When Is the Sharp Covariance Envelope Tight? Feature-Only Geometry for Volume-Sampled Least Squares

Prior analyses by Derezinski and Warmuth established all-size sampling identities, selected-OLS unbiasedness, and inverse moments for ordinary volume sampling, while their exact arbitrary-fixed-response loss and prediction-covariance formulas are at the rank-size endpoint s=d. We establish a Loewner envelope for centered coefficient covariance for every full-rank fixed pool, response, and legal budget d <= s <= m under ordinary indexed fixed-size volume sampling followed by selected unweighted least squares; its coefficient is globally sharp over the full-rank class. Global sharpness does not determine attainability on the pool in hand. Under positive loss, strict-interior budgets, and no coloops, a feature-only margin nu_A gives the exact fixed-design spectral phase: nu_A > 0 if and only if the normalized spectral envelope is strict for every compatible residual, whereas nu_A = 0 if and only if some compatible residual is spectrally tight; the same zero-margin residual is tight at every strict-interior budget. A residual-augmented change of measure supplies the response-aware mechanism and a one-sided quantitative slack bound, while support saturation proves the attainment direction. Critical equal-leverage geometry interprets the boundary, and sound lower certificates yield conservative same-primitive cardinality decisions. Frozen-feature examples show that the certificate is nonvacuous and measure the fixed-pool cost of its authorized reduction. The claims concern conditional centered, full-Gram-whitened coefficient covariance, not population generalization.

Hadamard Flattening and Gaussian Pooling Sketch for Least Squares with Coordinate-wise Guarantee

Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a smaller problem. Standard subspace embeddings guarantee that the cost of the regression is nearly preserved, but coordinate-wise accuracy of the solution is more delicate: we want the solution vector itself to be close to the optimal solution in $\ell_\infty$ norm. In particular, we want to find a vector $x'\in \mathbb{R}^d$ such that $\|x'-x^*\|_\infty\leq \fracε{\sqrt d}\cdot \|Ax^\star-b\|_2\cdot \|A^\dagger\|_{\rm op}$. Price, Song and Woodruff initiated the study of this problem and showed that the subsampled randomized Hadamard transform (SRHT) with $O(ε^{-2} d^{1+Θ(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee. A subsequent work of Song, Ye, Yin and Zhang claimed to improve the row count to $O(ε^{-2}d\log^3 n)$. Unfortunately, their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform, which combines a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression in which the noise is independent of the entire sketched design; this conditional independence is exactly what the earlier argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(ε^{-2}d\log d)$ rows, uses one Hadamard pass with a padded internal dimension $N=\widetilde{O}(n+ε^{-2}d^3)$, and is efficient to apply: the sketched pair $(SA, Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+ε^{-2}d^4)$ time.

A Novel Fourier Feature Network for Solving Partial Differential Equations

Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both. Similar to Extreme Learning Machines (ELMs), FENs employ a single-hidden-layer architecture to generate a set of basis functions. The target function is then approximated as a linear combination of these basis functions, with the coefficients determined using the least squares method. However, unlike ELMs, which often rely on affine transformations to improve representational power, FENs can achieve high-precision solutions without requiring such transformations on the input variables. To evaluate the representational capacity of these networks, we search for an optimal scaling factor within a predefined range for the randomly initialized and fixed weights and biases. By adjusting this scaling factor, we ensure a fair comparison between FENs and ELMs using various activation functions, such as $\text{sigmoid}$, $\tanh$, and $\text{swish}$. Our numerical experiments demonstrate that FENs consistently achieve higher accuracy than ELMs.

Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples

We develop a rigorous theory of discrete residual least-squares approximation for elliptic spectral equations $\mathfrak L_βu=f$ using linearized ReLU$^k$ neural networks on the sphere, where $\mathfrak L_β$ is a positive elliptic spectral multiplier of order $β$. Given a parameter set $Θ_n=\{θ_{j}^*\}_{j=1}^n\subset\mathbb S^d$, we approximate $u$ in the linearized network space $L_n^k(Θ_n)$ by the discrete residual on the collocation points $\{η_i^*\}_{i=1}^m$ \begin{equation*} u_{n,m}\in\arg\min_{v_n\in L_n^k(Θ_n)}\frac1m\sum_{i=1}^m\left(f(η_i^*)-\mathfrak L_βv_n(η_i^*)\right)^2. \end{equation*} With $k>\frac{d-1}{2}+β$, for antipodally quasi-uniform network parameter sets and any quasi-uniform collocation points with $m\gtrsim n$, we prove that \begin{equation*} \|u-u_{n,m}\|_{\mathcal H^β(\mathbb S^d)}\eqsim\|f-\mathfrak L_βu_{n,m}\|_{\mathcal L^2(\mathbb S^d)}\lesssim n^{-\frac{r}{d}} \begin{cases} \|f\|_{\mathcal W^{r,p}(\mathbb S^d)},&\frac{d}{p}<r\leq \frac{d}{2},~p>2,\\ \|f\|_{\mathcal H^r(\mathbb S^d)},&r>\frac{d}{2}. \end{cases} \end{equation*} We also establish a high-probability residual estimate, up to a logarithmic factor and an arbitrarily small smoothness loss, for i.i.d.\ uniformly distributed collocation points. The key analytical ingredient is a Bernstein inequality for linearized ReLU$^k$ network spaces. If $\underline h$ denotes the antipodal separation distance of the network parameters, then \begin{equation*} \|v_n\|_{\mathcal H^r(\mathbb S^d)}\lesssim\underline h^{-(r-s)}\|v_n\|_{\mathcal H^s(\mathbb S^d)},\qquad 0\leq s<r<k+\tfrac12. \end{equation*}

Sequential Preconditioned Conjugate Gradient Method for Linear Statistical Models

We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and warm-starts each subproblem from the previous solution. A final refinement stage is then performed on the full-scale problem. Since most iterations are carried out on smaller subproblems, the overall computational cost is significantly reduced. We establish the convergence theory, prove that SPCG attains OLS prediction accuracy, and derive per-subproblem iteration bounds and complexity estimates. Numerical experiments show that SPCG reaches the target prediction accuracy with fewer iterations and less CPU time than full-data PCG and Iterative Double Sketching (IDS).

In-context learning of closed form solution to simple linear regression task using transformer with linear self-attention

In-context learning is a remarkable property of transformers and has recently received a lot of interest. In many studies of in-context learning, it has been shown that transformers are capable of implementing solver for linear and non-linear regression problems, in which the most of them implement gradient descent algorithm. However, it is still unclear whether those implementations have actually been acquired through training. In this paper, we construct a transformer with linear self-attention, which in-context learns the least squares estimate in a simple regression task. The point here is that the closed form (analytical) solution is approximately obtained by using layer normalization rather than an approximate solution based on gradient descent algorithm. Then, we show an experimental example, in which our implementation is mainly used in the transformer trained with l1 regularization when the target output is the least squares estimate.

Hybrid Least Squares/Gradient Descent Methods for MIONets

In this paper, we propose an efficient hybrid least squares/gradient descent (LSGD) method for MIONets to accelerate training. This method generalizes the LSGD method for DeepONets. Since MIONet is the sum of the entrywise product of multiple branch networks and a trunk network, it can be viewed as a multilinear function with respect to the last layer parameters of each branch network. These sets of parameters can be optimized using the alternating least squares method, where we solve the LS system for a single branch network in turn. To handle the large-sized system matrix, we introduce Kronecker and Khatri-Rao products and tensor permutation matrices to factor the large matrix into small ones. Our method is compatible with a general type of $L^2$ loss with regularization terms for the last layer parameters of each branch, where linear operators can be applied to the MIONet output in each loss term.

Distributionally Robust Linear Regression With Block Lewis Weights

We present an algorithm for the group distributionally robust (GDR) least squares problem. Given $m$ groups, a parameter vector in $\mathbb{R}^d$, and stacked design matrices and responses $\mathbf{A}$ and $\mathbf{b}$, our algorithm obtains a $(1+\varepsilon)$-multiplicative optimal solution using $\widetilde{O}(\min\{\mathsf{rank}(\mathbf{A}),m\}^{1/3}\varepsilon^{-2/3})$ linear-system-solves of matrices of the form $\mathbf{A}^{\top}\mathbf{B}\mathbf{A}$ for block-diagonal $\mathbf{B}$. Our technical methods follow from a recent geometric construction, block Lewis weights, that relates the empirical GDR problem to a carefully chosen least squares problem and an application of accelerated proximal methods. Our algorithm improves over known interior point methods for moderate accuracy regimes and matches the state-of-the-art guarantees for the special case of $\ell_{\infty}$ regression. We also give algorithms that smoothly interpolate between minimizing the average least squares loss and the distributionally robust loss.