Maria Nikitina, Anton Bishuk, Oleg Bakhteevcs.LG stat.ML
This paper examines the relationship between the parameters of autoencoder models and the statistical properties of the data on which they are trained. Autoencoders are defined as models with an encoder-decoder architecture, trained to reconstruct input data through a compressed latent representation. It is proposed that the model parameters can be viewed as a dense vector representation of the corresponding sample. To test this hypothesis, a theoretical and experimental study is conducted in which a vector representation is formed based on the spectral characteristics of the autoencoder parameter matrices. Theoretical analysis shows that the singular values of the model parameter matrices are related to the eigenvalues of the covariance matrix of the training data, ensuring the transfer of information between the data space and the parameter space. Experimental results on the CIFAR-10 and FashionMNIST datasets confirm that the resulting vector representations allow for a high degree of accuracy in distinguishing between models trained on different data subsets, without resorting to complex vector generation algorithms or using the original samples. These results suggest that the parameters of trained autoencoders can be viewed as sample representations.
Lucas Degeorge, Paul Couairon, Arijit Ghosh +3cs.CV
Natural images follow a $1/f^2$ spectral distribution: most signal energy lies in the low spatial frequencies, while the perceptually important structures such as textures and edges occupy sparse high-frequency bands. Pixel-space reconstruction objectives, however, treat all spatial errors uniformly, causing low frequencies to dominate the optimization signal and delaying the learning of fine-scale details. In this work, we identify this objective-level spectral imbalance as a key inefficiency in training pixel-space flow models. To address it, we propose a Focal Log-Frequency Loss (f-loss), a spectrally balanced objective that equalizes the learning signal across frequencies, emphasizing high-frequency components that are otherwise underrepresented in pixel-space objectives. Building on this, we introduce a simple training strategy that combines frequency and pixel supervision: we first emphasize frequency-domain learning early to capture all frequencies, and then transition to standard pixel-space v-loss for spatial refinement. This balancing mitigates the low-frequency bias of pixel losses and aligns the training signal with the evolving needs of the model. Our approach is conceptually simple, requires no architectural changes, and acts as a drop-in replacement for flow matching losses. Across multiple model scales, it accelerates convergence by up to 40% while consistently improving FID and perceptual fidelity. We will release code and models.
We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with \(τ\geq m\), we prove that the Wishart splitting matrix \(W\) gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous \(99\%\) Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.
Yujun Ji, Somyajit Chakrabortycs.LG physics.comp-ph quant-ph
Open quantum systems lose energy and phase coherence through different dissipative processes, but these processes can produce overlapping dynamical signatures. The Liouvillian spectrum summarizes how such a system relaxes, yet it is not obvious how much of that spectrum is needed to distinguish the underlying dissipation rates. We study this question for amplitude damping and dephasing in a six-qubit Lindblad model whose spectrum can be derived analytically. We retain only the slowest non-steady spectral modes and ask how many are required before each dissipative rate becomes recoverable. We show that population modes contain no dephasing information, which creates a lower bound of D = 2^n retained modes for uniform dephasing identifiability in the relevant rate regime. The measured recovery threshold reaches this bound at n = 4,5,6, while n = 3 remains above it. At n = 6, least squares achieves a mean joint absolute error of order 10^-9, compared with 4.355 x 10^-4 for four tabular learning methods. Robustness tests show that this advantage weakens when the spectra are perturbed and when a transverse field breaks the commuting structure. These results show that the amount and structure of retained spectral information can determine whether dissipative parameters are recoverable, independently of the estimator used. The present conclusions apply to noise-free simulator spectra rather than measurement-derived spectra.
Robustness of segmentation models is commonly assessed through input-domain perturbations, while dependence on frequency content within learned feature representations remains less understood. We probe this dependence using targeted post-training low-pass interventions on internal representations of three segmentation architectures, ResNet50-UNet (CNN), VM-UNet (SSM), and Swin-UNETR (Transformer), across CVC-ClinicDB and ISIC2018, with headline evaluations performed on untouched held-out test sets. At cutoff rho=0.25, feature-domain low-pass filtering causes severe degradation on CVC: Dice drops by 100%, 73.2%, and 30.9% for CNN, SSM, and Transformer, respectively, compared with 9.4%, 10.3%, and 0.6% on ISIC. The cross-dataset difference is statistically significant for every architecture. Single-stage interventions further show that sensitivity is localized at architecture-specific depths: the CNN peaks at a mid/late encoder block, whereas the SSM peaks in an early encoder stage on both datasets. Native feature-domain spectral measurements show an inverse association between high-frequency energy and fragility on CVC; the relationship is only partial on ISIC and is therefore treated as a candidate correlate rather than a proven mechanism. Finally, Fourier augmentation improves robustness to input-space low-pass filtering but leaves feature-domain degradation essentially unchanged. These results show that feature-spectral robustness is strongly dataset-dependent, architecture-specific, and distinct from input-domain spectral robustness.
Directed graph learning naturally leads to trainable nonsymmetric propagation operators with distinct right and left spectral structures. Building on the two-sided cone Rayleigh framework for generalized pencils \[ B_θ-λG, \] we develop a learning-oriented methodology for spectral certification, sensitivity analysis, and control without requiring symmetry, nonnegativity, or cone preservation. In the positive-orthant setting, computable lower and upper cone bounds provide an a posteriori enclosure of a distinguished cone level, while smooth soft-min/max surrogates preserve rigorous one-sided bounds with explicit approximation errors and remain differentiable with respect to the trainable parameters. For a simple interior level, the right and left modes satisfy \[ Dλ_C(B)[H]=v_C^T H u_C, \] yielding first-order optimal graph-supported interventions under prescribed perturbation budgets and motivating adaptive spectral control. Numerical experiments demonstrate the applicability of the approach beyond cone-preserving operators and in directed learning settings. Signed nonsymmetric perturbations reveal a transition from interior eigenpairs to boundary complementary quasi-pairs, including non-spectral cone levels, while controlled experiments show that symmetrization can remove predictive information carried solely by edge direction. On the directed Cora citation network, adaptive recomputation of the right--left sensitivity reduces the distinguished spectral level by approximately $21.5\%$ under a cumulative edge-weight reduction budget of $0.5\%$, with no observed change in test accuracy for the trained model and data split considered.
Orthogonal optimisers such as Muon can substantially accelerate large language model pretraining relative to Adam, yet the mechanism remains incompletely understood. We investigate this through an out-of-sample spectral probing analysis of Transformer loss landscapes. At checkpoints along real training trajectories, we decompose each momentum buffer into its singular directions and estimate the loss-optimal step size along each direction on held-out data. The resulting spectral profile is anisotropic yet stable across batches and training stages, and consistent across the optimisers and model scales: a volatile head operating at the Edge-of-Stability supports a much smaller step size than the tolerant bulk, which permits substantially larger steps. This profile provides a unified spectral allocation account of why Muon outperforms Adam, which outperforms SGD. It also exposes a limitation of Muon's uniform scaling: it still underutilises the bulk. Guided by this finding, we introduce Spectral-Aware Muon (SAMuon), which holds the head at the Muon scale and amplifies the bulk using a static spectral prior. We provide two variants: the complete SAMuon follows the measured profile using a low-rank randomised SVD and the simplified SAMuon-lite uses a two-level approximation via rank-one power iteration. Neither method adds persistent optimiser state or notable extra FLOPs beyond Muon at scale, and the idealised exact-whitening versions of both retain Muon's asymptotic convergence rate under standard assumptions. Across "modded-nanogpt" models from 124M to 1B parameters, both variants outperform tuned AdamW and Muon (Scion implementation) baselines in all evaluated model-scale and batch-size configurations. SAMuon requires 13.3% to 24.0% fewer training tokens to reach the same validation loss as Muon, while SAMuon-lite retains most of this gain with near-zero wall-clock overhead.
The rapid advancement of Large Language Models (LLMs) makes it increasingly difficult to distinguish human writing from machine-generated text. Training-free detection offers a scalable solution, yet common confidence-based metrics mainly measure average token probabilities and often miss the signal fluctuations that characterize human writing, which we call "generative vitality". Spectral analysis offers a way to capture this vitality, but its mechanism and practical boundaries remain underexplored. In this paper, we analyze spectral detection from both theoretical and empirical perspectives. We connect spectral energy to variance in proxy log-probability trajectories and explain how broader human token choices create the fluctuations used by frequency-domain indicators. We further show that the strength of this signal depends on text length and sampling range: spectral evidence is clearest for long, continuous, constrained generation, while short, fragmented, mixed, and edited settings require complementary confidence and fluctuation views. These findings clarify when frequency-domain detection works and provide guidance for future multi-dimensional detector design.
Simulation is central to modern engineering and science, but the cost of numerical solvers for partial differential equations (PDEs) remains a bottleneck whenever fast or many-query evaluations are required. Neural emulators trained on solver-generated data promise significant speedups, yet they are usually framed as opaque alternatives to the very methods that produce their training signal. This thesis argues the two paradigms are more alike than different: neural architectures mirror classical discretizations, their errors are amenable to the same spectral analysis, and insight flows profitably in both directions. We approach the relationship by disentangling the multiple roles a solver plays in the emulator learning pipeline. Mode-wise Fourier analysis then provides a common language in which solver errors, architectural inductive biases, and training objectives can all be read off simultaneously. Taken together, this allows synthesizing three contributions. (1) APEBench, a comprehensive benchmarking suite for autoregressive neural emulators of PDEs that uses fast differentiable pseudo-spectral solvers in JAX. (2) Progressively Refined Differentiable Physics, an investigation of the effect of unconverged solvers on surrogate training. (3) Neural Emulator Superiority, an analysis of the influence of numerical errors and architectural inductive biases.
Instance encoding is a popular empirical technique for privacy enhancement when sharing data to an untrusted server. It transforms sensitive data through an encoding process before sharing, with the hope that the encoding process retains utility but makes it hard to reconstruct the original data. However, most work offers no theoretical guarantee that the encoding process is actually irreversible. A recent work derived a mean-squared error (MSE) bound limiting any adversary's reconstruction accuracy, offering one of the first theoretical results in this domain. This bound, however, has three critical limitations: it is often too loose, only works with randomized encoders (excluding many deterministic encoders practitioners use), and only bounds MSE. We introduce a family of new bounds that (1) are tighter, (2) applicable even to fully deterministic encoders, and (3) can extend beyond MSE to other norm-based similarity metrics, by properly accounting for the encoder's spectral structure. We evaluate our bounds across a range of encoders, datasets, and attacks, showing they hold consistently and improve upon the existing bound.
James Hartley, Zeropy Surio, Daniel Whitmore +2cs.LG
Analytic continual learning has emerged as a strong exemplar-free alternative to gradient-based class-incremental learning because it replaces iterative optimization with closed-form ridge updates. Yet the usual forgetting narrative, centered on stochastic gradient overwriting, does not explain why analytic methods still drift on old classes despite exact recursive solvers. We identify the culprit as spectral interference: the joint ridge classifier for all tasks shares the inverse autocorrelation operator $(R+λI)^{-1}$, so incoming task samples that load onto old dominant eigendirections dilute the spectrum and perturb old-class logits even when old labels are never revisited. Based on this view, we propose SPARCL, a spectral partitioned analytic continual learner that decomposes the running autocorrelation into a high-energy core and a residual complement, freezes old-class classifier components in the core subspace, and updates only the residual block through recursive least squares with an optional residual random-projection expansion. This yields a simple closed-form update with a provable invariance guarantee for the core contribution of old logits. Across CIFAR-100, CUB-200, ImageNet-R, and ImageNet-A under a frozen ViT-B/16 protocol, SPARCL closes most of the gap from classical analytic learners to strong representation matchers, while remaining complementary to sparse feature-decorrelation approaches such as Fly-CL.
Whether distinct neural architectures develop common collective dynamics remains an open question. Recent analysis of Transformer language models revealed a nearly flat, weakly infrared-enhanced time-scale density of states (TDOS) associated with near-marginal long-memory dynamics. Here we test whether a closely related organization emerges in Mamba, whose selective state-space dynamics provides a fundamentally different microscopic mechanism. Mamba allows relaxation dynamics to be resolved at three levels: the intrinsic spectrum of the learned state-space generator, its input-conditioned selective rescaling, and the collective TDOS of the complete block measured from its Jacobian. These spectra are not identical: selective dynamics and the remaining block transformations substantially reorganize the microscopic relaxation hierarchy. Nevertheless, the full block develops a reproducible slow-mode continuum whose infrared sector becomes progressively better resolved with increasing sequence length. Cumulative analysis yields $ρ(λ)\simλ^β$, with the long-sequence Mamba exponent stabilizing near $β_{\rm M}\simeq-0.17$. The corresponding memory dynamics follows $K(t)\sim t^{-(1+β)}$, close to the marginal $1/t$ regime. Despite fundamentally different microscopic dynamics, Transformer full-block spectra exhibit closely related infrared organization, with representative exponents of order $β_{\rm Tr}\sim-0.1$. These results separate explicit state-space memory from collective infrared organization and show that distinct sequence architectures can develop closely related near-marginal slow-mode dynamics. They extend infrared collective organization beyond Transformers and provide an independent test of the dynamical structure described by Cognitive Field Theory.
Javier Lopatin, Teja Kattenborn, Eya Cherif +1cs.CV cs.LG
Hyperspectral reflectance spectroscopy enables non-destructive estimation of plant functional traits, yet current deep learning approaches process spectra as one-dimensional sequences, which limits how they capture long-range inter-band dependencies. We asked whether transforming 1D spectra into 2D image representations improves multi-trait prediction with convolutional neural networks (CNN). We compared nine transformations using EfficientNet-B0 on the GreenHyperSpectra dataset (7,897 labeled spectra, eight traits, 400-2450 nm), benchmarked against published 1D CNN results on the same split. Trained from scratch, the simplest transformation, a direct Reshape of the spectrum into a 2D grid, performed best ($R^2 = 0.684 \pm 0.001$) and improved on the state-of-the-art 1D baseline ($R^2 = 0.587$, $+0.097$). We then pretrained a 2D masked autoencoder (MAE-2D) on 139,000 unlabeled spectral images. Linear probing, which freezes the encoder and trains only a multilayer perceptron head, reached $R^2 = 0.646$ and exceeded every 1D self-supervised counterpart, including the fine-tuned MAE-1D ($R^2 = 0.641$). Under cross-dataset evaluation all models lost most of their accuracy and none beat the 1D baseline significantly. To identify which wavelengths drive each prediction, we applied Integrated Gradients and Grad-CAM and unfolded band importance back to the spectral axis. Protein ($r = 0.45$) and leaf water ($r = 0.33$) agreed with sensitivities simulated by the PROSAIL radiative-transfer model, while carotenoids ($r = 0.06$) and leaf area index ($r = -0.11$) did not, showing that the model reads established leaf chemistry for traits with sharp absorption features. The representational advantage of 2D spectral images, rather than architectural complexity or ImageNet pretraining, drives the gain over 1D approaches.
Nominal LoRA rank is a design parameter; calibrated spectral evidence is a separate inferential quantity. This article develops a finite-sample framework for inferring effective rank structure in public foundation-model adapters. The theoretical core is an exact chi-square divergence for the fixed-dimensional Gaussian rank-one reference experiment, with an unknown signal direction integrated under a rotation-invariant reference prior. The resulting series yields a computable finite-sample Le Cam bound at concrete layer sizes, an explicit remainder bound for numerical truncation, and the rectangular Baik-Ben Arous-Peche (BBP) limit. A compact-manifold Laplace expansion shows that finite-sample likelihood evidence also depends on leading spectral gaps through the factor $s_1^{|m-n|}\prod_{i\ge2}(s_1^2-s_i^2)$, motivating joint calibration of clustered singular values. Building on these results, we introduce an empirical-null workflow for PEFT LoRA adapters: factor reconstruction, Monte Carlo $p$-values, stagewise and block testing, and module-wise and corpus-level BH reporting. In an audit of 26 public adapters, 684 modules, six architecture families, and 31,770 public-checkpoint spectra rows, calibrated effective rank is typically much smaller than nominal rank and differs systematically from 95\% energy retention. A measured RoBERTa-RTE slice on $n=24$ examples illustrates the measurement path from calibrated ranks to task evaluation, without treating the slice as a utility study. The main empirical finding is that calibrated effective rank is usually far below nominal rank, and that energy retention and statistical surprise answer different questions.
A multiplicative dual-encoder network computes a real-valued output for a pair of inputs as the inner product of their separate encodings. This architecture has been developed independently in operator learning, bipartite matching, contrastive vision-language models, retrieval, and other areas, yet no unified theory guides the basic design decisions: how many interaction modes to represent, how to normalize the encoders, and when the architecture should be avoided. We provide such a foundation by introducing the class of functions of low interaction rank, a class whose intrinsic complexity is measured by its interaction spectrum. Within this framework, approximation error decomposes into a spectral truncation term and an encoder-realization term; sample complexity is governed by the sum of the two encoder complexities rather than their product; and a usability criterion based on spectral decay determines when the architecture can succeed. The same framework exposes a central identifiability problem: the encoders are defined only up to a linear gauge symmetry that leaves the learned coordinates arbitrary. We show that normalization is gauge fixing and that whitening pins the interaction modes up to permutation and sign, thereby explaining the uninterpretability of contrastive dimensions and providing a constructive remedy. Experiments on synthetic kernels, operator learning, and CLIP models validate the theoretical predictions: spectral decay rates match the predicted scaling, whitening recovers the true modes, and independently trained CLIP models are related by a single rotation which, after removal by whitening, exposes interpretable concept axes. The code of this paper is provided at https://github.com/RS2002/Mul-Net .
In a $β$-VAE, increasing the regularization strength acts as a spectral cutoff by collapsing low-utility latent coordinates. In the linear Gaussian VAE, the collapse order matches the ranking of reconstruction utilities exactly, because both are set by the PCA spectrum. We ask which parts of this picture survive in fully connected nonlinear VAEs trained on WorldClim. We find that nonlinear interactions shift and broaden collapse onsets, so thresholds no longer coincide exactly with utilities. However, the common ordering is preserved over the resolved ranks, so the spectral cutoff still acts as a utility cutoff and the effective-description logic carries through. The resulting effective-dimension curves reveal a head--tail tradeoff: increasing depth concentrates utility into the first few coordinates but worsens tail fidelity.
Drake Brown, Yuhao Huang, Shih-Hsin Wang +1cs.LG cs.AI math.NA
Drifting models are a recent class of one-step generative models that evolve the model distribution during training using a predefined sample-based drift field. Although they avoid iterative inference, their kernel-based drift fields induce frequency-dependent training dynamics: In the linearized regime, each Fourier mode of the density residual decays at a rate determined by the kernel spectrum, leading to slow recovery of fine-scale structure. We propose Second-Order Drifting Models, which lift drifting dynamics into phase space by augmenting generated samples with artificial velocity variables. We show that the resulting density perturbations obey accelerated second-order dynamics in Fourier space, connecting drifting models to the celebrated Nesterov acceleration from optimization theory. This provides a principled mechanism for mitigating the spectral stiffness of first-order drifting while preserving one-step inference. We derive a practical semi-implicit training algorithm and evaluate it on synthetic distribution matching, sequential data generation, and robotic control. Across these settings, the second-order drifting model improves convergence behavior and achieves competitive or superior performance over first-order drifting baselines.
Kasun Dewage, Marianna Pensky, Suranadi De Silva +1cs.LG cs.AI cs.CL
We apply Marchenko-Pastur (MP) random matrix theory to pre-trained attention weights in order to separate each projection matrix into a random-like bulk and a set of spectral outliers. We validate this decomposition causally: zeroing the MP-identified outliers (signal) in Mistral-7B drives HellaSwag, MMLU, and PIQA close to random-chance performance, whereas zeroing a count-matched subset of bulk singular values causes smaller but non-negligible degradation. Across 11 pre-trained transformers we identify five recurring patterns: spectral outliers encode a dominant component of the learned structure; Q projections carry the most outliers; V projections under grouped-query attention lack a clean signal/noise separation; entry-level outliers form structured row-bands in Q and column-bands in O; and specific residual-stream dimensions persist as band outliers across layers in K and O. We close by outlining how these observations could inform parameter-efficient fine-tuning and structured pruning.
Latents from vision foundation models (VFMs) are semantically rich and well suited for visual understanding. Recent representation autoencoder methods such as RAE have shown that they can provide promising latent spaces for image generation. However, VFM latents remain difficult to model directly: DiT-generated latents exhibit spectral mismatch with encoder latents, especially in high-frequency components. Our channel-wise spectral analysis further reveals that these high-frequency components are diffusely distributed across latent channels and entangled with semantic information, making the latent space difficult for DiT to model. To address these challenges, we propose SPAE, latent adaptation framework for generation. Specifically, SPAE employs a compact bottleneck to distill stable semantic information while suppressing high-frequency components, thereby improving the alignment between DiT-generated latents and encoder latents. In addition, we apply a channel-wise masking strategy to promote the decoupling of semantic information and high-frequency details across bottleneck channels. Experiments show that SPAE achieves a favorable balance among visual understanding, generation quality, and reconstruction fidelity.
Transformer language models are usually analyzed through vector geometry, yet ordered context and rotary position encoding introduce explicit phase structure into query-key interactions. This paper develops a bounded spectral framework for examining rotary phase alignment, hidden-state continuity, and semantic drift without treating language models as literal physical wave systems. It first identifies ordered hidden-state sequences, rather than vocabulary indices, as valid domains for spectral decomposition. It then derives the Rotary Position Embedding (RoPE) attention score as a sum of magnitude-weighted cosine terms and proves a local stability lemma: uniformly bounded phase displacement limits degradation of the corresponding pre-softmax score. To extend phase analysis beyond native RoPE coordinates, the paper defines complex modal coordinates over fixed orthonormal direction pairs and introduces a weighted coherence functional for hidden-state trajectories. These constructions support a strict distinction between representational continuity and execution-boundary admissibility. Internal coherence may describe preservation of task-relevant relations, but it cannot authorize a consequential transition. Positioned against existing geometric, spectral, phase-modulation, representation-analysis, and mechanistic-interpretability accounts, the framework contributes a theoretical and methodological program for determining when spectral structure explains continuity and when governance must remain an external predicate over execution.
Large Language Models can produce fluent text that is false, unsupported by the available evidence, or inconsistent with information that appears to be internally represented by the model. We study hallucination detection from the geometry of hidden activations and introduce the D-Score, a simple spectral statistic computed from a single forward pass. For a fixed model, layer, and tolerance parameter, the D-Score counts how many singular directions of the hidden activation matrix have singular values that remain close to the leading one. We use this quantity as a hallucination score, classifying an input text as hallucinated when its D-Score is larger than a pre-defined quantity. The motivation is that, when a model processes a text that conflicts with information available in its own internal state, the hidden representation may encode both the asserted content and some form of counter-evidence, uncertainty, correction, or lack of support; this can make the hidden trajectory spread across additional singular directions. We formalize this intuition through a lightweight spectral argument and evaluate the resulting detector on FAVA-Annotation and RAGTruth. The experiments indicate that the D-Score is a strong hidden-state signal for hallucination detection, while requiring no external verifier, no retrieval step, and no multiple generations.
Transformer architectures have achieved remarkable success across diverse domains; however, directly applying their standard self-attention mechanism to recommendation often yields suboptimal performance, sometimes even trailing behind well-designed simple recommendation models. In this paper, we reveal that this performance bottleneck stems from severe embedding and attention collapse unique to recommendation scenarios. The heterogeneity and long-tail nature of recommendation data lead to a severe spectral collapse dominated by a few principal singular values. We further theoretically demonstrate that this triggers a vicious cycle in recommendation model's forward and backward propagation, which accelerates embedding and attention collapse and limits the model's scaling capability with increased depth. To address these issues, we propose SpecFormer, a novel Spectral-Aware Transformer designed for mitigating embedding and attention collapse in recommendation. Specifically, SpecFormer introduces 1) a Learnable Spectral Softening module to dynamically smooth the singular values distribution of the input token embeddings; 2) a Spectrum-softened Attention mechanism to model feature interaction under a more uniform spectral distribution space; 3) a Spectral Residual Position Encoding via Taylor expansion of singular values, explicitly providing a spectral inductive bias for feature interactions. Extensive experiments on one industrial and two public datasets demonstrate that SpecFormer significantly outperforms state-of-the-art baselines. Notably, SpecFormer has been successfully deployed in a real-world commercial recommender system and exhibits exceptional scaling capabilities: stacking SpecFormer layers actively improves the attention effective rank and recommendation performance.
LoRA fine-tuning can create intruder dimensions: new leading singular vectors of the updated weight matrix $W+BA$ that are nearly orthogonal to all pretrained singular vectors and that drive catastrophic forgetting. Since their discovery, no theory has predicted, layer by layer on measured spectra, when they appear. We derive a per-layer critical update strength $s^\ast=\barθ/(γσ_1(BA))$, computed from the measured spectrum of $W$ alone through the rectangular spiked-deformation transform, together with an exact secular-equation characterization of the updated spectrum, with no fitted parameters. In a pre-specified study spanning four dense Transformer families, a state-space model, a mixture-of-experts model, and an encoder-decoder (18 adapters, 9{,}840 layer scans), the law localizes the empirical threshold within a factor of two on $82\%$ of layers, separates intruder-bearing from intruder-free layers at deployment with a mean AUC of $0.89$, holds unchanged on six third-party adapters, and predicts where WikiText-2 perplexity begins to degrade; a combination of the two pre-specified edge evaluations reaches $98\%$ and is confirmed out-of-bag on the external adapters ($0.997$). Full fine-tuning disperses its update far below the threshold of every layer, which resolves the asymmetry between LoRA and full fine-tuning. Norm-matched interventions confirm that threshold-crossing layers, rather than update magnitude, carry the forgetting, and a spike-budget rule derived from the thresholds, requiring one SVD and no validation sweeps, reduces forgetting by $62\%$ on the most fragile model at no task cost.
Zermelo's algorithm is a classical method for computing the maximum likelihood estimator in the Bradley--Terry (BT) model, but its convergence can be slow in practice. To accelerate computation, Newman introduced a family of Zermelo-type fixed-point iterations parameterized by $α$, with Zermelo's algorithm recovered at $α=1$. Empirical evidence suggests that the choice $α=0$ often converges substantially faster, making it a promising alternative, yet the mechanism underlying this acceleration remains elusive. This paper provides theoretical insight into this phenomenon through a systematic local convergence analysis. We derive closed-form expressions for local convergence factors under synchronous and asynchronous updates and analyze their dependence on $α$ via spectral analysis of the associated Jacobian matrices. For synchronous updates, we show that the algorithm may fail to converge when $α<1$, and its local convergence factor is quasi-convex in $α$ under the population BT model. In contrast, asynchronous updates are always locally convergent, and their local convergence factor is provably monotonically increasing in $α$ under the population BT model of consistently ordered bipartite comparison graphs, establishing the optimality of $α=0$ in this setting. We further establish asymptotic approximation results for the population convergence factors under the BT model, justifying their practical relevance. Numerical experiments on synthetic and real-world datasets confirm the theory. Our analysis complements existing convergence results and shows that the acceleration of $α=0$ arises not only from the parameter choice but, more importantly, from the use of asynchronous updates.
Neural network misclassifications exhibit characteristic spectral instability in internal activations that is invisible at the output layer. This phenomenon is identified and formalized as Spectral Drift -- the frequency-domain distance between consecutive layer activations -- with empirical validation showing that failures exhibit significantly higher drift than correct predictions (1.9% increase, p<0.001). This spectral signature emerges during internal processing but becomes masked in final outputs, explaining why confidence-based detection methods struggle. This work introduces Self-Detecting Neural Networks (SDNN), a framework that monitors spectral dynamics across network depth using Short-Time Fourier Transform, wavelet decomposition, and statistical moments to capture multi-scale spectral features. A lightweight detector network (5% parameter overhead) learns to identify failure-indicative patterns via curriculum learning on progressively challenging distributions: natural misclassifications, distribution shifts, and adversarial perturbations. Experiments on CIFAR-10 demonstrate that SDNN achieves 79.0 +/- 25.3% AUROC across three seeds, substantially outperforming confidence-based baselines including MaxSoftmax (50.5%) and Energy Score (52.9%) by approximately 25-30 percentage points. Ablation studies reveal that wavelet decomposition and statistical features make consistent contributions, while STFT's role remains unclear. This work establishes spectral analysis of internal activations as a promising direction for neural network reliability, revealing diagnostic information inaccessible to output-based approaches.
Reinforcement learning with verifiable rewards (RLVR) is rapidly advancing the reasoning capabilities of language models, yet the optimization layer that converts reward feedback into weight-space updates remains poorly understood. Building on our prior analysis (Zhu et al., 2025), we study this missing layer through the singular structure of model weights and identify spectral inheritance: RLVR can reuse the base model's weight spectra while acquiring new behavior through changes in the associated input and output singular frames. We operationalize spectral inheritance as Isospectral Optimization (ISO), an RLVR-native, fixed-spectrum optimization framework with complementary offline and online instantiations. Offline, ISO-Merger combines the frame changes of shared-base specialists into a single fixed-spectrum model, requiring no post-merge data, rollouts, gradient updates, or on-policy distillation (OPD). It recovers complementary specialist capabilities and achieves the strongest aggregate performance among the compared data-free merging methods. Online, ISO-Optimizer applies a chosen base optimizer, including AdamW and Muon, to the frame variables while keeping the base spectra fixed. Across reasoning and coding tasks ranging from 1.5B to 8B parameters, ISO-Optimizer improves accuracy in the reported runs and reaches matched scores with substantially fewer training steps. On Qwen3-8B-Base, AdamW reaches an aggregate accuracy of 0.495 after 270 training steps. ISO-AdamW reaches the same accuracy after only 100 training steps and improves further to 0.509 after 210 training steps. Together, ISO offers a concrete answer to RLVR's missing optimization layer: rather than inheriting pre-training optimization wholesale, design post-training around the structure of reward-driven adaptation: inherit the spectrum, optimize the frames.
Manuel Fernandez, Yizhe Zhustat.ML cs.LG math.PR math.ST
We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability $p$, shared latent variables make the adjacency entries dependent. At the connectivity scale $np=Ω(\log n)$, the spherical adjacency matrix satisfies, with high probability,$\|A-\mathbb E A\|_{\mathrm{op}}=O\left(\sqrt{np\log n}+npτ\right)$, where $τ$ is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When $np\gg\log n$, vector and relative Gram-matrix errors vanish for$\log(1/p)\ll d\ll np\log(1/p)/\log n$ in the spherical model and $\log^2(1/p)\log n\ll d\ll np\log(1/p)/\log n$ in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.
Continuous-time generative frameworks construct probability paths between base and target domains by optimizing time-dependent velocity fields. While theoretical targets favor straight trajectories, empirical networks develop complex path deformations. This paper presents the Finite-Time Spectral Sensitivity (FTSS) g(t), a gradient-free, forward-pass metric that exposes flow geometry by tracking the root-mean-square singular value of the state-transition matrix. Serving as a continuous proxy for stable rank, g(t) reveals a distinct geometric pathology under data scarcity: while generalizing models maintain stable effective dimensions, overfitting causes a spectral collapse. We leverage this structural phenomenon to develop an internal geometric audit based on g(t). Our framework detects generative memorization using purely internal trajectory dynamics, removing the need for external membership queries or baseline data comparison.
Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from \(69.2\%\) to \(96.2\%\) across five-seed synthetic-corruption experiments and \(94.4\%\pm1.9\%\) on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from \(28.88\pm1.95\) under clean training to \(97.09\pm1.78\) at \(60\%\) corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.
The paper imports the Kontsevich Segal Witten criterion from quantum gravity into machine learning to evaluate complex linear maps Standard techniques analyze magnitude or positive definiteness whereas this method exclusively limits the collective phase of a spectrum The researchers create three distinct differentiable certificates comprising a determinant sector a subset product envelope and the full criterion The subset envelope prevents all exterior power eigenvalues from touching the negative real axis This constraint precisely matches the accept or reject choices of an exponential minor enumeration while reducing processing expenses drastically The team provides a differentiable enforcement application via a Schur parameterization The document also identifies crucial boundaries regarding where this system works The constraint cannot balance deep linear propagation since restricting the phase budget damages eigenvector conditioning Furthermore the technique remains completely blind to magnitude based targets like normalizing flow likelihoods Thus researchers must restrict this tool specifically to models that process the argument of a spectral product