Daily Papers

We formalize a transfinite Phi process that treats all possibility embeddings as operators on structured state spaces including complete lattices, Banach and Hilbert spaces, and orthomodular lattices. We prove a determinization lemma showing that lifting to sets or distributions yields a deterministic global dynamic, an ordinal stabilization theorem sending operator transforms to the fixed subspace by stage omega under normal spectral contraction, and a product of Riesz projections theorem for commuting layers. We establish a compositionality law for lifted maps, show closure of Phi packings, and present a quantitative application to sensory depletion that models tissue removal as a projection and derives strict decreases in the attainable fixed point under minimal monotonicity and positivity assumptions. We also state measurable conditions for probabilistic lifts, give explicit non normal and non commuting counterexamples, and provide finite dimensional and stochastic witnesses together with per theorem scope tables and a small reproducible code appendix.

Fix a strip width \(ρ>0\), a Diophantine frequency \(α\), an integer \(N\ge2\), and \(λ>1\). For the family \[ V_{\varepsilon,δ}(x) =2λ\cos(2πN x)+\varepsilonδ(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(ρ\) perturbations such that every \(δ\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(δ)>0\) for which \[ gathered Leb\left\{ E\inΣ(V_{\varepsilon,δ},α): ω(α,E;V_{\varepsilon,δ})=s \right\}>0, 0<\varepsilon<\varepsilon_0(δ), \qquad s=1,\ldots,N. gathered \] The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves \(s\) pairs of simple zeros on a positive-measure spectral set.

Alternating projections (AP) near a clean intersection of smooth manifolds can converge slowly when the Friedrichs angle is small. We develop the Fixed-Point Manifold Acceleration (FPMA) framework for {local convergence and polynomial acceleration of} smooth maps that fix a manifold pointwise. {Under suitable regularity and uniform normal spectral stability, FPMA establishes local R-linear convergence and a smooth limiting map inducing a local retraction. At fixed points, consistent polynomial acceleration with p(1)=1 preserves the selected limit's first-order response to normal initial perturbations whenever the transformed normal blocks are uniformly spectrally stable. Under sufficient smoothness, we give a curvature criterion for second-order induced retractions and show that stable consistent polynomial acceleration preserves both the quadratic retraction expansion and this criterion. {As a special case, we establish new local R-linear convergence guarantees for fixed-degree Chebyshev-accelerated exact AP and second-order induced retractions. For uniform Friedrichs angle θ_Fin(0,π/2), acceleration strictly improves the local convergence rate per AP evaluation. The large-degree limiting spectral factor at fixed θ_F predicts O(θ_F^{-1}) evaluations for fixed relative error reduction as θ_Fdownarrow0, versus O(θ_F^{-2}) for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}

One of the challenges in the study of generative adversarial networks is the instability of its training. In this paper, we propose a novel weight normalization technique called spectral normalization to stabilize the training of the discriminator. Our new normalization technique is computationally light and easy to incorporate into existing implementations. We tested the efficacy of spectral normalization on CIFAR10, STL-10, and ILSVRC2012 dataset, and we experimentally confirmed that spectrally normalized GANs (SN-GANs) is capable of generating images of better or equal quality relative to the previous training stabilization techniques.

We investigate the spectral statistics of random perturbations of semiclassical Schrödinger operators on compact manifolds, near energy levels corresponding to chaotic classical dynamics. The prototypical example is that of an operator h^2Δ+V(x) that is perturbed by a smooth potential h^αV_ω with αin (0,1), and V_ω is a random potential that decorrelates on distances h^β, with 0<β<2α. We show that for a generic perturbation, the spectral fluctuations of the smoothed counting function of eigenvalues obey a universal behavior at a certain mesoscopic scale, which is coherent with the predictions of random matrix theory.

We study the pseudospectrum of the Kerr black hole for gravitational perturbations with spin weight s=2, the case of greatest interest in gravitational-wave astronomy. In the hyperboloidal framework with two-dimensional Chebyshev collocation, we formulate the quasinormal mode problem as a non-self-adjoint eigenvalue problem. For sneq0 the fields Ψ^{(s)} and Ψ^{(-s)} are independent, so the conserved current method yields no single-field energy norm. We overcome this with the Teukolsky--Starobinsky identities, which relate Ψ^{(-2)} to a fourth-order operator on Ψ^{(2)} and yield an H^2 energy norm after symmetrization. With the H^1 norm known for s=0, this suggests the pattern H^{|s|}, verified for s=0 and s=2. All results use an ω-independent two-field hierarchy. We find that the pseudospectrum tightens by 1.1 to 3.5 decades at each step of L^2to H^1to H^2, yet stays large in every norm. The s=+2 and s=-2 operators are isospectral but not isopseudospectral: at twelve of fourteen probes the ψ_0 resolvent exceeds the ψ_4 one by 0.19 to 0.82 decades in the H^2 norm, while at the two probes furthest along the QNM line the ordering reverses by 0.18 to 0.19 decades. The gravitational pseudospectrum exceeds the scalar one by 0.5 to 5.2 decades in the window minimum and by 5.9 decades in L^2 at ωM=0.47-0.17,i, while along the damping direction the s=2 resolvent is non-monotonic and the s=0 resolvent smooth. QNM spectral instability is therefore a universal feature of Kerr, and is stronger for gravitational perturbations.

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius R around a known reference state ρ. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write Γ=d(Trsqrtρ)^2+d^2R. For dge2, 0<Rle1, and nge1 copies, the expected risk is Θ(min{R,Γ/n}). At failure probability 0<δle1/3, the minimax confidence radius is \[ Θ\!\left(\min\left\{R,\frac{Γ+\log(1/δ){n}}\right\}\right). \] For varepsilonle c_0R with a universal c_0>0, the copy complexity is Θ((Γ+log(1/δ))/varepsilon^2). For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.

We consider fermions in a zero-temperature superconducting anti-de Sitter domain wall solution and find continuous bands of normal modes. These bands can be either partially filled or totally empty and gapped. We present a semi-classical argument which approximately captures the main features of the normal mode spectrum.

We present an end-to-end pipeline for estimating stellar parameters from Sloan Digital Sky Survey Data Release 12 spectra using a fully connected multitask neural network with residual blocks, whose hyperparameters are tuned via Bayesian optimization. The preprocessing pipeline includes per-spectrum standardization, RobustScaler normalization of the target variables -- effective temperature T_{eff}, metallicity [Fe/H], and surface gravity log g -- and data augmentation via Gaussian noise injection. On a held-out test set, the model achieved Mean Absolute Errors (MAE) of 59.76~K for T_{eff}, 0.103~dex for [Fe/H], and 0.130~dex for log g. Normalized against the full-scale range of each parameter, these results represent range-normalized errors between 1% and 3%, achieved with a highly efficient model complexity of approximately 540,000 trainable parameters. These results demonstrate that a compact residual multitask architecture, combined with principled signal preprocessing, provides a parameter-efficient solution for nonlinear parameter estimation in large-scale spectral datasets. In particular, the proposed model achieves competitive performance with substantially lower complexity than deeper neural network baselines.

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

Let F be a graph, and let B_r(F) be the class of r-uniform Berge-F hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for B_{r}(C_3)-free linear r-uniform hypergraphs and upper bounds for the spectral radii of B_{r}(K_{2,t})-free or {B_{r}(K_{s,t}),B_{r}(C_{3})}-free linear r-uniform hypergraphs, where C_{3} and K_{s,t} are respectively the triangle and the complete bipartite graph with one part having s vertices and the other part having t vertices. Our work implies an upper bound for the number of edges of {B_{r}(K_{s,t}),B_{r}(C_{3})}-free linear r-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

While spectral-based optimizers like Muon operate directly on the spectrum of updates, standard adaptive methods such as AdamW do not account for the global spectral structure of weights and gradients, leaving them vulnerable to two empirical issues in large language model (LLM) training: (i) the optimizer updates can have large spectral norms, potentially destabilizing training and degrading generalization; (ii) stochastic gradient noise can exhibit sparse spectral spikes, with a few dominant singular values much larger than the rest. We propose SPECTRA, a general framework addressing these by (i) post-spectral clipping of updates to enforce spectral-norm constraints; (ii) optional pre-spectral clipping of gradients to suppress spectral noise spikes. We prove that post-clipping constitutes a Composite Frank-Wolfe method with spectral-norm constraints and weight regularization, recovering Frobenius and ell_{infty}-norm regularization with SGD-based and sign-based methods. We further analyze how pre-clipping mitigates sparse spectral spikes. We propose efficient soft spectral clipping via Newton-Schulz iterations, avoiding expensive SVD. Experiments on LLM pretraining show SPECTRA uniformly improves validation loss for various optimizers, including AdamW, Signum, and AdEMAMix, with the best-performing variants achieving state-of-the-art results. Models trained with SPECTRA exhibit smaller weight norms, confirming the link between spectral clipping and regularization.

Spectral rendering reproduces the wavelength-dependent appearance that RGB rendering cannot: metamerism, colour shifts under spectrally rich illumination, coloured glass, metallic reflection and subsurface scattering. It requires a spectrally aware renderer, but the rendering ecosystem is predominantly RGB: game engines, cloud render farms, GPU rasterisers and most offline renderers consume RGB alone, with no access to the internal wavelength propagation that spectral rendering relies on. We introduce learned spectral codes, a compact linear representation that enables general spectral rendering through unmodified RGB pipelines. A trained encoder compresses any visible-range spectral quantity into a small latent vector; the codes are packed as RGB triplets and processed by an off-the-shelf RGB renderer. The codec preserves the arithmetic of light transport, exactly for scaling and addition and approximately for element-wise products, so two standard RGB passes suffice to produce results closely matching a full spectral rendering. The same codec handles reflectance, illumination, coloured glass, conductors, and participating media: one pipeline across material classes, with no per-material modification to the renderer. Where spectral asset data is unavailable, lightweight neural upsamplers bridge legacy RGB inputs to the latent space. Across diverse materials and illumination, the method yields substantial quality improvements on the cases where RGB rendering fails most visibly, at rendering cost comparable to efficient spectral-sampling techniques and with no changes to the renderer.

The {K_{1,1}, K_{1,2},C_m: mgeq3}-factor of a graph is a spanning subgraph whose each component is an element of {K_{1,1}, K_{1,2},C_m: mgeq3}. In this paper, through the graph spectral methods, we establish the lower bound of the signless Laplacian spectral radius and the upper bound of the distance spectral radius to determine whether a graph admits a {K_2}-factor. We get a lower bound on the size (resp. the spectral radius) of G to guarantee that G contains a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Then we determine an upper bound on the distance spectral radius of G to ensure that G has a {K_{1,1}, K_{1,2},C_m: mgeq3}-factor. Furthermore, by constructing extremal graphs, we show that the above all bounds are best possible.

Spectra are a prevalent yet highly information-dense form of scientific imagery, presenting substantial challenges to multimodal large language models (MLLMs) due to their unstructured and domain-specific characteristics. Here we introduce SpecVQA, a professional scientific-image benchmark for evaluating multimodal models on scientific spectral understanding, covering 7 representative spectrum types with expert-annotated question-answer pairs. The aim comprises two aspects: spectra scientific QA evaluation and corresponding underlying task evaluation. SpecVQA contains 620 figures and 3100 QA pairs curated from peer-reviewed literature, targeting both direct information extraction and domain-specific reasoning. To effectively reduce token length while preserving essential curve characteristics, we propose a spectral data sampling and interpolation reconstruction approach. Ablation studies further confirm that the approach achieves substantial performance improvements on the proposed benchmark. We test the capability of prominent MLLMs in scientific spectral understanding on our benchmark and present a leaderboard. This work represents an essential step toward enhancing spectral understanding in multimodal large models and suggests promising directions for extending visual-language models to broader scientific research and data analysis.

Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an O(n^3) Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form h_θ(A_q),R, where A_q is a normalized magnetic operator, h_θ a learnable scalar spectral response, and R a block of random probes. Because the PE is a matrix function of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that k = O(log(1/varepsilon)) block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with 1/s Monte-Carlo error, and the undirected q{=}0 case improves heterophilous benchmarks over no-PE and polynomial baselines.

For a signing σ of a d-regular graph, the spectrum of A_σ depends only on the signs of cycles. We study the affine mathbb F_2 family of signings making every short even cycle unbalanced, and show that averaging over it converts the sign problem of the Bilu-Linial conjecture into a counting problem: a master identity expresses the family-averaged trace as a parity-weighted sum over wrap classes confined to the span W of the constraint cycles, and the family-averaged Ihara L-function diagonalizes so that every prime whose parity escapes W contributes the Ramanujan rate d-1 automatically. Uniform averaging over all signings, by contrast, provably cannot certify a spectral radius below the Kesten profile. We prove matched upper and lower bounds for the confined walk counts, a doubling injection from below, and from above an ear-decomposition encoding in which the number of fresh runs of a non-backtracking walk equals the cycle rank of its support, combined with a window lemma for bicycle-free graphs and a rank bound via the Moore bound for irregular graphs. Consequences include varepsilon-versions of the Bilu-Linial conjecture: every d-regular graph that is subcritical at scale log n, and every d-regular graph bicycle-free at radius Cloglog n/δ, admits a signing in the parity family with ρ(A_σ)le2d-1(1+Cδlog(1/δ))(1+o(1)). We further identify the necessary hypotheses exactly (K_d-trapping; tree-burst gadgets), give an exact certificate on the hypercube, and record a decisive obstruction to two-sided interlacing: mathbb E_σdet(xI-A_σ^2) is not real-rooted, already for the quadrilateral, where it equals (x^2-4x+2)^2+4.

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues \(d|χ_ϕ(u)|^2\), turning stability into an explicit worst-direction design problem; write \(λ\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[λ^{-1}]=\infty\), and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly \(d^2\) outcomes in every integer dimension has floors \(Θ(d^{-3})\) for odd \(d\) and \(Θ(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(λ\ge4/9\). Our main result treats every prime-power dimension of characteristic \(p\ge5\). A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to \([L_q,U_q]\) with \(U_q/L_q\to1\). Its SIC-normalized minimum tends to one, and \(λ(ϕ_q)/Λ_q^\star\to1\) for the global finite-field WH max--min optimum \(Λ_q^\star\), without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at \(I/d\), while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

The spectral behaviour of the synaptic matrix, representing the neuronal connection strengths, is an important tool to analyze the stability and transient dynamics of a typical brain as well as its learning process and memory capacity. The complexity of the brain due to large number of neurons as well as underlying transient mechanisms e.g. homeostasis, seizure or synaptic plasticity can lead to networks with time-varying degree and type of sparsity. This renders an exact determination of the synaptic matrix not only technically difficult but also meaningless, leaving its statistical analysis as the best available theoretical approach. This motivates us to pursue a spectral analysis of the synaptic matrix models with different type of sparsity and thereby analyze latter's role on various aspects of network dynamics and stability. Our results have potential relevance for detemining the type of synaptic sparsity required to induce a specific brain function or desired transient mechanism e.g for pharmacological effects or physiological modulators.

Whitening loss offers a theoretical guarantee against feature collapse in self-supervised learning (SSL) with joint embedding architectures. Typically, it involves a hard whitening approach, transforming the embedding and applying loss to the whitened output. In this work, we introduce Spectral Transformation (ST), a framework to modulate the spectrum of embedding and to seek for functions beyond whitening that can avoid dimensional collapse. We show that whitening is a special instance of ST by definition, and our empirical investigations unveil other ST instances capable of preventing collapse. Additionally, we propose a novel ST instance named IterNorm with trace loss (INTL). Theoretical analysis confirms INTL's efficacy in preventing collapse and modulating the spectrum of embedding toward equal-eigenvalues during optimization. Our experiments on ImageNet classification and COCO object detection demonstrate INTL's potential in learning superior representations. The code is available at https://github.com/winci-ai/INTL.

Diffusion models typically suffer from error accumulation during iterative sampling, commonly referred to as exposure bias. We reveal systematic frequency-dependent discrepancies between training and inference, which can be interpreted as frequency-dependent SNR error. Crucially, the direction of this mismatch varies across models and timesteps, indicating that fixed correction rules do not generalize. We propose Spectral Alignment (SPA), a lightweight, guidance-based method that calibrates the power spectrum of intermediate predictions to a pre-computed prior. Our approach consists of two stages: (1) offline fitting of a parametric spectrum model from training data, and (2) inference-time guidance via efficient FFT-based gradient computation. SPA introduces minimal computational overhead (3-4\%) and is complementary to Classifier-Free Guidance (CFG). We demonstrate consistent improvements across diverse architectures, from pixel-space models (DDPM, ADM) to latent diffusion models (SD2.0, SDXL) and flow-matching models (SD3.5, FLUX). Our implementation is available at https://github.com/SonyResearch/SPA.

We study spectral approximations of the dynamic programming semigroup for finite-horizon optimal control on a connected compact Lie group G, and of the associated first-order Hamilton-Jacobi-Bellman equation. The Bellman operator is monotone and non-expansive in the supremum norm, while the Peter-Weyl decomposition of L^{2}(G), on which every Fourier method on G rests, is orthogonal, and the mismatch is quantitative. The natural sup-norm error recursion of the Galerkin iteration is amplified at every step by the Lebesgue constant of the spectral projection, which grows logarithmically on S^{1} and polynomially on compact Lie groups of rank one, including SO(3), and in computation the iteration violates elementary bounds within a few steps. We restore the dynamic programming structure at the discrete level by replacing the orthogonal projection with spectral filters of Markov type. An auxiliary heat-kernel/vanishing-viscosity scheme yields qualitative sup-norm convergence for Lipschitz data. The main result is a Fejér-type filter on G, finite-rank, positivity preserving and non-expansive, together with a convergence theorem at the rate O(sqrtδ+ε+1/(δN^2)) for Lipschitz data, where δ is the time step, N the spectral resolution and ε the viscosity. The viscosity may be zero, and the coupling δ=N^{-1} then gives the rate N^{-1/2}. The proof interprets the filter as a small random perturbation of the controlled dynamics, requires neither a priori regularity of the value function nor a consistency argument in the viscosity sense for the filtering step, and extends to a fully discrete realization based on positive cubature, with exact Wigner transport on SO(3). Numerical experiments confirm the predicted rates and filter bias and quantify the frame dependence of two chart-based baselines.

Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored the multiscale architectures and various operator designs, they are limited to learning the operators as a whole in the coordinate space. In real physical science problems, PDEs are complex coupled equations with numerical solvers relying on discretization into high-dimensional coordinate space, which cannot be precisely approximated by a single operator nor efficiently learned due to the curse of dimensionality. We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Going beyond the coordinate space, LSM enables an attention-based hierarchical projection network to reduce the high-dimensional data into a compact latent space in linear time. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space that approximates complex input-output mappings via learning multiple basis operators, enjoying nice theoretical guarantees for convergence and approximation. Experimentally, LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks covering both solid and fluid physics. Code is available at https://github.com/thuml/Latent-Spectral-Models.