We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hatθ^\top x$ approximately follows the convolution of the (generally non-Gaussian) distribution of $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.

This paper studies high-dimensional M-estimation in the proportional asymptotic regime (p/n -> gamma > 0) when the noise distribution has infinite variance. For noise with regularly-varying tails of index alpha in (1,2), we establish that the asymptotic behavior of a regularized M-estimator is governed by a single geometric property of the loss function: the boundedness of the domain of its Fenchel conjugate. When this conjugate domain is bounded -- as is the case for the Huber, absolute-value, and quantile loss functions -- the dual variable in the min-max formulation of the estimator is confined, the effective noise reduces to the finite first absolute moment of the noise distribution, and the estimator achieves bounded risk without recourse to external information. When the conjugate domain is unbounded -- as for the squared loss -- the dual variable scales with the noise, the effective noise involves the diverging second moment, and bounded risk can be achieved only through transfer regularization toward an external prior. For the squared-loss class specifically, we derive the exact asymptotic risk via the Convex Gaussian Minimax Theorem under a noise-adapted regularization scaling. The resulting risk converges to a universal floor that is independent of the regularizer, yielding a loss-risk trichotomy: squared-loss estimators without transfer diverge; Huber-loss estimators achieve bounded but non-vanishing risk; transfer-regularized estimators attain the floor.

In the present paper we study the performance of linear denoisers for noisy data of the form $\mathbf{x} + \mathbf{z}$, where $\mathbf{x} \in \mathbb{R}^d$ is the desired data with zero mean and unknown covariance $\mathbfΣ$, and $\mathbf{z} \sim \mathcal{N}(0, \mathbfΣ_{\mathbf{z}})$ is additive noise. Since the covariance $\mathbfΣ$ is not known, the standard Wiener filter cannot be employed for denoising. Instead we assume we are given samples $\mathbf{x}_1,\dots,\mathbf{x}_n \in \mathbb{R}^d$ from the true distribution. A standard approach would then be to estimate $\mathbfΣ$ from the samples and use it to construct an ``empirical" Wiener filter. However, in this paper, motivated by the denoising step in diffusion models, we take a different approach whereby we train a linear denoiser $\mathbf{W}$ from the data itself. In particular, we synthetically construct noisy samples $\hat{\mathbf{x}}_i$ of the data by injecting the samples with Gaussian noise with covariance $\mathbfΣ_1 \neq \mathbfΣ_{\mathbf{z}}$ and find the best $\mathbf{W}$ that approximates $\mathbf{W}\hat{\mathbf{x}}_i \approx \mathbf{x}_i$ in a least-squares sense. In the proportional regime $\frac{n}{d} \rightarrow κ> 1$ we use the {\it Convex Gaussian Min-Max Theorem (CGMT)} to analytically find the closed form expression for the generalization error of the denoiser obtained from this process. Using this expression one can optimize over $\mathbfΣ_1$ to find the best possible denoiser. Our numerical simulations show that our denoiser outperforms the ``empirical" Wiener filter in many scenarios and approaches the optimal Wiener filter as $κ\rightarrow\infty$.

Gaussian mixtures are a popular model in high-dimensional statistics since, besides being an universal approximator, they often lead to mathematically tractable problems.