We establish a sharp phase transition for fitting random vectors by an ellipsoid. The random vectors have independent subgaussian coordinates with mean zero, variance one, and a common fourth moment, and the number of vectors is proportional to the square of the dimension. We identify an explicit satisfiability threshold such that, with high probability, a positive definite ellipsoid passes through every data point below the threshold, whereas no positive semidefinite fit exists above it. We also determine the optimal squared fitting error throughout the unsatisfiable regime. In particular, the threshold depends on the coordinate distributions only through their common fourth moment, revealing a fourth moment universality phenomenon. For standard Gaussian data the threshold is $1/4$, resolving the ellipsoid fitting conjecture.
Theodor Misiakiewicz, Garrett G. Wenmath.PR cond-mat.dis-nn cs.DS math.ST stat.ML
Let $x_1,\ldots,x_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$. An \emph{ellipsoid fit} is a matrix $S \succeq 0$ such that $x_i^\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\{ x : x^\top S x = d\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \to \infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \sim d^2/4$. We prove this conjecture. If $\lim \sup n/d^2 = α^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigenvalues in a fixed interval $[λ_- , λ_+] \subset (0,\infty)$ depending only on $α^*$. Conversely, if $\lim \inf n/d^2 > 1/4$, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension $d(d+1)/4$ of the positive semidefinite cone.
Transformer layers generate state-dependent interaction networks: token representations determine the attention matrix, which in turn updates the representations. We study this feedback in a minimal normalized self-attention dynamics and identify the overlap gap as the central quantity governing its attractor structure in the thermodynamic limit. When tokens form internally aligned clusters and their similarity to members of the same cluster exceeds that to every other cluster by a nonvanishing amount, inter-cluster attention is exponentially suppressed as the dimension increases. This mechanism produces a high-dimensional manifold of clustered fixed points, ranging from a few macroscopic clusters to extensive microscopic fragmentation, and also controls their stability against perturbations. Starting from an unstructured Gaussian state, we find that clustered states nucleate from the diffuse background only above a finite threshold in attention sharpness, giving rise to a dynamical attention-condensation transition.
High dimensional statistical theory has established the importance of constant aspect ratio, when the number of dimensions ($d$) and samples ($n$) satisfy $n,d\to\infty$ with $n/d\to γ\in(0,\infty)$, in understanding the limits of canonical estimation problems. In particular, for estimating the top eigenvector of a $d\times d$ population covariance matrix from $n$ iid samples, the BBP phase transition gives a precise threshold -- a simple functional of the aspect ratio -- such that the top sample principal component attains nonzero asymptotic correlation with the truth only when the leading population eigenvalue exceeds it. In this paper, we show that for online / streaming algorithms the story is very different, and constant aspect ratio is insufficient for nonzero overlap. We study Oja's algorithm, the most popular method for online PCA. Let $Σ=θ^2 v_0v_0^\top+I\in\mathbb{R}^{d\times d}$, and run Oja's algorithm with step size $δ/d$ on $n$ iid samples $X_k\sim\mathcal{N}(0,Σ)$, with output $\hat v_n$. Then, as $n,d\to\infty$ with $n/d\log d\toγ\in(0,\infty)$, we establish a phase transition: $|\langle\hat v_n,v_0\rangle|\to 0$ when $γ<γ_*$, and $\toρ_*$ when $γ>γ_*$. Here $ρ_*=ρ_*(θ,δ)=\sqrt{(θ^2-δ/2)_+/θ^2(1+δ/2)}$ and $γ_*=γ_*(θ,δ)=1/2δ(θ^2-δ/2)_+$. Further, at criticality, when $n=[γ_*d\log d+ηd]$ and $d\to\infty$, $η\in\mathbb{R}$, the correlation is random: $|\langle\hat v_n,v_0\rangle|\stackrel{w}{\to}ρ_*|G|\exp(η/2γ_*)/\sqrt{ρ_*^4+G^2\exp(η/γ_*)}$ where $G\sim\mathcal{N}(0,1)$. This is in stark contrast to ordinary high dimensional PCA, where nonzero overlap is possible at constant $n/d$ and improves as $n/d$ increases.
Using the language of Wilsonian renormalization group theory (RG), we treat the Transformer's attention mechanism as a perturbation of the trained MLP residual-stack fixed point and ask whether it constitutes a relevant, marginal, or irrelevant operator. We derive a fixed-point shift formula and obtain four testable predictions for the fixed-point geometry, effective rank profile, layer specificity, and perturbation decay spectrum. Testing these on synthetic Markov chain sequences with controlled correlation length, we find: (1) For large chains(long correlation), attention is strongly relevant: it closes a residual loss gap the MLP cannot bridge and drives a phase transition in representation space, with effective rank jumping above input dimensionality at layer 1 and stabilizing at a high-dimensional plateau. (2) For short chains(short correlation), attention is irrelevant: the Transformer converges to the same loss and fixed-point geometry as the MLP, though it contracts perturbations faster. (3) The transition is dominated by the first-layer head (L0H0), which accounts for more than 4 times the representational shift of any subsequent head, consistent with the prediction that the relevant operator acts before the MLP begins integrating out positional variation. (4) Perturbation decay experiments reveal a regime reversal: in the long correlation regime the Transformer selectively preserves slow Markov modes (5.4 times the dynamic range in decay length vs. 1.3 times for the MLP); in the short correlation regime it suppresses all modes faster than the MLP, with no spectral selectivity. Together, these results show that the relevance of attention is not a property of the architecture but of the spectral structure of the data-generating process, and that a first-order RG perturbation framework provides a predictive account of that difference.
Grokking -- the delayed generalization of neural networks long after they have memorized their training data -- wastes thousands of training epochs and is notoriously unpredictable. Building on the recent result that Transformer attention is formally isomorphic to a thermodynamic system, we treat the variance of attention logits as a specific heat Cv and show that its peak reliably precedes the generalization transition. We introduce CvAdamW, a drop-in AdamW variant that monitors Cv online and injects thermal energy by dynamically scaling weight decay when a phase transition is detected. Through a strictly iterative development process we identify three failure modes -- initialization noise, mini-batch micro-ripples, and slingshot blinding -- and resolve them with a memorization gate and an exponential-moving-average shock absorber. On modular arithmetic (a+b mod 97), CvAdamW enables grokking at epoch 2802 in a 4000-epoch budget where the baseline never groks. We further propose a scale-invariant z-score reformulation that removes task-specific hyperparameters, and evaluate it across 10 paired seeds. A paired analysis shows the cold-start variant reduces mean grokking latency by 257 epochs (6.0%; median 166 epochs; Wilcoxon p=0.049, Cohen's d=0.68, bootstrap 95% CI [53,489]), improving 8 of 10 seeds; on this single task Cv peaks before grokking in all 10 seeds. Our results indicate that neural networks may expose detectable precursors of impending generalization transitions, and that a physically motivated, proportional intervention can facilitate generalization within a fixed compute budget. Code and data are public.
Grokking -- the delayed onset of generalization long after a network has fit its training set - -is usually studied in models too large to read completely and reported from single training runs. We instead study a publicly released ~11,856-parameter Llama-style transformer (Glimmer-1-Base) on modular arithmetic, small enough to enumerate its weights, attention, and full input-output map, and we measure grokking as a multi-seed rate rather than a single outcome. In this fully-tractable regime grokking is a conditional, fragile phase transition. It is gated by training-set coverage, whose threshold tracks output cardinality (the modulus) more than task structure, an ordering that holds above the transition and across a ten-fold change in domain size. Weight decay reproduces the Omnigrok inverted-U at 12K parameters, a positive control on the rate measurement. Grokking also sits on a numerical knife-edge: two perturbations of the floating-point environment -- CPU thread count (reduction order) and CPU-versus-GPU execution -- each flip a minority of same-seed outcomes without a detectable shift in the aggregate rate. Decomposition into sub-task specialists helps chiefly by making coverage cheap rather than by adding supervision. Methodologically, multi-seed control under a fixed numerical environment overturns three dramatic single-run narratives in our own data, each a seed confound. The unit of evidence for grokking must therefore be a multi-seed rate under a pinned numerical environment, checked where possible against a direct reading of the model.
The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.
The rapid scaling of over-parameterized machine learning architectures, particularly LLMs, raises a profound crisis: do these systems exhibit genuine intelligence, or are they merely sophisticated statistical pattern matchers? Classical flat Euclidean statistics cannot differentiate continuous interpolation from the autonomous discovery of novel causal laws. To resolve this, we introduce Statistically Meaningful Geometry (SMG), a framework modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. We prove that under persistent out-of-distribution (OOD) stimuli governed by unmodeled causal mechanisms, continuous optimization fails. Unmodeled variance is rejected by the visible horizontal base manifold, leaking into the unobservable vertical fiber space and generating an accumulation of Active Acausal Tension. Driven by the statistical manifold's non-linear curvature, this tension inevitably strikes a conjugate focal boundary ($T_{\text{crit}} = π^2 / K_{\text{max}}$), triggering localized volumetric collapse and a catastrophic matrix singularity ($[G_f]^{-1} \to \infty$). We demonstrate this geometric breakdown acts as the strict non-equilibrium trigger for a Gauge Symmetry Break (GSB). The system purges hidden tension from unobservable gauge redundancies, spontaneously crystallizing a new, mathematically independent horizontal coordinate axis. This non-parametric phase transition registers as a discrete $+1.0$ integer step-jump in observable Structural G-Entropy. By decoupling parameter charts and subjecting emergent axes to a Minimal Energy Path Criterion and a Causal Invariance Filter, we distinguish genuine discovery from malignant hallucinations. Ultimately, SMG provides a parameter-free, falsifiable dashboard to mathematically certify true intelligence, transforming AI for Science into an engine of autonomous paradigm shifts.
Water looks unchanged as it warms, then at a critical point it boils. We ask whether long-horizon language agents show an analogous transition in their implicit world models. In some parameter settings, changing state load by a small amount, or adding a single step of horizon, leaves behavior nearly unchanged; near a critical boundary, the same small change causes a sudden world collapse. We study this effect in a deterministic task family with exact per-step gold state. A large grid search over state cardinality, dependency density, horizon, branching, observation mode, and mutation rate reveals a phase diagram: a solved plateau, a narrow transition band, and a collapse floor. Per-step traces show the mechanism: world-state fidelity fails before action validity, so the agent is not merely choosing a bad action; it is acting from a corrupted world. Stronger models translate the critical boundary but do not remove the qualitative transition. These results make world-model collapse a measurable bottleneck for long-horizon agents.
Can a vision model truly see an object, or does it only fit surface-level visual cues? Following Wittgenstein's view that the limits of language are the limits of the world, we view a model's recognition ability as bounded by the descriptive system it has learned. In current vision models, this system is often realized through learned feature representations that exploit local statistical cues. We therefore ask whether a model can still classify correctly when such local cues provide no stable basis for distinction. We formalize this question with syntactic distance, which measures class separability through the symmetry of the operations mapping one class to the other: positive distance exposes exploitable local features, whereas zero distance requires global semantics rather than local rules. We construct a visual self-referential task in maximum-variance binary noise: positive samples contain a closed square, while negative samples contain an otherwise identical square with one flipped boundary pixel. The two classes differ in global semantics but have zero syntactic distance, making local statistical shortcuts unreliable. Experiments on ResNets and Vision Transformers reveal a consistent phase-transition phenomenon, with accuracy collapsing to random guessing once the image scale crosses a critical point and does not recover within the tested range. Larger training sets and models only delay this collapse, while globally attentive ViTs reach it earlier. These results reveal a structural capability boundary of current architectures on global-concept tasks, suggesting that general intelligence may require creating new language, not reusing an existing one.
Foundation models are often used as fixed black-box predictors for downstream tasks with limited labeled data, but their predictions may be biased and unsafe to trust blindly. We study this setting through black-box assisted nonparametric regression: a learner observes labeled samples and can query a fixed predictor $f_0$, while the target $f^*$ is close to $f_0$ in $L_2(P_X)$ up to an unknown radius $δ$. We give a finite-sample minimax characterization showing a phase transition at $δ_c(n) \asymp n^{-β/(2β+d)}$, with leading risk $\min\{δ^2, n^{-2β/(2β+d)}\}$. We then analyze a Safe Residual Estimator: it learns a correction around $f_0$, initializes the residual head at zero so the initial predictor equals $f_0$, and uses holdout selection to revert to $f_0$ when the learned correction is not supported by validation data. Here, "safe" means avoiding negative transfer, i.e., performing worse than the black-box predictor alone. The estimator matches the leading minimax term up to an additive validation-selection cost. Synthetic regression experiments verify the predicted phase transition, while CIFAR-100 with CLIP and AG News with Qwen3-8B provide practice-facing evidence that the same residual-correction tradeoff is useful beyond the formal squared-loss regression setting.
When a person's records appear in k independent data silos, each protected by (epsilon, delta)-differential privacy, standard composition yields a valid (k*epsilon, k*delta)-DP guarantee for the joint output. This worst-case bound, however, does not answer the concrete inference question: at what k can an adversary actually identify a target person? This paper develops the information-theoretic framework needed to answer that question. We introduce cross-silo person-level DP (XSP-DP), a Pufferfish-style privacy notion whose adjacency relation captures all records of a single person across all silos simultaneously, and verify that the standard basic composition bound carries over to this adjacency model. Within this framework we prove that de-anonymization undergoes a phase transition at k* = Theta(log n / epsilon^2) (population size n, per-silo RR parameter epsilon): a Fano lower bound shows any estimator fails for k << k*, while a matching maximum-likelihood upper bound shows the attack succeeds for k >> k*. An explicit XOR + randomized-response construction demonstrates information synergy: each silo's output is individually uninformative about the target, yet the joint mutual information is strictly positive. For non-coordinated binary randomized-response mechanisms, we prove that de-anonymization is inevitable once k exceeds the threshold, establishing that cross-silo coordination is necessary. These results provide a baseline threat model and Theta-level threshold for cross-silo inference attacks under local DP.
A filtered approximate-nearest-neighbor (ANN) query returns the k nearest vectors among those satisfying an attribute predicate P of selectivity s. The best execution strategy -- pre-filter, post-filter, or in-filter -- changes with s, so a system must estimate s and choose. We model this as an argmax over a landscape with phases (regions where each strategy wins) separated by boundaries, and show that selectivity-estimation error produces plan regret -- recall lost versus the oracle strategy -- only in the critical regions around those boundaries. The regret is a wedge of log-width equal to the multiplicative estimation error epsilon and height equal to the local cliff |V'(s*)| epsilon; the flip-margin 1/|V'(s*)| is the condition number of a sibling cardinality-estimation study reappearing as the local boundary theory. The two phase boundaries follow from independent mathematics: order statistics place the post-filter cliff at s ~ k/K, and site percolation places the in-filter cliff at s_c ~ 0.83/M for graph degree M (corpus-size independent). Criticality exists only under a constrained budget B < sqrt(k n). Under pre-registered decision rules we confirm, on synthetic sweeps and real SIFT1M, that regret concentrates ~290x at the boundary and that the regret curves obey a finite-size scaling collapse onto one universal wedge across two decades of corpus size. A real approximate index does not mis-locate the boundary, but a biased cost model opens a persistent miscalibration band that estimation-error robustness cannot fix. The contribution is a characterization, not a new index. Code and the full pre-registration are public.
We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+σ^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $σ= o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(Π^\star)$, i.e., $\|X^\star-Π^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-Π^\star\|_F^2$ transitions from $o(n)$ for $σ= \tilde{o}(n^{-1/2})$ to $Ω(n)$ for $σ= \tildeΩ(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.
Invariant learning can fail even when the invariant structure is statistically identifiable. We show a conditional computational barrier: under a black-box samplable supervised sparse recovery primitive motivated by average-case sparse-recovery reductions, there exist \emph{samplable} multi-environment instances with a one-dimensional predictive invariant subspace ($k=1$) that are learnable with polynomial samples by exhaustive search, while any polynomial-time constant-accuracy recovery algorithm would contradict the primitive. We further quantify environment diversity by a separation parameter $γ$, which controls identifiability and the curvature of invariance objectives. Under sufficient diversity and local Gaussian regularity, the minimax risk is $\mathbb{E}[\dist(\hat{V},V_{\mathrm{inv}})^2]=Θ(k(d-k)/(n|\mathcal{E}|))$, and under label-induced shifts a phase transition occurs at $n^*\propto k(d-k)/(|\mathcal{E}|γ^2)$ with refined estimation error scaling proportional to $1/γ^2$. Synthetic and real datasets illustrate the predicted gaps and transitions and motivate simple diversity diagnostics.
Gilad Lerman, Teng Zhangcs.IT cs.LG math.ST stat.ML
Robust Subspace Recovery (RSR) aims to identify an underlying d-dimensional subspace from a dataset heavily corrupted by outliers. Complexity-theoretic results establish a threshold for the problem's computational hardness based on the dimension-scaled signal-to-noise ratio (DS-SNR): the problem is SSE-hard when the DS-SNR is strictly less than 1, and solvable via practical algorithms when it is greater than 1 under general position assumptions. However, the exact behavior of practical algorithms at the critical boundary DS-SNR = 1 has remained unknown. This work resolves the behavior of Tyler's M-estimator (TME) at this critical boundary, consequently establishing a sharp phase transition. Specifically, we prove that TME converges exactly to the true subspace for DS-SNR \geq 1 under a new stability condition, which is less restrictive than the general position assumptions used in prior literature. Our analysis utilizes a decomposition of the TME iterates within a majorization-minimization framework.
We propose a statistical-field framework for text generated by large language models (LLMs), treating token embeddings as continuous spin variables on a one-dimensional chain. Defining a susceptibility from the connected two-point correlator and an order parameter from the ensemble-averaged embedding field, we vary the \texttt{softmax} temperature $T$ and observe a sharp susceptibility peak near a characteristic $T_c$ with power-law-like scaling, a concurrent rapid change in the order parameter, and a collapse onto a single semantic direction below $T_c$. The intrinsic dimension estimated by the two nearest neighbor (TwoNN) method independently corroborates these findings, reaching a minimum near $T_c$. Results are robust across model scales (Qwen3: 0.6B--32B) and prompt categories. While the phenomenology closely resembles a continuous phase transition, the non-equilibrium nature of autoregressive generation warrants further investigation. Our framework provides quantitative tools for probing the collective statistical structure of LLM outputs and suggests connections between decoding strategies and critical phenomena.
Kyunghoo Mun, Matthew Rosenzweigmath.AP math-ph math.PR stat.ML
We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $β_*^{(d)}>0$ such that \begin{equation*} \frac{I_{d/2+1}(β_*^{(d)})}{I_{d/2}(β_*^{(d)})}=\frac1d, \end{equation*} where $I_ν$ is the modified Bessel function of the first kind. For $0<β\le β_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation*} K_\#^{(d)}(β)=\frac{β^{d/2}}{2^{d/2}Γ(d/2)I_{d/2}(β)}, \end{equation*} and the phase transition is continuous. For $β>β_*^{(d)}$, the uniform density is not globally minimizing at $K_\#^{(d)}(β)$, so the critical coupling satisfies $K_c<K_\#^{(d)}(β)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner--Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk--Hecke/Bessel coefficient computation and a degree-two quartic obstruction.