Papers

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.

We study extremal absolute central moments of Gaussian quadratic forms under a fixed Frobenius norm. For a nonzero real symmetric matrix $M$ and $G\sim N(0,I_n)$, we construct an explicit centered difference of Gamma variables with the same mean, variance, and third centered moment as $G^{\mathsf T}MG-\operatorname{tr}M$. After normalization, replacing the quadratic form by this Gamma difference does not decrease $\mathbb{E} f$ for every $C^2$ test function $f$ such that $f''$ is convex and $f$, $f'$, and $f''$ have polynomial growth. In particular, it gives an explicit upper bound for every absolute moment of order $p\ge3$. We then prove that, for every $p\ge4$, this bound is maximized by the centered square $g^2-1$ of a single standard Gaussian variable $g$. The resulting sharp inequality is \[ \bigl\|G^{\mathsf T}MG-\operatorname{tr}M\bigr\|_p \le \|g^2-1\|_p\,\|M\|_{\mathrm F}, \] with equality if and only if $M$ has rank one.

Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds of order $\kappa\sqrt d$, up to logarithmic factors. A spectral-gap lower bound at the corresponding scale $(\kappa \sqrt{d})^{-1}$ would give an upper bound of order $\kappa \sqrt{d}$ on the Monte Carlo asymptotic variance relative to independent sampling, uniformly over all square-integrable functions. However, when $\kappa$ is bounded away from~1, the best fixed-step spectral gap that can be guaranteed uniformly over this target class is at most of order $\max\{\log(\kappa d)/(\kappa d), e^{-cd}\}$ for some positive universal constant~$c$. We show that uniform randomization of the step size improves this worst-case guarantee. At each iteration, the algorithm draws $h\sim\operatorname{Unif}(0,H)$ and performs one ordinary MALA transition. Choosing $H$ of order $[L\sqrt{d(1+\log d+\log\kappa)}]^{-1}$ yields a right spectral-gap lower bound of order \[ \frac{1}{\kappa\sqrt{d(1+\log d+\log\kappa)}}, \] uniformly over the target class. Thus, given $\kappa > 1$, for all square-integrable functions, the ratio of the Monte Carlo asymptotic variance relative to independence sampling has an upper bound of order $\sqrt{d \log d}$. This contrasts with fixed-MALA, where the ratio can be as bad as $d/\log d$ in thew worst-case scenario. The spectral gap also gives geometric convergence of the lazy kernel from every initial density in $L^2(\pi)$, central limit theorems, and nonstationary mean-square error bounds. This work was developed with substantial assistance from ChatGPT.

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.

The Lehmer transform encodes a positive dataset as a single analytic curve, the ratio of its power sums at consecutive orders read as a function of the order. As the order sweeps the real line the curve generates every location statistic between the sample extremes, and its logarithm is the unit increment of the free energy of an exponentially tilted ensemble; the statistical theory of the curve flows from that identity. The slope of the curve is a Jeffreys divergence, the arithmetic-geometric gap is a relative entropy, and multiplicativity over independent factors makes deconvolution of multiplicative noise a pointwise division. A quantitative rigidity theorem locates the frontier of identifiability at geometric growth of the curve, with resonant log-periodic families of distinct laws sharing one curve beyond it. The empirical curve is an exact re-coordinatization of the order statistic, hence minimal sufficient, and its influence function, exact leading bias and limit theory are developed in full. At heavy tails the curve undergoes a phase transition whose two knees, one unit apart, sit at the tail index and its unit shift, a whole-sample diagnostic complementing threshold-based estimators.

The purpose of this paper is to rigorously quantify the statistical and learning-theoretic properties of adversarial training models for classification. Equivalently, we establish the statistical properties of empirical optimal partial transport. Precisely, first we provide two types of central limit theorems (CLT): CLT centered at the expected empirical value, and CLT centered at the population one with smoothing. These results are based on the uniqueness of optimal potential for various equivalent optimal transport formulations, and the empirical process theory argument. For the binary setting, we indeed prove the uniqueness of optimal potential by leveraging the connection between optimal partial transport and the derived multi-marginal optimal transport formula. As byproducts, we also obtain the stability of a saddle point of the adversarial training model, and the sample complexity and concentration probability of the generalization error.

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-\(\alpha\) 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 \(O_Q(\log(1/\alpha))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(\eta>0\), they are equivalent to an alarm family \((T_\gamma)_{\gamma\ge1}\) satisfying \(\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma\) for every null law and every stopping time \(\sigma\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(\gamma\) and \((1+\eta)\gamma+1\), and having robust Lorden delay \(O_Q(\log\gamma)\).

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.

We study finite-sample valid tests of the one-sided mean hypothesis $H_0:\mu\leq 1$ against $H_1:\mu>1$ for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null $\mathbb{E}[X_i\mid\mathbf{X}_{-i}]\leq 1$, and which also gives conditions that allow combining dual certificates for p-values to show that their pointwise minimum is also a valid p-value. The framework proves finite-sample validity of Wang and Zhao's nonparametric likelihood-ratio statistic $T_{\mathrm{nplr}}$, yields a new p-value $T_{\mathrm{bin}+}$ extending the Clopper–Pearson binomial test to general nonnegative random variables, and shows that the pointwise minimum $\min\{T_{\mathrm{nplr}},T_{\mathrm{bin}+}\}$ is itself a valid and more powerful p-value. We establish sharp optimality results for such testing problems in two regimes: both $T_{\mathrm{nplr}}$ and $T_{\mathrm{bin}+}$ attain a universal detectability boundary for the null $H_0$ without moment or tail assumptions, and $T_{\mathrm{bin}+}$ attains a nonparametric power lower bound under $n^{-1/2}$-local alternatives to $H_0$. Efficient algorithms and numerical experiments demonstrate substantial finite-sample power gains over existing valid methods.

Consider $n$ independent random vectors sampled from uniform distribution on $(p-1)$-dimensional unit sphere. This paper investigates the limiting joint distribution for the minimum and the maximum values of their pairwise angles. It proves that the minimum and the maximum angles are asymptotically independent when both $n$ and $p$ tend to infinity, which solves an open problem raised in Cai, Fan and Jiang (2013) [Journal of Machine Learning Research 14, 1837-1864]. Cai, Fan and Jiang (2013) obtained the limiting marginal distributions for both the minimum and the maximum angles under assumption $\lim_{n\to\infty}{\ln n}/p=\beta$ according to whether $\beta=0$, $\beta\in (0,\infty)$, or $\beta=\infty$. This paper presents unified limits for both joint distributions and marginal distributions regardless of the relative divergence rate of $n$ and $p$. The paper also derives the limiting distributions for some statistics based on the minimum and the maximum angles,

Bayesian pseudo-posteriors based on moment conditions, such as Bayesian empirical likelihood (EL) and Bayesian exponentially tilted empirical likelihood (ETEL), provide a robust route to Bayesian inference when the model is specified only through moment restrictions, but their computation is often prohibitive. In this work, we address two barriers to Bayesian pseudo-inference under moment restrictions. First, scalable sampling is challenging for tall data because the pseudo-likelihoods are not log-additive across observations, so standard likelihood-subsampling methods do not apply. Second, in simulation-based and latent-variable problems, the moment function itself may be defined through an intractable expectation, making full-data pseudo-posterior inference infeasible. We develop a new family of subsampled pseudo-posteriors that replaces the full-data pseudo-posterior with an aggregate of mini-batch pseudo-posteriors. We show that naive mini-batching induces a shifted-mixture distortion, yielding aggregates that are correctly centered but overly diffuse. To remove this distortion, we introduce moment-function-level control variates that align mini-batch moment equations around a reference estimator and recover the full-data posterior shape. For Bayesian ETEL, we establish finite-sample total variation bounds to the full-data pseudo-posterior, allowing discontinuous and intractable moment functions. Numerical experiments show that the method applies broadly across moment-condition-based Bayesian pseudo-inference, closely approximates full-data pseudo-posteriors, and substantially reduces computational cost.

Suppose that the eventual use of data is not known when the data are reduced or collected. This note considers two simple boundary cases. In a finite statistical experiment, a statistic preserves the Bayes risk for every finite later decision problem if and only if it is sufficient. Hence, when the minimal sufficient statistic is one-to-one, exact preservation of all later decision problems permits no nontrivial reduction. We then consider adaptive sampling from $m$ independent Gaussian streams when an external query specifies the coordinate to be classified only after sampling stops. Under coordinatewise error control, the optimal symmetric average sample size is exactly $m$ times the one-coordinate optimum. A change-of-measure argument gives the corresponding pointwise lower bound in terms of binary relative entropy.

We consider a rectangular random matrix formed by adding a rank-one term to a matrix with independent entries. The direction on the left side of that term is estimated by the leading left singular vector, and the error is multiplied by the square of the size of the added term. For entries with mean zero and variance one, we identify the exact tail condition for the following statement to hold for every deterministic sequence of directions: the largest error over all sizes of the added term converges to the same fixed value determined by the limiting ratio of rows to columns. This condition (weaker than the existence of fourth moments) requires that the tail probability, multiplied by the fourth power of the threshold, to tend to zero. If it fails, convergence to this value already fails when both directions are coordinate vectors, even at a size fixed before the matrix is drawn. For regularly varying tails of order between two and four, the largest error tends to infinity.

We develop a framework for simultaneous change-point inference of high-dimensional functional time series. The observations are modeled as temporally dependent vectors whose coordinates take values in possibly different separable Hilbert spaces, thereby covering a broad class of functional data. Heterogeneous mean changes may occur at coordinate-specific locations, and the contemporaneous dependence across coordinates is left unrestricted. Our procedure is based on coordinatewise cumulative-sum statistics and a residual block multiplier bootstrap that provides a common critical value for the global test and all coordinatewise decisions. We establish a nonasymptotic Gaussian approximation, quantitative nonasymptotic bounds for strong family-wise error control under arbitrary mixtures of changed and unchanged coordinates, covariance-adaptive detection guarantees, and simultaneous high-probability bounds for change-point localization. The bounds accommodate high-dimensional regimes in which the number of functional coordinates grows exponentially in a power of the sample size. We investigate finite-sample performance in simulations and illustrate the method using river discharge curves and high-frequency financial log returns.

We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $\Pi_n^T=\E[T(\Theta)\vert \mathcal F_n]$, where $\Theta$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of $\Pi^T$, we show that the posterior statistic after additional observations is larger in convex order when the current posterior is based on fewer observations. We also prove preservation of convexity: the expected value of a convex function of the future posterior statistic is convex in the current posterior statistic. Together, these two properties provide structural tools for establishing time-monotonicity results in dynamic Bayesian decision and optimal stopping problems. If the exponential family contains an infinitely divisible distribution, the results extend to a continuous-time observation model through a family of Lévy processes.

We study regression under bounded responses in terms of excess mean squared error. When the comparator class is finite, this setting is known as model selection aggregation, and achieving minimax excess risk requires improper learning algorithms. Contrary to this, in the universal learning framework no improperness is needed, as simple empirical risk minimization achieves the best-possible exponential learning rate. Hence, the two frameworks suggest different optimal algorithmic principles. This poses the question of best-of-both-worlds guarantees: Are minimax and universal exponential rates achievable by the same algorithm? For finite hypothesis classes, we answer this question in the affirmative by showing that the $Q$-aggregation estimator - which is known to achieve minimax optimal tails - achieves exponential universal rates. A wide range of other estimators and algorithmic principles (ERM, sequential averaging, pruning, and star estimation) do not achieve both. For countably infinite hypothesis classes, we answer the question in the negative by showing that there is an inherent trade-off between achieving exponential universal and minimax uniform rates. This trade-off is exactly traced by combining optimal algorithms from each world using $Q$-aggregation. Besides these results, we prove several additional structural results about universal rates in learning with squared loss.

We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are $\alpha$-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the $\alpha$-TV-LDP. We prove that the deconvolution estimator of $r$-Sobolev smooth functions attains the pointwise rate $(n\alpha)^{-\frac{2r-1}{2r}}$ up to log factors which is faster than $(n\alpha^2)^{-\frac{2r-1}{2r+1}}$ under the classical $\alpha$-LDP and closer to the nonprivate minimax rate $n^{-\frac{2r-1}{2r}}$. Next, we use a Goldenshluger-Lepski procedure to build a free of the smoothness adaptive procedure and show optimality of our rates in the convolution model of our privatisation scheme. We illustrate the benefits of this simple privacy mechanism by implementing a neural network estimator which does not need to add more noise in the optimization steps. Numerical results show significant improvement of the estimation rate over the Laplace and the private-SGD mechanisms.

Testing whether two independent samples arise from the same underlying distribution is a fundamental statistical problem. We propose a new class of two-sample distribution tests based on a family of probability metrics $\Delta_{n,p}$, constructed from repeatedly integrated quantile functions. On their respective domains, these metrics are proved to be genuine distributional distances. The case $n=1$ recovers the $p$-Wasserstein distance, which requires finite $p$-th moments; for $n\geq2$, the proposed metrics are well defined and require only finite first moments. The asymptotic properties of the plug-in statistic are established, including strong consistency and limiting distributions under the null and fixed alternatives. A permutation calibration for finite-sample inference is also proposed. We further derive an asymptotic power function under local alternatives. Finally, the finite-sample performance of the proposed tests is examined through simulation studies, and their reduced sensitivity to extreme upper-tail observations is illustrated through a real data application.

The statistical analysis of heavy-tailed data has received considerable attention because extreme observations frequently arise in many practical applications. The Pareto type-I distribution is a fundamental heavy-tailed model used in economics, finance, actuarial science, insurance, reliability, and extreme value analysis. In this paper, we propose novel goodness-of-fit tests for the Pareto distribution using a mean residual life characterization. The test statistic is constructed using U-statistic theory, and its asymptotic behaviour is established under both the null and alternative hypotheses. Its finite-sample performance is evaluated through Monte Carlo simulations using maximum-likelihood and method-of-moments estimation and compared with existing tests. The results show that the proposed test controls the nominal significance level and performs competitively in terms of power across a broad range of alternatives. Finally, the proposed methodology is illustrated using the Danish fire insurance loss and pollution datasets.

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 $\sigma^d$, 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 $O(\sigma^2)$. We then establish a minimax lower bound of order $(n\sigma^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2\alpha+d)}\lesssim\sigma\le\sigma_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2\alpha/(2\alpha+d)}$.

Through the seminal works of Peligrad and Bradley, the existing sufficient conditions for the central limit theorem for $\rho$-mixing sequences are known to be very close to necessary. Nevertheless, we show that further improvements to these conditions are achievable when the $\rho$-mixing coefficients are non-summable, bridging a long-standing gap in the borderline case. The key idea is an iterative block-size technique that enables precise evaluations of the slowly varying variance components. Finally, known examples and new extensions–including iterated logarithmic mixing rates–are investigated to demonstrate the sharpness of our criteria, revealing a clear parameter structure and posing related open problems.

We study estimation and inference for a semiparametric class of time series models that specify only the conditional expectation, which is a known link function applied to a linear combination of past observations and covariates. The class covers count, binary, bounded and conditionally heteroskedastic responses within a single formulation, and the parameter is estimated by a quasi-likelihood estimating equation based on the first conditional moment. Under stationarity and a weak-dependence condition expressed through the functional dependence measure, we establish two results. First, using a Fuk–Nagaev inequality for weakly dependent sequences, we show that the estimator is localized in a shrinking neighbourhood of the true value with probability $1-o(n^{-1/2})$. Second, combining a Berry–Esseen bound for weakly dependent sequences with a Gaussian anti-concentration argument to control the remainder of the linear expansion, we obtain a Berry–Esseen bound for linear projections of the estimator, uniform over projection directions. From the projected bound we derive studentized confidence intervals with explicit coverage error and a conservative Bonferroni test for linear hypotheses on the parameters. For real data analysis, we extend the Beta autoregression for double-bounded data to an arbitrary link given by the inverse of a distribution function, and apply it to ten pairwise realized correlations of large-cap technology-stock returns, using Nasdaq and Dow~Jones index returns as covariates.

Let a finite population of n labelled examples carry a class-weighted loss, with pi*n in a rare positive class weighted by N0/N1. We study estimation of total risk from a subsample K << n under designs allocating K0 and K1 draws to the two strata. We derive the exact finite-population variance of the weighted risk estimator under class-conditional sampling without replacement and solve for the optimal allocation. The class multiplier inflates positive-stratum dispersion by the imbalance ratio, causing that ratio to cancel from the optimal allocation and making equal, rather than proportional, allocation the natural default. Simple random sampling is dominated by an explicit between-stratum term; an exact bias identity shows that cluster-representative selection has no general unbiasedness guarantee; and a Serfling bound transfers the allocation result to selection error over a finite candidate set. Under the implemented truncation, the realised allocation ratio is gamma=min(2*pi/f,1), independent of n, yielding the parameter-free efficiency prediction A(pi,f)=gamma/[pi(1-pi)(1+gamma)^2]. A separate measurability result bounds the record occupied by a labelled example, determining the required train-test separation and controlling departure from block independence under absolute regularity. We test these predictions on forecasting the onset of statistically explosive price regimes, dated ex post by the Phillips-Shi-Yu procedure, using 350 U.S. equities from 2004-2011 with under 1% positive rows and five purged forward blocks. The predicted ordering of the four designs holds, and at the 10-day horizon the five design points are ordered exactly as predicted by A (Spearman rho=1, exact p=0.0167). The predicted dependence on pi across horizons does not hold; we identify the channels lying outside the design-based argument.

In this paper, we study the autocovariance matrix estimation and inference problems under heavy-tailedness, high-dimensionality, general nonlinear temporal dependence, and potentially nonstationarity of time series. We consider two types of tail-robust autocovariance matrix estimation methods: the element-wise Huber's $M$-estimator and a computationally more efficient element-wise truncated estimator. Both estimators are designed to achieve sharp error bounds in matrix max-norm. The nonasymptotic properties of these estimators are proved based on new variants of Bernstein-type inequalities under functional dependence for the potentially nonstationary processes which may be of independent interest. Moreover, we prove a high-dimensional Gaussian approximation result, as a limiting distribution, for our element-wise truncated autocovariance estimator. A Gaussian multiplier bootstrap result is also given to facilitate the practicality. Our theoretical results are nonasymptotic, which gives explicit error bounds in terms of the sample size, dimensionality, moments, and the strength of temporal dependence. Numerical evidence is provided to support our theoretical results. Finally, we illustrate the benefits of the proposed methodology for detecting change points in monthly macroeconomic data.

This paper considers the energy landscape of the t-SNE algorithm. While this algorithm has enjoyed broad adoption, the non-convexity of the associated energy has made it difficult to rigorously understand what the algorithm captures in many settings. In particular, a number of well-known numerical examples, several of which are reproduced in this article, suggest a complicated energy landscape with many local minimizers that do not respect the topology or clustering structure of the underlying data. This work seeks to provide first steps towards a rigorous explanation of these phenomena. Specifically, for a general family of energies, which include both the original t-SNE algorithm and recently identified large data limits, and for densities in feature space which obey a continuous symmetry, we construct infinite families of distinct critical points. These critical points are based upon identifying pairs of discrete symmetries, one in the original feature space and the other in the target embedding space, which are preserved under gradient dynamics. These critical configurations exhibit many characteristics, such as topology breaking and spurious clustering, which are often observed empirically. Finally, numerical and analytical examples are given throughout as a means of illustrating the approach.

In his seminal paper, Bell [Physics Physique Fizika 1, 195 (1964)] considers the correlations that result from space-like separated measurements on a pair of entangled particles. He uses relativity theory to motivate the Bell causal structure, then shows the existence of quantum correlations that cannot be explained classically within this causal structure. Classical explanations of such quantum correlations are possible in alternative causal structures, for instance, those that allow superluminal causal influences, but, as shown in [New Journal of Physics 17 033002 (2015)], all such alternative explanations require fine tuning (causation without correlation). Here we discuss the existence of spurious quantum correlations — correlations that look quantum in one causal structure, but have a natural classical explanation in another. More precisely, there are causal structures that admit non-classical quantum correlations, but for which the same correlations have a classical explanation in another causal structure without fine tuning. The realisation in the other causal structure can also be achieved without breaking any natural constraints on the causal structure that follow from relativity theory. However, similarly to non-classical quantum correlations in the Bell causal structure, we find other causal structures with non-classical quantum correlations that do not have a classical causal explanation in any alternative causal structure without fine tuning.

We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein–von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a $\sqrt{N}$-scale.

Probability plays two distinct roles in modern data-analytic practice: (i) as an internally coherent, filtration-relative language for sequential forecasting and, together with a stated loss or utility, decision making; and (ii) as a foundation for empirical claims such as stabilization, calibration, and repeated-sampling validity. The first role is relative to an assessment law $\mathbb Q$, whereas the second is evaluated under a governing law $\mathbb P$ and a declared repetition regime. In classical settings the roles are often aligned by conditional correctness under $\mathbb P$ together with i.i.d.\ or other stable-law assumptions under a fixed design, or by exchangeability. In adaptive and AI-mediated settings they can diverge because forecasts operate within feedback loops while reliability is evaluated under a potentially different law and regime. We synthesize relevant results from prequential forecasting, calibration, martingale and game-theoretic validity, adaptive data analysis, conformal prediction, and long-context AI reliability into a formal account of role separation. The resulting Role Separation Principle distinguishes internal coherence from conditional-mean correctness under the evaluation law and from target-specific reference and stability conditions. Two separation results show that a proper coarsening of the limiting-frequency distribution need not identify the governing law, and that even conditionally correct forecasts under that law need not accompany stabilization. We then develop a unified stabilization framework spanning filtration-based, shift-based, policy–environment, and design-based regimes and operationalize it through a regime-conditional audit checklist. Applied illustrations clarify calibration claims, synthetic-data validity, and structural mismatches relevant to some high-confidence unsupported AI outputs.

Here, we report high-order functional network connectivity as a promising way for studying the brain connectome. Traditional functional connectivity approaches capture only pairwise relationships between brain regions, overlooking complex multivariate dependencies that underlie cognition and behavior. First, we demonstrated that high-order interactions capture more information and can distinguish between resting-state and task-state brain activity. Second, we introduce a matrix-based entropy-functional method for estimating triadic interactions, which are statistical dependencies among triplets of brain regions, and apply it to large-scale functional brain networks. The resulting triadic networks revealed distinct community patterns that complement those observed in traditional pairwise functional connectivity analyses and simultaneously capture additional connection information. Despite the potential combinatorial explosion of triadic configurations, the networks exhibited constrained and hierarchical structures that allowed computation and interpretation. These findings position triadic connectivity as a promising next-step functional connectivity framework for probing brain network organization and high-order neural interactions, while also highlighting key biological and technical challenges that require careful consideration.

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.