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.
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.
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.
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.
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.