The Loss Does Not See the Basis, but Adam Does

Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Sparsity Induced Identifiability in Matrix Tri-Factorisation

Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction. Compared to conventional two-factor models, matrix tri-factorisation provides greater modelling flexibility, while sparsity constraints often improve both interpretability and recovery performance. Although the role of sparsity has been extensively studied for two-factor matrix factorisation, rigorous theoretical guarantees for general real-valued matrix tri-factorisation remain largely unexplored. To address this gap, we establish, to the best of our knowledge, the first rigorous theoretical study for sparsity-induced identifiability in general real-valued matrix tri-factorisation. Our analysis is enabled by a novel decomposition strategy that transforms the original problem into two coupled auxiliary factorisation problems, while preserving the structural information necessary to the recovery of the original factor matrices from the observations. Building upon this decomposition, we derive recovery guarantees and structural consistency results that characterise how coefficient sparsity influences the sufficient recovery conditions, convergence behaviour, spectral approximation error, high-probability bounds, and structure preservation. Comprehensive Monte Carlo experiments validate the proposed theory and demonstrate close agreement between the theoretical results and empirical observations.

Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across computational imaging, with medical applications increasingly demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in clinical pipelines. This study introduces Quasi-SVD, a differentiable, fully parallelized matrix factorization framework for GPUs. Rather than enforcing orthogonality on both factors, it guarantees exact orthogonality for a single Lie-parameterized factor while recovering the remaining components through soft constraints, enabling efficient parallel decomposition without iterative singular-vector optimization. This asymmetric design, provably sufficient for valid factorisation, achieves reconstruction fidelity of SSIM = 0.89-0.94 and accelerates computation by 3-20x relative to cuSOLVER and randomised SVD, enabling throughput above 25 FPS. Performance is evaluated on two medical imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, and (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices. Across both regimes and multiple imaging instruments, the proposed framework demonstrates robust domain transfer and throughput exceeding 25 FPS at clinical matrix scales, a rate sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, Quasi-SVD makes structured matrix factorisation practical for real-time imaging.

Reassessing Muon for Matrix Factorization

Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training. Its empirical success has motivated a growing body of theoretical work that interprets Muon as steepest descent under the spectral norm. Yet it remains unclear which of Muon's advantages stem from its update rule itself and which are artifacts of the scale, architecture, and data of modern deep networks. In this work, we isolate the optimizer from these confounding factors by studying Muon on a simple, well-understood, and spectrally structured problem: low-rank matrix factorization. Through a controlled comparison against carefully tuned adaptive baselines, we find that Muon does not consistently outperform AdamW in this setting and that several previously reported advantages are sensitive to hyperparameter choices. Our results provide a more nuanced picture of when spectrum-aware orthogonalization is beneficial and argue for evaluating modern optimizers on controlled problems in addition to end-to-end benchmarks.

Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima

An important quantity in the theory of gradient descent (GD) is the \emph{sharpness}, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a \emph{single scalar output}, providing a normal form for large-step GD in a neighbourhood of an \emph{isolated} flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with \emph{vector-valued outputs} (including regression with arbitrarily many observations), and (2) to a neighbourhood of a \emph{manifold} of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges, including the solution of a singular partial differential equation via a novel method that may be of independent interest. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

On the Complexity of Entrywise Power Matrix Factorization

Given a nonnegative matrix $X$, a factorization rank $r$ and a real parameter $p$, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the component-wise exponent. EPMF includes the modulus model ($p=1$) and component-wise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the signing problem. We first show that the signing problem, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank of Fawzi et al. (Math. Prog., 2015). We then show that the signing problem can be solved in polynomial-time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is linear in the input size $X$. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.

Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics

Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank $r$ is delicate and can substantially slow optimization. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank $r$ larger than necessary and enjoys faster convergence as $r$ increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.

Muon learns balanced solutions in matrix factorization without slow saddle-to-saddle dynamics

Matrix factorization (i.e., problems of the form $\min_{\mathbf{P},\mathbf{Q}} \|\mathbf{M}^\star - \mathbf{P}^\top\mathbf{Q}\|_\mathrm{F}^2$) is a minimal learning problem that exhibits both nonlinear parameter dynamics and representation learning. In this setting, we study how parameter trajectories under the Muon optimizer differ from those of gradient descent. We identify three main dynamical differences: 1) Muon avoids the slow saddle-to-saddle dynamics from small initialization. Muon instead learns all the top modes of $\mathbf{M}^\star$ at the same rate, with the smaller modes converging first. 2) Muon remains stable even when the learning rate exceeds the critical threshold set by the local loss sharpness. This frees the learning rate from the condition number of the problem, enabling rapid convergence via exponential learning rate annealing. 3) Once the weights are aligned with each other and the target, Muon flow conserves the matrix quantity $\sqrt{\mathbf{P}^\top \mathbf{P}}-\sqrt{\mathbf{Q}^\top \mathbf{Q}}$, while gradient flow is known to conserve the matrix $\mathbf{P}^\top\mathbf{P} - \mathbf{Q}^\top\mathbf{Q}$. Despite having distinct conserved quantities, both optimizers find the so-called \textit{balanced} solution from vanishing initialization. When training from small random initialization, the weights spontaneously align early in training. We derive the alignment rates in simple settings and show that they predict the empirical alignment rates in general. Finally, we exploit structural properties of Muon to construct a learning rate schedule that achieves near-perfect alignment in only two optimization steps.

Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization

We analyze Bregman ADMM for nonconvex linearly constrained problems under two-sided relative smoothness, a condition that replaces the standard Lipschitz gradient assumption with a Hessian comparison relative to a Bregman kernel. This setting covers polynomial objectives arising in matrix and tensor models for which a global Lipschitz-gradient constant need not exist. We show that on an invariant open state-space domain, one iteration of Bregman ADMM defines a smooth primal--dual fixed-point map whose strict-saddle KKT points are unstable fixed points; consequently, from random initialization the iterates converge to a strict saddle with probability zero. Combined with existing first-order convergence results, this yields almost-sure second-order stationarity of limiting KKT points. We extend the analysis to a multi-block star consensus formulation for distributed optimization. The technical novelty lies in a determinant reduction with a Bregman-specific symmetrization and scaling step in the two block spectral argument, together with a null space cancellation exploiting the star graph structure in the consensus case. Numerical experiments on distributed matrix factorization illustrate the theory, and a symmetric tensor factorization example demonstrates the broader Bregman proximal splitting idea beyond the separable consensus setting.

Learning Interpretable Text Signals for Structured Responses

Textual data are often collected alongside structured response variables, but prediction and interpretation are commonly treated as separate tasks. This paper studies rating prediction as an initial case of interpretable text-response modelling, where the aim is to learn textual representations that are both semantically meaningful and aligned with an external response. We propose a joint non-negative matrix factorisation and binomial regression model, in which the document-topic representation is learned from both text reconstruction and rating prediction. Simulation experiments and a real-world review dataset show that the model can recover stable response-relevant textual signals and achieve competitive performance against linear and ridge regression baselines. The framework provides a practical step towards interpretable modelling of text-linked outcomes, with potential extensions to other response types beyond bounded ratings.

You Don't Need to Run Every Eval

A modern model release reports scores on 40+ benchmarks and the same evaluations were run many more times before it: to track training progress, compare design choices, and select the checkpoint for the release. But do we need to run every eval? We compile a public score matrix of 84 frontier models on 133 benchmarks (2,604 cells, 23.3% filled) and find it is approximately rank-2: a model's scores across all benchmarks are largely determined by just two numbers. We confirm this in two ways: (i) scores hidden from the matrix are best recovered using two factors, and (ii) two factors already explain over 90% of the variation among models on the benchmarks they share. Building on this, we design BenchPress: a logit-space rank-2 matrix completion method that recovers held-out scores to within 4.6 points. Using BenchPress, we find a subset of five benchmarks {GPQA-D, HLE, Codeforces, MMLU-Pro, ARC-AGI-1} that can recover the rest of a model's public scorecard to within 3.93 points. For a tighter evaluation budget, a cheaper set {GPQA-D, MMLU-Pro, Aider Polyglot, MATH-500, AIME 2026} can predict a model's evals to within 4.55. Finally, we identify what affects prediction reliability and combine these factors with predictor disagreement to quantify when predictions can be trusted. We release the score matrix, the code for reproducing all of our experiments on Github.

A Bayesian Boolean Matrix Factorization with Application to Copy Number Analysis in Cancer

Binary data factorization is common, but real-valued methods ignore discreteness and yield hard-to-interpret factors. Boolean Matrix Factorization (BooMF) instead decomposes a binary matrix into two lower-rank binary matrices via logical AND and OR, expressing the data as a Boolean disjunction of interpretable patterns. In cancer genomics, BooMF can reveal coordinated feature changes that may drive tumor evolution, unlike rotational or additive decompositions. Most existing BooMF methods are heuristic, greedy, sensitive to initialization, prone to local optima, and do not support principled model selection or uncertainty quantification. We introduce Bayesian Boolean Matrix Factorization (BBMF), a fully conjugate generative model with sparsity-inducing priors. It enforces Boolean constraints, yields interpretable latent factors with coherent uncertainty quantification, and admits Gibbs sampling with closed-form full conditionals. Because cancer evolution often involves widespread, near-simultaneous chromosome-number changes (e.g., whole-genome duplication followed by instability and selection), Boolean factorizations capture these patterns more naturally than additive models. Applied to arm-level copy-number alteration data in multiple myeloma, where entries indicate presence/absence of chromosomal-arm amplifications, BBMF finds a small set of interpretable bicliques linking patient subsets to recurrently co-altered chromosomal arms, providing a compact, biologically meaningful summary of tumor heterogeneity and demonstrating BBMF's utility for uncovering discrete latent structure in complex binary data.

On solving symmetric multi-type orthogonal non-negative matrix tri-factorization problem

We study the symmetric multi-type orthogonal non-negative matrix tri-factorization problem, where several symmetric non-negative matrices are simultaneously approximated by factors of the form $GS_{i}G^{\top}$, with a shared non-negative and orthogonal factor $G$. This model is motivated by clustering and network analysis, where non-negativity improves interpretability and orthogonality gives a natural assignment-type structure to the latent factor. Since the resulting optimization problem is highly non-convex, we develop two heuristic algorithms for computing high-quality local solutions. The first one is a fixed point method derived from the Karush-Kuhn-Tucker conditions after adding a penalty term for the orthogonality constraint. The second one is a three-stage ADAM-based method that combines non-negativity-preserving optimization, orthogonalization, and restricted ADAM refinement on the feasible set. We evaluate both methods on synthetic data, including noisy instances, and on citation network benchmarks. The synthetic experiments show that both algorithms recover factorizations close to the optimum and remain stable under noise. On real networks, the learned embeddings are competitive with or better than standard baselines such as SVD, node2vec, and classical link prediction heuristics in link prediction, node clustering, and node classification tasks.

Graph Regularized Non-negative Reduced Biquaternion Matrix Factorization for Color Image Recognition

Non-negative reduced biquaternion matrix factorization (NRBMF) uses the product of reduced biquaternion (RB) matrices to incorporate the non-negativity constraints of color image pixels into the factorization process. However, NRBMF mainly focuses on reconstruction accuracy and does not explicitly exploit the local geometric structure of image data, which may limit the discriminative ability of the obtained low-dimensional coefficient representations. To address this issue, we propose a graph regularized non-negative reduced biquaternion matrix factorization (GNRBMF) model for color image recognition. The proposed model incorporates a graph Laplacian regularizer into the reduced biquaternion coefficient matrix, encouraging nearby samples in the original space to have similar coefficient representations. Meanwhile, GNRBMF retains the non-negativity property of NRBMF in the reduced biquaternion algebra. To solve the optimization problem, a component-wise alternating projected gradient algorithm is derived, and its convergence properties are analyzed. Experimental results on three color image datasets show that the proposed GNRBMF model achieves competitive or superior recognition performance compared with several methods in most tested settings.

The Bandit's Blind Spot: The Critical Role of User State Representation in Recommender Systems

With the increasing availability of online information, recommender systems have become an important tool for many web-based systems. Due to the continuous aspect of recommendation environments, these systems increasingly rely on contextual multi-armed bandits (CMAB) to deliver personalized and real-time suggestions. A critical yet underexplored component in these systems is the representation of user state, which typically encapsulates the user's interaction history and is deeply correlated with the model's decisions and learning. In this paper, we investigate the impact of different embedding-based state representations derived from matrix factorization models on the performance of traditional CMAB algorithms. Our large-scale experiments reveal that variations in state representation can lead to improvements greater than those achieved by changing the bandit algorithm itself. Furthermore, no single embedding or aggregation strategy consistently dominates across datasets, underscoring the need for domain-specific evaluation. These results expose a substantial gap in the literature and emphasize that advancing bandit-based recommender systems requires a holistic approach that prioritizes embedding quality and state construction alongside algorithmic innovation. The source code for our experiments is publicly available on https://github.com/UFSCar-LaSID/bandits_blind_spot.