Neural networks are widely used for image-related tasks but typically demand considerable computing power. Once a network has been trained, however, its memoryand compute-footprint can be reduced by compression. In this work, we focus on compression through tensorization and low-rank representations. Whereas classical approaches search for a low-rank approximation by minimizing an isotropic norm such as the Frobenius norm in weight-space, we use data-informed norms that measure the error in function space. Concretely, we minimize the change in the layer's output distribution, which can be expressed as (W fW)Σ1/2 F where Σ1/2 is the square root of the covariance matrix of the layer's input and W, fW are the original and compressed weights. We propose new alternating least square algorithms for the two most common tensor decompositions (Tucker-2 and CPD) that directly optimize the new norm. Unlike conventional compression pipelines, which almost always require post-compression fine-tuning, our data-informed approach often achieves competitive accuracy without any fine-tuning. We further show that the same covariance-based norm can be transferred from one dataset to another with only a minor accuracy drop, enabling compression even when the original training dataset is unavailable. Experiments on several CNN architectures (ResNet-18/50, and GoogLeNet) and datasets (ImageNet, FGVC-Aircraft, Cifar10, and Cifar100) confirm the advantages of the proposed method.

We present an algorithm based on the alternating direction method of multipliers (ADMM) for solving nonlinear matrix decompositions (NMD). Given an input matrix $X \in \mathbb{R}^{m \times n}$ and a factorization rank $r \ll \min(m, n)$, NMD seeks matrices $W \in \mathbb{R}^{m \times r}$ and $H \in \mathbb{R}^{r \times n}$ such that $X \approx f(WH)$, where $f$ is an element-wise nonlinear function. We evaluate our method on several representative nonlinear models: the rectified linear unit activation $f(x) = \max(0, x)$, suitable for nonnegative sparse data approximation, the component-wise square $f(x) = x^2$, applicable to probabilistic circuit representation, and the MinMax transform $f(x) = \min(b, \max(a, x))$, relevant for recommender systems. The proposed framework flexibly supports diverse loss functions, including least squares, $\ell_1$ norm, and the Kullback-Leibler divergence, and can be readily extended to other nonlinearities and metrics. We illustrate the applicability, efficiency, and adaptability of the approach on real-world datasets, highlighting its potential for a broad range of applications.

We study the minimax estimation error for distributed covariance matrix estimation in the vertical-split (feature-split) setting, where two agents each observe different coordinates of $m$ i.i.d. sub-Gaussian samples and communicate a limited number of bits to a central server. While Rahmani et al. [2025] established nearly tight bounds for dense (unstructured) cross-covariance matrices, we investigate whether imposing elementwise $s$-sparsity on the cross-covariance $C_{21}$ can reduce the required communication and sample complexity. In contrast to the horizontal-split setting, where Braverman et al. [2016] showed that sparsity does not reduce communication cost for mean estimation, we prove that sparsity does help for cross-covariance estimation in the vertical split. Specifically, we establish minimax lower bounds showing that the communication budget per agent scales as $B_k = Ω(σ^4 d_k\, s' \log(d_1 d_2/s')/\varepsilon^2)$ and the sample complexity for cross-covariance estimation as $m = Ω(σ^4\, s' \log(d_1 d_2/s')/\varepsilon^2)$, where $s' = s \wedge d_{\min}$. For the $1$-sparse case, this yields an exponential improvement from $d_1 d_2$ to $\log(d_1 d_2)$ compared to the dense rate. Our lower bounds are established via Fano's method with an explicit sparse packing using a Varshamov--Gilbert-type argument for signed partial permutation matrices combined with the Conditional Strong Data Processing Inequality of Rahmani et al. [2025]. We show the bounds are tight with a matching achievable scheme, based on covering-net quantization and entry-wise hard thresholding, that attains the $s$-sparse lower bound up to polylogarithmic factors.

Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured-and unstructured-grid problems.

In this section, we provide further intuition about the proposed AdaQN method. As shown in Figure 5, in AdaQN we need to ensure that the approximate solution of the ERM problem with m samples denoted by wm is within the superlinear convergence neighborhood of BFGS for the ERM problem with n = 2msamples. Here, w m and w n are the optimal solutions of the risks Rm and Rn corresponding to the sets Sm and Sn with mand nsamples, respectively, where Sm Sn. The statistical accuracy region of Rm is denoted by a blue circle, the statistical accuracy region of Rn is denoted by a red circle, and the superlinear convergence neighborhood of BFGS for Rn is denoted by a dotted purple circle. As we observe, any point within the statistical accuracy of w m is within the superlinear convergence neighborhood of BFGS for Rn.

The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the particular noise structure of SGD to its linear stability (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum θ is linearly stable for SGD, then it must satisfy H(θ) F O( B/η), where H(θ) F,B,η denote the Frobenius norm of Hessian at θ, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum exponentially fast. Hence, for minima accessible to SGD, the sharpness--as measured by the Frobenius norm of the Hessian--is bounded independently of the model size and sample size. The key to obtaining these results is exploiting the particular structure of SGD noise: The noise concentrates in sharp directions of local landscape and the magnitude is proportional to loss value. This alignment property of SGD noise provably holds for linear networks and random feature models (RFMs), and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are also justified by extensive experiments on CIFAR-10 dataset.