Matrix

Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.

The Brier score conflates two distinct properties of probabilistic predictions: reliability (calibration error) and resolution (discriminatory power). We introduce the Manokhin Probability Matrix, a BCG-style two-dimensional diagnostic framework that separates them. Classifiers are placed on a 2x2 grid by Spiegelhalter Z-statistic and AUC-ROC expected rank, then assigned to one of four archetypes: Eagle (good on both axes), Bull (strong discrimination, poor calibration), Sloth (well-calibrated, weak discriminator), and Mole (poor on both). Each archetype carries a distinct prescription. We populate the matrix from a large-scale empirical study spanning 21 classifiers, 5 post-hoc calibrators, and 30 real-world binary classification tasks from the TabArena-v0.1 suite. The assignment is unambiguous. CatBoost, TabICL, EBM, TabPFN, GBC, and Random Forest are Eagles. XGBoost, LightGBM, and HGB are Bulls; Venn-Abers calibration cuts log-loss by 6.5 to 12.6% on Bulls but degrades Eagles by 2.1%. SVM, LR, LDA, and the empirical base-rate predictor are Sloths. MLP, KNN, Naive Bayes, and ExtraTrees are Moles. A theoretical asymmetry follows: no order-preserving post-hoc calibrator can add discriminatory power (Proposition 1), so calibration is the fixable part and discrimination is the hard part. The practical rule is direct: do not optimise aggregate Brier score without first decomposing it; optimise discrimination first, then fix calibration post-hoc. Code and raw experimental data are available at https://github.com/valeman/classifier_calibration.

Matrix-based optimizers have demonstrated immense potential in training Large Language Models (LLMs), however, designing an ideal optimizer remains a formidable challenge. A superior optimizer must satisfy three core desiderata: efficiency, achieving Muon-like preconditioning to accelerate optimization; stability, strictly adhering to the scale-invariance inherent in neural networks; and speed, minimizing computational overhead. While existing methods address these aspects to varying degrees, they often fail to unify them, either incurring prohibitive computational costs like Muon, or allowing radial jitters that compromise stability like RMNP. To bridge this gap, we propose Nora, an optimizer that rigorously satisfies all three requirements. Nora achieves training stability by explicitly stabilizing weight norms and angular velocities through row-wise momentum projection onto the orthogonal complement of the weights. Simultaneously, by leveraging the block-diagonal dominance of the Transformer Hessian, Nora effectively approximates structured preconditioning while maintaining an optimal computational complexity of $\mathcal{O}(mn)$. Furthermore, we prove that Nora is a scalable optimizer and establish its corresponding scaling theorems. With a streamlined implementation requiring only two lines of code, our preliminary experiments validate Nora as an efficient and highly promising optimizer for large-scale training.

The standard sparse-autoencoder (SAE) interpretability protocol labels each feature from its top-activating contexts and validates by single-feature steering. We propose the pairwise matrix protocol, co-varying steering coefficient with joint condition, and report three findings the standard one-corner protocol misses on Qwen3-1.7B-Instruct, replicated on Gemma-2-2B-it. First, a feature labelled "AI self-disclaimer" from its top contexts produces an inverted U-shape under a coefficient sweep: at c=+500 the model substitutes a fluent contemplative-philosopher voice for the disclaimer. Two further features anchor the criterion (one monotonic, one pure breakdown). Second, three near-orthogonal cluster-specific features that individually steer a philosophy-of-mind register, jointly suppressed at c=-500, damage grounded composition on recipes and engine explanations as well as introspective prompts; single-feature suppression at the same magnitude leaves controls intact. Third, a matched-geometry comparison of single-feature, joint, and random-direction perturbations (norm ~1.55, cosine ~0.64) yields three distinct output regimes: single-feature substitutes strategy filler, random direction substitutes diverse content, joint suppression alone produces placeholder text. Coherence loss is direction-pattern-dependent, not magnitude-dependent. All three findings reproduce on Gemma with model-specific damage signatures; the matched-geometry control is CI-separated by ~10x. The pipeline also locates a top causally responsible feature in Llama-3.1-8B-Instruct.

In this work, we formulate a new multi-task active learning setting in which the learner's goal is to solve multiple matrix completion problems simultaneously. At each round, the learner can choose from which matrix it receives a sample from an entry drawn uniformly at random. Our main practical motivation is market segmentation, where the matrices represent different regions with different preferences of the customers. The challenge in this setting is that each of the matrices can be of a different size and also of a different rank which is unknown. We provide and analyze a new algorithm, MAlocate that is able to adapt to the unknown ranks of the different matrices. We then give a lower-bound showing that our strategy is minimax-optimal and demonstrate its performance with synthetic experiments.

EigenDecomposition (ED) is at the heart of many computer vision algorithms and applications. One crucial bottleneck limiting its usage is the expensive computation cost, particularly for a mini-batch of matrices in deep neural networks. Our previous work proposed a dedicated QR-based ED algorithm for batched small matrices (dim${<}32$). This short paper targets the limitation and proposes a batch-efficient Divide-and-Conquer based ED algorithm for larger matrices. The numerical test shows that for a mini-batch of matrices whose dimensions are smaller than $64$, our method can be much faster than the Pytorch SVD function.

We consider the problems of computing the optimal rank-$1$ Hankel and Toeplitz-structured approximation of arbitrary matrices under $L_2$ and $L_1$-norm error. Such problems arise naturally in engineered systems, including the basic few-shot signal Direction-of-Arrival (DoA) estimation problem that is of importance to modern autonomous systems applications. We develop accurate and computationally efficient structured matrix decomposition algorithms for both formulations and then derive analytically grounded small-sample-support DoA estimators for practical sensing system deployments. The resulting estimators under the $L_2$ and $L_1$ norms are formally shown to be maximum-likelihood optimal under white Gaussian and Laplace noise, respectively. The estimators are further validated through extensive simulation studies and real-world data experiments in few-shot DoA inference.

Maintaining long-term accuracy of stereo camera calibration parameters is important for autonomous systems' perception. This work proposes Online Tracking of Essential Matrix by Stochastic Optimization (TESO). The core mechanisms of TESO are: 1) a robust loss function based on kernel correlation over tentative correspondences, 2) an adaptive online stochastic optimization on the essential manifold. TESO has low CPU and memory requirements, relies on a few hyperparameters, and eliminates the need for data-driven training, enabling the usage in resource-constrained online perception systems. We evaluated the influence of TESO on geometric precision, rectification quality, and stereo depth consistency. On the large-scale MAN TruckScenes dataset, TESO tracks rotational calibration drift with 0.12 deg precision in the Y-axis (critical for stereo accuracy) while the X- and Z-axes are five times more precise. Tracking applied to sequences with simulated drift shows similar precision with respect to the reference as tracking applied to no-drift sequences, indicating the tracker is unbiased. On the KITTI dataset, TESO revealed systematic inconsistencies in extrinsic parameters across stereo pairs, confirming previous published findings. We verified that intrinsic decalibration affected these errors, as evidenced by the conflicting behavior of the rectification and depth metrics. After correcting the reference calibration, TESO improved its rotation precision around the Y-axis 20 times to 0.025 deg and its depth accuracy 50 times. Despite its lightweight design, direct optimization of the proposed TESO loss function alone achieves accuracy comparable to that of neural network-based single-frame methods.

We consider the problem of minimizing the convergence time for decentralized federated learning (DFL) in wireless networks under broadcast communications, with focus on mixing matrix design. The mixing matrix is a critical hyperparameter for DFL that simultaneously controls the convergence rate across iterations and the communication demand per iteration, both strongly influencing the convergence time. Although the problem has been studied previously, existing solutions are mostly designed for decentralized parallel stochastic gradient descent (D-PSGD), which requires the mixing matrix to be symmetric and doubly stochastic. These constraints confine the activated communication graph to undirected (i.e., bidirected) graphs, which limits design flexibility. In contrast, we consider mixing matrix design for stochastic gradient push (SGP), which allows asymmetric mixing matrices and hence directed communication graphs. By analyzing how the convergence rate of SGP depends on the mixing matrices, we extract an objective function that explicitly depends on graph-theoretic parameters of the activated communication graph, based on which we develop an efficient design algorithm with performance guarantees. Our evaluations based on real data show that the proposed solution can notably reduce the convergence time compared to the state of the art without compromising the quality of the trained model.

Training instability remains a critical challenge in large language model (LLM) pretraining, often manifesting as sudden gradient explosions that waste significant computational resources. We study training failures in a 5M-parameter NanoGPT model scaled via $μ$P, identifying two key phenomena preceding collapse: (1) rapid decline in weight matrix stable rank (ratio of squared Frobenius norm to squared spectral norm), and (2) increasing alignment between adjacent layer Jacobians. We prove theoretically that these two conditions jointly cause exponential gradient norm growth with network depth. To break this instability mechanism, we propose MSign, a new optimizer that periodically applies matrix sign operations to restore stable rank. Experiments on models from 5M to 3B parameters demonstrate that MSign effectively prevents training failures with a computational overhead of less than 7.0%.

We study the problem of differentially private second moment estimation and present a new algorithm that achieve strong privacy-utility trade-offs even for worst-case inputs under subsamplability assumptions on the data. We call an input $(m,α,β)$-subsamplable if a random subsample of size $m$ (or larger) preserves w.p $\geq 1-β$ the spectral structure of the original second moment matrix up to a multiplicative factor of $1\pm α$. Building upon subsamplability, we give a recursive algorithmic framework similar to Kamath et al 2019, that abides zero-Concentrated Differential Privacy (zCDP) while preserving w.h.p. the accuracy of the second moment estimation upto an arbitrary factor of $(1\pmγ)$. We then show how to apply our algorithm to approximate the second moment matrix of a distribution $\mathcal{D}$, even when a noticeable fraction of the input are outliers.