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arXiv2026

Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization

In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.

cs.IT↗

arXiv2026

Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials

We prove a sharp lower bound for growing-degree polynomial estimation in the Gaussian planted submatrix model. The observation is $$ \boldsymbol{Y}= \fracλ{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, $$ where the coordinates of $\boldsymbolθ$ are independent $\mathsf{Ber}(ρ)$ variables and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every fixed $λ>0$ and $ρ\in(0,1)$, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $ρ-q_{\mathsf{amp}}/λ$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree result of Montanari and Wein~\cite{montanari2025equivalence} for the Bernoulli prior. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\to\infty$ within this range. It therefore resolves the Bernoulli rank-one case of the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high}. The proof constructs a low-degree certificate using \emph{conditional} joint cumulants of the signal coordinates and their products. Specifically, we condition on an auxiliary Gaussian channel $\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows. Most of the arguments in this paper were generated using GPT-6 Astra.

math.ST↗

arXiv2026

Tensor network representations of discrete maximum entropy distributions via mean polytopes

We present tensor network representations for discrete maximum entropy distributions under expectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution's support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.

math.ST↗

arXiv2026

Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation

Hilbert-space embeddings enable inference for populations of persistence diagrams, but separation between individual diagrams need not survive population averaging. We develop a framework for inference on population mean embeddings, with particular attention to the additive landmark representations PLACE and PALACE. Treating each diagram as one independent observation, we apply Hilbert-space limit theory to obtain covariance estimators, two-sample tests, and confidence balls under suitable moment conditions, without requiring a lower-distortion bound. For additive embeddings, we identify the population mean as an embedding of the mean counting measure and show that geometric separation of these measures alone cannot guarantee uniform testing power. We then introduce a model with latent template diagrams, missing features, and location perturbations. Under common or feature-specific prevalence conditions, a diagram-level lower-distortion certificate yields explicit lower bounds on population mean separation. These margins provide finite-sample uniform power guarantees, and an additional information-divergence comparison gives matching sample-complexity bounds over restricted scale ranges. Confidence sets yield lower bounds on transport separation of population mean measures and exclusion guarantees for specified structured alternatives. We also quantify how orthogonal truncation changes the certified signal and the approximation allowance needed for confidence sets targeting the full embedding, relating sample size, retained coordinates, and template separation. Simulations examine calibration, power, and coverage, and an analysis of resting-state connectivity from the Autism Brain Imaging Data Exchange illustrates the procedures.

math.ST↗

arXiv2026

Sharp Structure-Agnostic Minimax Risk for Partial Linear Models

We characterize the sharp structure-agnostic minimax risk for coefficient estimation in the partial linear model when the outcome and treatment nuisances are learned by two distinct black-box learners, which resolves the open problem in double machine learning posed by Gu (2025). For each nuisance $q\in\{μ,π\}$, we characterize the available learner by an approximation-error budget $a_q$ and a stochastic-error budget $s_q$, with the latter controlled through localized Rademacher complexity. Writing $\mathcal E_n$ for the minimax mean-squared error, we show that \[\mathcal E_n\asymp1\wedge\left\{\frac1n+\left(a_μa_π+\min\left\{a_πs_μ+s_π^2,\,a_μs_π+s_μ^2\right\}\right)^2\right\}.\] The main new ingredient is a novel lower bound for the general two-learner problem. Our proof constructs four finite-mixture testing experiments using orthogonal code functions. Across these experiments, the hidden perturbations are placed outside both learner classes, outside only the treatment learner class, outside only the outcome learner class, or inside both learner classes. These four configurations capture, respectively, the interaction between the two approximation errors, the two asymmetric interactions between one learner's approximation error and the other learner's learning error, and the joint estimation difficulty of learning both nuisances. Combining the four resulting lower bounds yields the displayed rate, which matches the latest upper bound in Gu (2026). Our result shows that standard double machine learning can overstate the intrinsic difficulty of target estimation and provides a target-specific principle for learner selection: approximation error and stochastic complexity must be jointly balanced across the two nuisance learners rather than optimized separately.

cs.LG↗

arXiv2026

Diagonal Attenuation: A Finite-Sample Correction for PCA

Principal component analysis (PCA) can rotate away from its population target when a covariance matrix is estimated from limited data. We introduce diagonal attenuation, which preserves sample cross-covariances while reducing coordinatewise sample variances. The method is revealed exactly by averaging a linear full-output reconstruction loss over random input masks; studying the correction directly extends it beyond the range attainable by masking. We isolate the part of the random coupling between retained and omitted population directions that is contributed by sample-variance errors, and show how attenuation can reduce the resulting rotation. Under balanced marginal variances, we derive an explicit expected-risk theorem, uniform over the attenuation path for all sufficiently large finite samples, and obtain the asymptotically risk-minimizing strength. For general covariances, we characterize when attenuation leaves the population PCA subspace unchanged and give a risk theorem that also accounts for changing eigengaps and the population cost when the target moves. Simulations track this tradeoff from exact preservation back to PCA. Across local image patches, speech spectra, and smartphone acceleration, both mask-derived and direct attenuation improve PCA under two fitting-sample budgets, and one of them has the largest mean gain among seven methods in every data--budget cell. The full path selects strengths beyond the mask-derived boundary on $63\%$--$95\%$ of the subsamples.

stat.ML↗

arXiv2026

Low-rank matrix recovery landscapes beyond RIP with application to rank-one measurements

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm--regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

math.OC↗

arXiv2026

Statistical Inference for Privatized Data with Unknown Sample Size

We develop both theory and algorithms to analyze privatized data in unbounded differential privacy (DP), where even the sample size is considered a sensitive quantity that requires privacy protection. We show that the distance between the sampling distributions under unbounded DP and bounded DP goes to zero as the sample size $n$ goes to infinity, provided that the noise used to privatize $n$ is at an appropriate rate; we also establish that Approximate Bayesian Computation (ABC)-type posterior distributions converge under similar assumptions. We further give asymptotic results in regimes where the privacy budgets vary, establishing similarity of sampling distributions as well as showing that the MLE in the unbounded setting converges to the bounded-DP MLE. To facilitate valid, finite-sample Bayesian inference on privatized data under unbounded DP, we propose a reversible jump MCMC algorithm which extends the data augmentation MCMC of Ju et al. (2022). We also propose a Monte Carlo EM algorithm to compute the MLE from privatized data in both bounded and unbounded DP. We apply our methodology to analyze a linear regression model as well as a 2019 American Time Use Survey Microdata File which we model using a Dirichlet distribution.

math.ST↗

arXiv2026

DDPM Score Matching and Distribution Learning

Score estimation is the backbone of score-based generative models (SGMs), especially denoising diffusion probabilistic models (DDPMs). A key result in this area shows that with accurate score estimates, SGMs can efficiently generate samples from any realistic data distribution (Chen et al., ICLR'23; Lee et al., ALT'23). This distribution learning result, where the learned distribution is implicitly that of the sampler's output, does not explain how score estimation relates to classical tasks of parameter and density estimation. This paper introduces a framework that reduces score estimation to these two tasks, with various implications for statistical and computational learning theory: Parameter Estimation: Koehler et al. (ICLR'23) demonstrate that a score-matching variant is statistically inefficient for the parametric estimation of multimodal densities common in practice. In contrast, we show that under mild conditions, denoising score-matching in DDPMs is asymptotically efficient. Density Estimation: By linking generation to score estimation, we lift existing score estimation guarantees to $(ε,δ)$-PAC density estimation, i.e., a function approximating the target log-density within $ε$ on all but a $δ$-fraction of the space. We provide (i) minimax rates for density estimation over Hölder classes and (ii) a quasi-polynomial PAC density estimation algorithm for the classical Gaussian location mixture model, building on and addressing an open problem from Gatmiry et al. (COLT'26). Lower Bounds for Score Estimation: Our framework offers the first principled method to prove computational lower bounds for score estimation across general distributions. As an application, we establish cryptographic lower bounds for score estimation in general Gaussian mixture models, conceptually recovering Song's (NeurIPS'24) result and advancing his key open problem.

stat.ML↗

arXiv2026

Exact affine conditioning beyond Gaussians: a unique characterization of the ensemble Kalman update

The analysis step of the stochastic ensemble Kalman filter, called the ensemble Kalman update (EnKU), is widely used for approximating posterior distributions in inverse problems and data assimilation. The EnKU approximates the posterior distribution $π_{X\mid Y=y_\star}$ by pushing forward the joint distribution $(X,Y)\simπ$ through an affine map $L^{\mathrm{EnKU}}_{π,y_\star}(x,y)$ that depends only on the covariance structure of $π$ and the observation $y_\star$. While the EnKU yields the exact posterior for Gaussian $π$ in the mean-field, this property alone does not uniquely determine the EnKU. In fact, there are infinitely many affine maps $L_{π, y_\star}$ that achieve such exact conditioning. In this paper, we offer a novel characterization of the EnKU among all such affine maps. We first exhaustively characterize the set ${E}^{\mathrm{EnKU}}$ of joint distributions for which the EnKU yields exact conditioning, showing that it is much larger than the set of Gaussians. Next, we show that except for a small class of highly symmetric distributions within ${E}^{\mathrm{EnKU}}$, the EnKU is the {unique} exact affine conditioning map. Further, we characterize the largest possible set of distributions ${F}$ for which a distribution-dependent, weakly observation-dependent, affine map exists, a class of transports that naturally includes the EnKU. We show that ${F}={E}^{\mathrm{EnKU}}\cup{S}_{\mathrm{nl-dec}}$ with a small symmetry class ${S}_{\mathrm{nl-dec}}$, meaning that for affine conditioning beyond the Gaussian setting, the EnKU has an exact set that is essentially maximally large.

math.ST↗

arXiv2026

Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $σ^d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.

stat.ML↗

arXiv2026

Bias-Corrected Subspace Intersection: Minimax-Optimal Shared Subspace Estimation in Multi-View Data

Estimating a low-dimensional subspace shared across noisy data matrices is a fundamental problem in multi-view matrix estimation. We study this problem under the two-view JIVE model, where each data matrix contains shared and view-specific low-rank components. We demonstrate that standard plug-in subspace intersection, including AJIVE, suffers from a second-order bias caused by direction-dependent leakage of the empirical singular vectors. We propose bias-corrected subspace intersection (BCSI), which removes this bias before estimating the shared subspace. We establish finite-sample risk bounds for BCSI that accommodate unequal view dimensions, signal strengths, and view-specific ranks and require no condition-number assumptions on the signal matrices. When the shared and view-specific ranks are comparable, these bounds match our minimax lower bounds up to universal constants. The resulting minimax rate contains a new second-order term, arising from quadratic leakage perturbations relative to the shrinking spectral gap when the view-specific subspaces are nearly aligned. This term is absent from previous JIVE minimax lower bounds. Numerical experiments demonstrate the advantage of BCSI over AJIVE when the leakage bias is pronounced. Along the way, we establish a nonasymptotic concentration result for the bias-corrected leakage Gram matrix of a rectangular spiked matrix, which may be of independent interest.

stat.ME↗

arXiv2026

A complete characterization of sequential testability and change detectability in i.i.d. models

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-$α$ test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size $O_Q(\log(1/α))$. Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every $η>0$, they are equivalent to an alarm family $(T_γ)_{γ\ge1}$ satisfying $\Prob_{P^\infty}(T_γ\leσ)\le \E_{P^\infty}σ/γ$ for every null law and every stopping time $σ$. In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between $γ$ and $(1+η)γ+1$, and having robust Lorden delay $O_Q(\logγ)$.

math.ST↗

arXiv2026

On the Equality of the ELBO to a Sum of Entropies at Stationary Points of Learning

The variational lower bound (a.k.a. ELBO or free energy) is the central objective for many established as well as for many novel algorithms for unsupervised learning. Such algorithms usually increase the bound until parameters have converged to values close to a stationary point of the learning dynamics. Here we show that (for a very large class of generative models) the variational lower bound is at all stationary points of learning equal to a sum of entropies. Concretely, for standard generative models with one set of latents and one set of observed variables, the sum consists of three entropies: (A) the (average) entropy of the variational distributions, (B) the negative entropy of the model's prior distribution, and (C) the (expected) negative entropy of the observable distribution. The obtained result applies under realistic conditions including: finite numbers of data points, at any stationary point (including saddle points) and for any family of (well behaved) variational distributions. The class of generative models for which we show the equality to entropy sums contains many standard as well as novel generative models including standard (Gaussian) variational autoencoders. The prerequisites we use to show equality to entropy sums are relatively mild. Concretely, the distributions defining a given generative model have to be of the exponential family, and the model has to satisfy a parameterization criterion (which is usually fulfilled). Proving equality of the ELBO to entropy sums at stationary points (under the stated conditions) is the main contribution of this work.

stat.ML↗

arXiv2026

Geometric Optics Approximation Sampling: A Reflector-Induced Transport Map Framework

In this paper, we propose Geometric Optics Approximation Sampling (GOAS), a reflector-induced transport-map framework for sampling from target measures. Once a reflecting surface is constructed, the associated transport map is explicitly determined by the physical law of reflection. As a concrete realization, we develop a supporting-hyperellipsoid construction that requires only a discrete approximation of the target measure and does not require gradient information of the target density. The formulation accommodates both density-based and sample-based target representations. A softmin smoothing technique is introduced to obtain a smooth approximate transport map from this piecewise hyperellipsoidal construction. We establish well-posedness and stability of the reflector-induced push-forward measure and derive quantitative error estimates in the maximum mean discrepancy metric, and convergence of continuous statistical observables, including fixed-order moments. Numerical experiments on an analytically tractable example, strongly non-Gaussian targets, sample-based target approximations, and Bayesian inverse problems demonstrate the accuracy and flexibility of GOAS.

math.NA↗

arXiv2026

Spectral Convergence of Random Feature Method in Multiple Dimensions

We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.

math.NA↗

arXiv2026

Towards a Statistical Understanding of Mixture-of-Experts

Mixture-of-experts (MoE) architectures increase model capacity by combining a collection of expert predictors through input-dependent routing, while often activating only a small subset of experts for each input. Despite their growing importance in modern large-scale models, the statistical roles of their design choices, especially routing, sparse activation, and shared experts, remain only partially understood, as existing theory has largely focused on parametric or correctly specified MoE models. In this paper, we view MoE as a form of localized aggregation and show how this localization reshapes the approximation-estimation-computation tradeoff. We derive oracle risk bounds for learning dense and sparse routing with evolving experts, separating approximation, expert-learning, and router-estimation errors, and characterize how sparse Top-K routing can retain the benefits of localized aggregation while controlling per-input computation. We also interpret gating through the geometry of input space, relating routing performance to regions of local expert advantage, and show how shared experts, as adopted in architectures such as DeepSeekMoE, can extract common predictive structure so that routed experts focus on residual local variation. Together, these results provide a unified statistical framework for understanding MoE through input-dependent expert aggregation, in which expert specialization and computational tradeoffs are governed by local predictive structure.

stat.ML↗

arXiv2026

Symmetries and Causality: Causal Effect Identification Beyond IID Data

In the natural sciences, symmetries and cause-effect relationships are ubiquitous. Yet for complex machine-learning tasks, like world-modeling in reinforcement learning, they appear difficult to harness. We propose a formal description of statistical systems based on symmetries in data leaving causal mechanisms invariant. The result is an abstract, simple and general mathematical language for causal reasoning. This paper provides formal descriptions of models and queries, setting up this language, and the formal infrastructure and strategies for their mathematically rigorous identification from data within this formalism. This approach reproduces and matches standard theoretical results on IID data and transport of experimental and non-experimental data. But its main purpose is to unify and substantially extend the scope of causal reasoning, in going beyond IID data and in approaching complex causal queries not captured by do- or soft-interventions. This new perspective on causally relevant aspects of data-modeling additionally sheds new light on well-known structures like c-components or hedges but also includes aspects of missing data and is inherently well-suited for the description of transfer and robustness properties.

math.ST↗