the state-of-the-art results compared to other alternatives. However, they falter significantly in Class-Incremental Learning scenarios characterized by Long-Tailed distributions. While the ill-conditioning of the autocorrelation (Gram) matrix is a known limitation of RLS, we demonstrate that class imbalance exacerbates this issue into a distinct spectral pathology: "tail" classes suffer from severe spectral collapse, rendering their subspaces numerically indistinguishable from noise. Standard Ridge Regression (

L_2

) fails to address this effectively as it applies isotropic regularization - a uniform penalty that is insufficient to stabilize the tail without over-shrinking the head. To address this, we propose Geometry-Spectral Rectification (GSR), a theoretically grounded framework that treats long-tailed learning as a spectral regularization problem. Unlike standard isotropic regularization (Ridge) which uniformly penalizes all eigenvalues, GSR acts as an anisotropic spectral filter, selectively inflating the collapsed eigenvalues of tail classes. We construct a structured, data-dependent spectral perturbation matrix

Δ

that selectively inflates collapsed tail eigen-directions of the Gram matrix. Theoretical analysis proves that GSR guarantees an improved stable rank for the Gram matrix, ensuring numerical stability. Extensive experiments show that GSR establishes a new state-of-the-art for analytic CIL, offering a superior trade-off between computational efficiency and robust generalization in long-tailed settings.

Quyen Tran, Hai Nguyen, Quan Dao +4

1Rutgers University · 2Tuft University · 3HUST +1

Class-Incremental Learning Spectral Regularization Long-Tailed Classification Continual Learning

Jul 24, 2026cs.LG

Beyond Negative-Ridge Endpoints: Mixed-Sign Spectral Regularization via Negative-Shifted Gradient Descent

In overparameterized linear regression, many weak spectral directions act like a ridge penalty on the signal-bearing spectrum; negative ridge is the natural correction, pushing filters above one. The stable negative-ridge endpoint, however, is structurally limited: its pole must stay below the smallest nonzero empirical eigenvalue, and it anti-shrinks smaller eigenvalues more than larger ones. Early-stopped negative-shifted gradient descent escapes this constraint. Its filter is smooth at the would-be pole and mixed-sign-capable: above-ridgeless directions form a leading prefix, with lower directions shrunk or exposure-controlled while stopping sets the crossover. In a Gaussian spike-plus-flat model we discover a Marchenko-Pastur barrier: the shift that cancels the implicit penalty lies a bulk width above the smallest empirical eigenvalue, and the stopped path improves on every admissible endpoint by a polynomial factor in risk under explicit conditions. Our main theorem permits a general high-effective-rank tail: its trace sets the implicit floor, its squared spectrum controls exposure, and the floor-critical path recovers all head scales at once, beyond positive shrinkage and, once scales separate, every uniform rescaling of ridgeless. Handling the noncontractive shifted dynamics is the central technical challenge; localized Duhamel integrals control them. A finite-grid hold-out inequality transfers the separations to the validation-selected algorithm.

Peng Zhao

Department of Applied Economics and Statistics · University of Delaware

Spectral Regularization Early Stopping Gradient Descent Linear Regression Ridge Regression Implicit Regularization

Jul 23, 2026cs.LG

Regularized Optimization on Grassmann Manifold: Theory, Algorithm and Applications

Spectral methods are among the most widely used techniques for community detection, clustering, and graph learning. Their performance, however, critically depends on the accurate estimation of the underlying spectral subspace and can deteriorate substantially in the presence of noise, outliers, or model perturbations. To address this limitation, we propose a Regularized Projection Matrix Approximation (RPMA) framework for robust estimation of rank-

K

projection matrices. RPMA extends classical spectral projection by incorporating a regularization term, producing projection estimates that are more robust, sparse, and interpretable. We formulate the proposed model as an optimization problem on the manifold of rank-

K

projection matrices and exploit its geometric equivalence to the Grassmann manifold. Based on this manifold characterization, we derive the first- and second-order optimality conditions, establish the local stability of the regularized leading eigenspace, and characterize the stability of the critical-point landscape under sufficiently small regularization. To efficiently solve the resulting nonconvex optimization problem, we develop a Riemannian gradient projection algorithm with backtracking line search, together with a more efficient Cayley--Sherman--Morrison--Woodbury (Cayley--SMW) gradient method that avoids repeated eigendecompositions. Extensive experiments on both synthetic and real-world datasets demonstrate that RPMA substantially improves the recovery accuracy of projection matrices and consistently outperforms conventional spectral projection methods for community detection and clustering under noisy environments.

Zhuan Liang, Zheng Zhai

Department of Statistics, Faculty of Arts and Sciences at Beijing Normal University, Zhuhai

Spectral Regularization Riemannian Optimization Nonconvex Optimization Graph Clustering Spectral Methods

Jul 22, 2026cs.LG

An Isotropy-Preserving Spectral Cap for Muon: Theory and Three Case Studies

Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward faster (t^{1/2} versus t^{1/4}). We further observe that the spectral-norm perturbation has a non-negative second-order term. This implies that a lightweight "spectral cap" -- which projects out only the first-order growth of the single top singular direction from each update -- can control the output covariance W K_X W^T without freezing training: the weight keeps learning through non-top directions, top-direction rotation, and top switching. We relate this cap to the min-entropy (H-infinity) of the singular-value spectrum. We then study three systems trained with Muon: a nanoGPT feed-forward projection, a 64-expert mixture-of-experts router, and the query/key projections of a bf16 FlashAttention block. In each case the cap increases isotropy and, at the margins -- a router collapsing to a single expert, and the near-divergence of one attention head -- prevents a concrete failure, while leaving validation loss essentially unchanged. We emphasize that the scale-invariance assumption is strong and that these small-scale results are preliminary; comments are welcome.

Jiachun Li

Spectral Regularization Muon Optimizer Neural Network Training Dynamics

Jul 13, 2026cs.LG

How to Tame Grokking: Representation Geometry as a Control Signal

Grokking is a phenomenon in which neural networks initially memorize training data and only later exhibit strong generalization after prolonged optimization. Despite extensive recent study, the factors influencing the emergence and timing of grokking remain incompletely understood. We investigate the relationship between representation geometry and delayed generalization. We find that dimensionality collapse consistently precedes the onset of grokking in all evaluated settings. Motivated by these observations, we introduce Geometric Dimensionality Regularization (GeomDR), a simple spectral regularizer that modifies the effective dimensionality of hidden representations during training. Across modular addition, modular division, and permutation composition tasks, GeomDR consistently alters grokking dynamics and can substantially accelerate the onset of generalization depending on the intervention schedule and target dimensionality. In several settings, grokking is accelerated by up to 52 times relative to standard AdamW training. Similar qualitative effects are observed in both multilayer perceptrons and transformers. Together, these results suggest that representation geometry can serve as an effective control signal for grokking and provide evidence that geometric interventions offer a practical approach for studying and influencing delayed generalization in neural networks.

Maksim A Kazanskii

Independent Researcher

Neural Network Generalization Spectral Regularization Representation Collapse Neural Representation Geometry Neural Network Training Dynamics Grokking

Jul 10, 2026stat.ML

Spectrally Deconfounded Gradient Boosting

Flexible machine-learning methods can be sensitive to hidden confounding: they may learn associations induced by unobserved confounders rather than stable signals. Spectral deconfounding mitigates this problem by shrinking high-variance directions of the covariate matrix that, under dense confounding, carry latent confounder information. Existing work has largely focused on linear models. We develop a nonlinear spectral deconfounding framework for gradient boosting. Our approach replaces the ordinary squared-error loss by a spectral loss, which alters the boosting dynamics by slowing down learning in confounding-aligned directions. We show that deconfounding is not achieved by the spectral loss alone, but by the interaction between spectral shrinkage and regularization, especially in terms of early stopping. Moreover, we provide a mixed-model interpretation that connects LAVA-type shrinkage to random-effects adjustment and yields an empirical-Bayes procedure for tuning the spectral loss. We also extend the method to general likelihoods and nonlinear confounding using Laplace approximations and kernel random effects. Across synthetic and real-world experiments, spectrally deconfounded boosting improves estimation of the target function under hidden confounding and is substantially more scalable than existing nonlinear spectral deconfounding baselines.

Andrea Nava, Peter Bühlmann, Fabio Sigrist

Seminar for Statistics, ETH Zürich, Zürich, Switzerland · IFZ, Lucerne University of Applied Sciences and Arts, Rotkreuz, Switzerland

Spectral Regularization Gradient-Boosted Decision Trees

Jul 4, 2026cs.LG

PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric

A(x)=Λ(x)U(x)

. The diagonal factor

Λ(x)

controls anisotropic scaling, while the orthogonal factor

U(x)

is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Varvara Nazarenko, Timur Lidzhiev, Alexander Tarakanov

Faculty of Computer Science, HSE University, Moscow, Russia · AI VK, Moscow, Russia

Spectral Regularization Representation Learning Metric Learning Spectral Methods

Jul 2, 2026cs.LG

Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Model

We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q

\sim

N(0, I), p

\sim

N(

Δ

, I), with linear features, where

Δ

is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.

Francis Bach

SIERRA · Inria - Ecole Normale Sup´erieure PSL Research University

Spectral Regularization Density Ratio Estimation

Jun 27, 2026math.NA

Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation

Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.

George Coote, Matthew J. Colbrook

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom

Spectral Regularization Koopman Operator Learning Dictionary Learning

Jun 22, 2026cs.LG

Differential Spectral Damping Gap Adaptive Regularization for Ill-Conditioned Kernel Methods

Kernel methods requiring matrix inversion -- particularly Least-Squares Twin Support Vector Machines (LSTSVM) -- suffer from exponential eigenvalue decay in their system matrices, producing severely ill-conditioned problems where standard Tikhonov regularization applies uniform damping regardless of eigenvector reliability. We propose Differential Spectral Damping (DSD), a regularization formula that adapts its penalty to localized eigengap structure: preserving eigenvectors with large spectral gaps (reliable per Davis-Kahan perturbation theory) while aggressively suppressing those with small gaps (directionally corrupted beyond recovery). We motivate DSD through a principled design procedure grounded in the Davis-Kahan

\sin(Θ)

theorem, systematically deriving the requirements for a reliability-aware damping function and selecting the exponential form for its smoothness, differentiability, and natural saturation properties. Through rigorous paired testing with fairly optimized baselines (including gradient-optimized Tikhonov receiving equal optimization opportunity), we demonstrate that DSD improves LSTSVM classification accuracy by +4.8 percentage points on real-world GINA (

d=970

, Cohen's

d = 4.49

,

p < 0.0001

), +10.4 percentage points at

d=200

, and +2.6 percentage points on Madelon (

d=500

) -- all using only principled spectral initialization while Tikhonov receives grid search. For pre-image reconstruction on manifold data, DSD ties Tikhonov at high perturbation noise (

p=0.99

) but slightly underperforms at lower noise levels; both reduce naive inversion error by

66\times

. We characterize the precise operating regime (

d \geq 100

, condition number

> 10^3

) and document where simpler methods suffice, providing practitioners with clear deployment guidance.

Praveg Vashishtha

Department of Computer Science and Engineering, Indian Institute of Technology Patna, India · Sopra Banking Software India

Spectral Regularization Support Vector Machines Kernel Methods

Jun 20, 2026cs.IR

The Pitfall of Scaling Up: Uncovering and Mitigating Popularity Bias Amplification in Scaling Transformer-based Recommenders

We identify a critical pitfall in scaling transformer-based sequential recommenders: while increasing model size improves recommendation accuracy, it simultaneously amplifies popularity bias. This bias drives systems to over-recommend popular items at the expense of niche ones, which not only undermines fairness but also degrades the broader ecosystem by reinforcing the Matthew effect and filter bubbles. Consequently, this bias amplification emerges as a fundamental obstacle to sustainable model scaling. Through comprehensive theoretical and empirical analyses, we uncover the root cause of this amplification. Our findings reveal that as model depth increases, the two core components of the transformer architecture, i.e., attention aggregation and feed-forward projections, synergistically induce severe spectral collapse in model predictions, which directly translates to the amplification of popularity bias. To address this challenge, we propose SPRINT (Scalable Popularity Regularization IN Transformers), which mitigates spectral collapse during scaling by constraining (i) the maximum column-sums of the attention score matrices and (ii) the spectral norms of the feed-forward parameters. Extensive experiments demonstrate that SPRINT significantly improves both accuracy and long-tail fairness. Crucially, it yields more favorable scaling behaviors when expanding model sizes from 0.05M to 0.34B parameters. The code is available at https://github.com/Tiny-Snow/GenRec.

Weiqin Yang, Yue Pan, Chongming Gao +4

Zhejiang University · State Key Laboratory of Blockchain and Data Security, Zhejiang University · College of Computer Science and Technology, Zhejiang University +2

Spectral Regularization Sequential Recommendation Algorithmic Fairness Recommender Systems

Jun 20, 2026math.NA

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative

L^2

error

O(10^{-3})

can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses

\mathrm{det} F

well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4

\times

wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

Jaemin Oh, Youngkyu Lee, Jerome Darbon +1

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Spectral Regularization PDE Solving Hyperelasticity Neural Operators

Jun 10, 2026cs.SD

Lung-SRAD: Spectral-Aware Regularized Audio DASS with Dual-Axis Patch-Mix Contrastive Learning for Respiratory Sound Classification

Recent respiratory sound classification (RSC) studies largely rely on CLS-token driven self-attention architectures such as the Audio Spectrogram Transformer (AST). While effective at modeling global context, recent analyses suggest a low-pass filtering behavior that may reduce sensitivity to localized abnormal patterns. In this work, we investigate State Space Models (SSMs) as an alternative backbone for RSC. Using the Distilled Audio State Space model, we analyze intermediate representations through spectral response curves and observe stronger preservation of mid-to-high spatial-frequency components. Based on these observations, we introduce spectral-aware layer regularization using Gaussian convolution applied to selected layers. We further propose Dual-Axis Patch-Mix contrastive learning tailored to SSM-based audio models for robust representation learning. Experiments on the ICBHI benchmark show that our approach achieves 64.48% score, outperforming the AST baseline by 5%. Code is available at https://github.com/RSC-Toolkit/Lung-SRAD.

Hemansh Shridhar, Miika Toikkanen, June-Woo Kim

RSC LAB, MODULABS, Republic of Korea · Department of Electronic Engineering, Wonkwang University, Republic of Korea · AI Convergence Research Institute, Wonkwang University, Republic of Korea

Contrastive Learning Spectral Regularization Audio Understanding Audio Classification State Space Models Respiratory Sound Classification

Jun 10, 2026cs.LG

Spectrally Regularized Latent Flow Matching for Turbulence Generation

Latent diffusion and flow matching have emerged as leading approaches for synthetic turbulence generation, yet they systematically under-represent dissipation-range amplitudes. We introduce a latent flow matching framework with a spectrally regularized compression stage that directly targets this failure mode. On a 256^2 DNS dataset at Re_f \approx 2250, replacing an MSE-trained VAE with a zone-weighted log-spectral objective raises deep-dissipation retained spectral power from 25% to 94% in reconstruction and from 20% to 79% in unconditional generation. The improved latent representation also yields a substantially better sampling cost-fidelity tradeoff: the MSE-trained latent space imposes a fundamental quality ceiling near DD bias -0.70 that no integrator or step-count can overcome, while the spectrally regularized latent space reaches DD bias -0.117 at just 20 function evaluations. Mechanistically, encoder-decoder swap experiments show that the improvement is driven primarily by encoder-induced latent reorganization rather than decoder capacity, while a support-amplitude decomposition reveals that MSE-trained models behave as conservative suppression models, minimizing pointwise error by attenuating intermittent high-wavenumber structure. Both pipelines recover the second-order structure function and the correct sign of S_3, indicating the correct cascade direction without explicit supervision. A small residual gap in the magnitude of S_3 suggests that phase-coherent triadic organization remains a complementary axis to amplitude fidelity for future generative turbulence models.

Khalid Rafiq, Aditya G. Nair

Department of Mechanical Engineering University of Nevada, Reno, NV, USA

Spectral Regularization Flow Matching Flow-Based Generative Modeling Computational Fluid Dynamics

Jun 5, 2026cs.LG

Closed-Form Spectral Regularization for Multi-Task Model Merging

Model merging combines several independently fine-tuned experts into a single multi-task model without any training data, reducing the storage, serving, and decentralized-development costs of large foundation models. State-of-the-art merging methods formulate merging as a layer-wise quadratic interference minimization problem. Although this problem admits an exact closed-form pseudoinverse solution, that solution underperforms hundreds of iterations of gradient descent in practice. The iterative loop dominates the cost of the pipeline, yet its effectiveness has remained unexplained. We revisit this regime and show that the iterative solver does not primarily act as an optimizer; rather, it serves as an implicit spectral regularizer for an ill-posed normal equation, where small-eigenvalue directions of the per-layer interference operator amplify proxy noise. Building on this finding, we formalize multi-task model merging as a noisy linear inverse problem and propose a spectral filtering estimator parameterized by a per-direction filter. We instantiate this estimator with SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent, with a hard top-K truncation that suppresses noise-amplifying small-eigenvalue directions. Furthermore, we propose SWUDI-A, an adaptive variant that replaces the global rank hyperparameter with per-layer rank rules, further improving robustness across architectures. Both variants share a single symmetric eigendecomposition per linear layer and require no training data or optimizer state. Across four general benchmarks and a multimodal merging benchmark spanning VQA, Geometry, Chart, OCR, Grounding, and modality merging, our proposed spectral solvers match or outperform state-of-the-art merging methods. Crucially, they reduce wall-clock time by 28-72x and peak GPU memory by up to 50%.

Yongxian Wei, Runxi Cheng, Xingxuan Zhang +4

Shenzhen International Graduate School, Tsinghua University, Shenzhen 518071, China · Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China · Sun Yat-sen University, Shenzhen 510275, China +1

Spectral Regularization Multi-Task Learning Spectral Methods Model Merging Implicit Regularization

Jun 3, 2026cs.LG

Low-Rank Decay for Grokking in Scale-Invariant Transformers: A Spectral-Geometric View

Modern Transformer architectures frequently employ normalization mechanisms such as RMSNorm and Query-Key Normalization, making parts of the model approximately scale-invariant with respect to weight magnitudes. In this regime, standard Frobenius-norm weight decay acts purely along the radial direction of the weight space and cannot directly simplify the function represented by the normalized layer. We study grokking in small algorithmic tasks through this lens and propose \emph{Low-Rank Decay} (LRD), a nuclear-norm-like spectral regularizer whose subgradient -- the polar factor

UV^\top

-- retains a tangential component even in the scale-invariant setting. This distinction has a concrete dynamical consequence: after the model memorizes the training set and task gradients vanish, L2 decay can no longer reshape the weight spectrum, whereas LRD continues to compress singular values in an

\ell_1

-like fashion. On modular arithmetic tasks, we find that LRD induces rapid effective-rank collapse in Query/Key matrices and expands the data-fraction boundary at which delayed generalization (grokking) occurs. We further provide a spectral-geometric interpretation through the ``needle-to-fan'' expansion of the nuclear-norm subdifferential near low-rank strata.

Mingyu Li

Beijing Normal University

Spectral Regularization Weight Decay Transformer Nuclear Norm Minimization Grokking

May 29, 2026math.NA

Stochastic Rounding Increases Small Singular Values

Over the past half-dozen years, stochastic rounding (SR) has regained significant attention as a quantization scheme for low-precision floating-point arithmetic, with applications spanning numerical analysis and modern machine learning systems. Recent work has shown that SR acts as an implicit regularizer by increasing the smallest singular value of extremely tall-and-thin (or, symmetrically, short-and-fat) matrices. In this work, we substantially sharpen and extend this understanding in two directions. First, we show that the regularization effect of SR is not restricted to extreme aspect ratio regimes: it persists for matrices with constant aspect ratio. Second, we demonstrate that SR does not merely regularize the smallest singular value, but instead lifts entire clusters of singular values at the tail of the spectrum. Together, these results provide a more general characterization of stochastic rounding as a spectral regularizer, revealing that its effects extend beyond extremal aspect ratios and act on a broader portion of the singular value spectrum.

Linkai Ma, Tingzhou Yu, Petros Drineas

Department of Mathematics University of Alberta

Spectral Regularization Implicit Regularization

May 28, 2026cs.LG

Ridge Regression from Poisson Resetting: A Renewal Perspective on Spectral Regularization

We connect stochastic resetting from non-equilibrium statistical physics with ridge regularization in statistical learning. For linear gradient flow, resetting to the origin at rate

r

produces stationary mean

(X^\top X+rI)^{-1}X^\top y

, exactly the ridge estimator with penalty

λ=r

. This uses the known Laplace-transform relationship between ridge regression and exponential-time averaging of gradient flow, with the exponential time now interpreted as the stationary age associated with Poisson resetting. We then extend this identity to general renewal reset laws: the exponential reset time distribution is the unique renewal law whose stationary mean reproduces scalar ridge in every eigendirection as an exact filter identity for every positive curvature, while non-exponential renewal laws generate alternative spectral filters. At the fluctuation level, we study a separate additive Ornstein-Uhlenbeck extension with constant diffusion, interpreted as a stylized SGD approximation. In this setting, the equality holds only at the level of the mean, since the reset process has a nonzero stationary covariance from accumulated OU noise and reset-timing variance, whereas deterministic ridge is a fixed estimator with the same center. Stylized experiments compare the deterministic renewal-induced filters directly and illustrate when filters induced by non-exponential reset-time laws can differ predictively from ridge. The results for the stationary mean and the induced spectral filters are established for continuous-time gradient flow with isotropic resetting on quadratic objectives; the covariance and risk formulas additionally assume additive noise with state-independent covariance.

Petar Jolakoski

Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia

Spectral Regularization Linear Regression Ridge Regression

May 9, 2026cs.CL

FragileFlow: Spectral Control of Correct-but-Fragile Predictions for Foundation Model Robustness

Robust adaptation of LLMs and VLMs is often evaluated by average accuracy or average consistency under perturbations. However, these averages can hide a structured failure mode: a prediction may remain correct while probability mass already flows from particular true classes toward systematic wrong competitors near the decision boundary. In this paper, we formalize this phenomenon as margin-aware error flow and introduce FragileFlow, a plug-in regularizer that uses a calibrated margin buffer to identify correct-but-fragile predictions and organize their off-class probability mass into a class-wise vulnerable-risk matrix. Theoretically, we provide the first PAC-Bayes upper bound for this margin-aware error-flow object, showing how empirical spectral control yields a conservative route to deterministic worst-class robustness under a stability condition. Experiments on multiple-choice LLM benchmarks and few-shot CLIP adaptation show that FragileFlow consistently improves the proposed theory-facing risk measures over matched baselines, yields perturbed worst-class accuracy gains in most settings, and preserves clean accuracy across comparisons.

Zhuoyun Li, Boxuan Wang, Jinwei Hu +2

School of Computer Science and Informatics, University of Liverpool, UK

VLM Robustness Spectral Regularization VLM Adaptation Neural Network Robustness Adversarial Robustness Language Model Robustness

May 4, 2026cs.LG

Detecting Adversarial Data via Provable Adversarial Noise Amplification

The nonuniform and growing impact of adversarial noise across the layers of deep neural networks has been used in the literature, without a formal mathematical justification, to detect adversarial inputs and improve robustness. In this work, we study this phenomenon in detail and present a formal adversarial noise amplification theorem. We specify a set of sufficient conditions under which the adversarial noise amplification is mathematically guaranteed. Based on theoretical observations, we propose a novel training methodology with a custom spectral loss function and a specific architectural design to enhance the amplification signal for detecting adversarial data. Finally, we introduce a new, lightweight detection mechanism that leverages the enhanced amplification signal and operates entirely at inference time. To validate our approach, we demonstrate the detector's efficacy against both state-of-the-art attacks and a purpose-built adaptive attack, confirming that enhanced amplification can serve as a robust and reliable signal for adversarial defense.

Furkan Mumcu, Yasin Yilmaz

University of South Florida

Spectral Regularization Adversarial Robustness Adversarial Example Detection

Dec 5, 2025cs.CV

Delving into Latent Spectral Biasing of Video VAEs for Superior Diffusability

Latent diffusion models pair VAEs with diffusion backbones, and the structure of VAE latents strongly influences the difficulty of diffusion training. However, existing video VAEs typically focus on reconstruction fidelity, overlooking latent structure. We present a statistical analysis of video VAE latent spaces and identify two spectral properties essential for diffusion training: a spatio-temporal frequency spectrum biased toward low frequencies, and a channel-wise eigenspectrum dominated by a few modes. To induce these properties, we propose two lightweight, backbone-agnostic regularizers: Local Correlation Regularization and Latent Masked Reconstruction. Experiments show that our Spectral-Structured VAE (SSVAE) achieves a

3\times

speedup in text-to-video generation convergence and a 10% gain in video reward, outperforming strong open-source VAEs. The code is available at https://github.com/zai-org/SSVAE.

Shizhan Liu, Xinran Deng, Zhuoyi Yang +3

Zhipu AI, Beijing, China · University of Chinese Academy of Sciences, Beijing, China · Tsinghua University, Beijing, China

Variational Autoencoders Spectral Regularization Video Diffusion Models Latent Diffusion Models