LOFT: Low-Rank Orthogonal Fine-Tuning via Task-Aware Support Selection

Orthogonal parameter-efficient fine-tuning (PEFT) adapts pretrained weights through structure-preserving multiplicative transformations, but existing methods often conflate two distinct design choices: the subspace in which adaptation occurs and the transformation applied within that subspace. This paper introduces LOFT, a low-rank orthogonal fine-tuning framework that explicitly separates these two components. By viewing orthogonal adaptation as a multiplicative subspace rotation, LOFT provides a unified formulation that recovers representative orthogonal PEFT methods, including coordinate-, butterfly-, Householder-, and principal-subspace-based variants. More importantly, this perspective exposes support selection as a central design axis rather than a byproduct of a particular parameterization. We develop a first-order analysis showing that useful adaptation supports should be informed by the downstream training signal, motivating practical task-aware support selection strategies. Across language understanding, visual transfer, mathematical reasoning, and multilingual out-of-distribution adaptation, LOFT recovers principal-subspace orthogonal adaptation while gradient-informed supports improve the efficiency-performance trade-off under matched parameter, memory, and compute budgets. These results suggest that principled support selection is an important direction for improving orthogonal PEFT.

Intrinsic Muon: Spectral Optimization on Riemannian Matrix Manifolds

Muon and related norm-constrained matrix optimizers have become central to large-scale learning problems. They are formulated as a linear maximization oracle (LMO) over an ambient matrix-norm ball in unconstrained Euclidean space. However, these do not generalize cleanly to manifold-valued parameters such as low-rank factorizations, orthogonality constraints, or symmetric positive definite (SPD) matrices. Naively restricting the Muon LMO to the tangent space (i) breaks quotient symmetries and (ii) couples the tangent-space constraint with an ambient norm bound, thereby obstructing closed-form solutions on various manifolds of interest. We resolve both issues with a single observation: every Riemannian metric canonically lifts a unitarily invariant Euclidean norm to an intrinsic norm on each tangent space, and the resulting intrinsic norm constrained LMO is symmetry preserving. Building on this, we introduce intrinsic Muon (iMuon), a unified framework that yields closed-form updates on the fixed-rank, SPD, Stiefel, and Grassmann manifolds for any unitarily invariant norm, including the spectral, Frobenius, and nuclear norms. We establish convergence guarantees for both deterministic and stochastic iMuon with rate constants that depend only on the manifold dimension. Notably, on the fixed-rank manifold this constant depends only on the rank, making the rate independent of factor conditioning and removing the runtime factor-rescaling required by prior work. Experiments on LoRA finetuning of LLMs, image classification, and subspace learning illustrate the efficacy of the proposed approach.

FedSEA: Achieving Benefit of Parallelization in Federated Online Learning

Online federated learning (OFL) has emerged as a popular framework for decentralized decision-making over continuous data streams without compromising client privacy. However, the adversary model assumed in standard OFL typically precludes any potential benefits of parallelization. Further, it fails to adequately capture the different sources of statistical variation in OFL problems. In this paper, we extend the OFL paradigm by integrating a stochastically extended adversary (SEA). Under this framework, the loss function remains fixed across clients over time. However, the adversary dynamically and independently selects the data distribution for each client at each time. We propose the \algoOFL{} algorithm to solve this problem, which utilizes online stochastic gradient descent at the clients, along with periodic global aggregation via the server. We establish bounds on the global network regret over a time horizon \(T\) for two classes of functions: (1) for smooth and convex losses, we prove an \(\mathcal{O}(\sqrt{T})\) bound, and (2) for smooth and strongly convex losses, we prove an \(\mathcal{O}(\log T)\) bound. Through careful analysis, we quantify the individual impact of both spatial (across clients) and temporal (over time) data heterogeneity on the regret bounds. Consequently, we identify a regime of mild temporal variation (relative to stochastic gradient variance), where the network regret improves with parallelization. Hence, in the SEA setting, our results improve the existing pessimistic worst-case results in online federated learning.

UniPROT: Uniform Prototype Selection via Partial Optimal Transport with Submodular Guarantees

Selecting prototypical examples from a source distribution to represent a target data distribution is a fundamental problem in machine learning. Existing subset selection methods often rely on implicit importance scores, which can be skewed towards majority classes and lead to low-quality prototypes for minority classes. We present $\methodprop$, a novel subset selection framework that minimizes the optimal transport (OT) distance between a uniformly weighted prototypical distribution and the target distribution. While intuitive, this formulation leads to a cardinality-constrained maximization of a \emph{super-additive} objective, which is generally intractable to approximate efficiently. To address this, we propose a principled reformulation of the OT marginal constraints, yielding a partial optimal transport-based submodular objective. We prove that this reformulation enables a greedy algorithm with a $(1-1/e)$ approximation guarantee relative to the original super-additive maximization problem. Empirically, we showcase that enforcing uniform prototype weights in UniPROT consistently improves minority-class representation in imbalanced classification benchmarks without compromising majority-class accuracy. In both finetuning and pretraining regimes for large language models under domain imbalance, UniPROT enforces uniform source contributions, yielding robust performance gains. Our results establish UniPROT as a scalable, theoretically grounded solution for uniform-weighted prototype selection. Our code is publicly available at GitHub\footnote{Code: https://github.com/efficiency-learning/UniPROT}

Generalized infinite dimensional Alpha-Procrustes based geometries

This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis.

A Riemannian Approach to Ground Metric Learning for Optimal Transport

Optimal transport (OT) theory has attracted much attention in machine learning and signal processing applications. OT defines a notion of distance between probability distributions of source and target data points. A crucial factor that influences OT-based distances is the ground metric of the embedding space in which the source and target data points lie. In this work, we propose to learn a suitable latent ground metric parameterized by a symmetric positive definite matrix. We use the rich Riemannian geometry of symmetric positive definite matrices to jointly learn the OT distance along with the ground metric. Empirical results illustrate the efficacy of the learned metric in OT-based domain adaptation.

Riemannian Federated Learning via Averaging Gradient Stream

In recent years, federated learning has garnered significant attention as an efficient and privacy-preserving distributed learning paradigm. In the Euclidean setting, Federated Averaging (FedAvg) and its variants are a class of efficient algorithms for expected (empirical) risk minimization. This paper develops and analyzes a Riemannian Federated Averaging Gradient Stream (RFedAGS) algorithm, which is a generalization of FedAvg, to problems defined on a Riemannian manifold. Under standard assumptions, the convergence rate of RFedAGS with fixed step sizes is proven to be sublinear for an approximate stationary solution. If decaying step sizes are used, the global convergence is established. Furthermore, assuming that the objective obeys the Riemannian Polyak-{\L}ojasiewicz property, the optimal gaps generated by RFedAGS with fixed step size are linearly decreasing up to a tiny upper bound, meanwhile, if decaying step sizes are used, then the gaps sublinearly vanish. Numerical simulations conducted on synthetic and real-world data demonstrate the performance of the proposed RFedAGS.

Submodular Framework for Structured-Sparse Optimal Transport

Unbalanced optimal transport (UOT) has recently gained much attention due to its flexible framework for handling un-normalized measures and its robustness properties. In this work, we explore learning (structured) sparse transport plans in the UOT setting, i.e., transport plans have an upper bound on the number of non-sparse entries in each column (structured sparse pattern) or in the whole plan (general sparse pattern). We propose novel sparsity-constrained UOT formulations building on the recently explored maximum mean discrepancy based UOT. We show that the proposed optimization problem is equivalent to the maximization of a weakly submodular function over a uniform matroid or a partition matroid. We develop efficient gradient-based discrete greedy algorithms and provide the corresponding theoretical guarantees. Empirically, we observe that our proposed greedy algorithms select a diverse support set and we illustrate the efficacy of the proposed approach in various applications.

Riemannian coordinate descent algorithms on matrix manifolds

Many machine learning applications are naturally formulated as optimization problems on Riemannian manifolds. The main idea behind Riemannian optimization is to maintain the feasibility of the variables while moving along a descent direction on the manifold. This results in updating all the variables at every iteration. In this work, we provide a general framework for developing computationally efficient coordinate descent (CD) algorithms on matrix manifolds that allows updating only a few variables at every iteration while adhering to the manifold constraint. In particular, we propose CD algorithms for various manifolds such as Stiefel, Grassmann, (generalized) hyperbolic, symplectic, and symmetric positive (semi)definite. While the cost per iteration of the proposed CD algorithms is low, we further develop a more efficient variant via a first-order approximation of the objective function. We analyze their convergence and complexity, and empirically illustrate their efficacy in several applications.

SLTrain: a sparse plus low-rank approach for parameter and memory efficient pretraining

Large language models (LLMs) have shown impressive capabilities across various tasks. However, training LLMs from scratch requires significant computational power and extensive memory capacity. Recent studies have explored low-rank structures on weights for efficient fine-tuning in terms of parameters and memory, either through low-rank adaptation or factorization. While effective for fine-tuning, low-rank structures are generally less suitable for pretraining because they restrict parameters to a low-dimensional subspace. In this work, we propose to parameterize the weights as a sum of low-rank and sparse matrices for pretraining, which we call SLTrain. The low-rank component is learned via matrix factorization, while for the sparse component, we employ a simple strategy of uniformly selecting the sparsity support at random and learning only the non-zero entries with the fixed support. While being simple, the random fixed-support sparse learning strategy significantly enhances pretraining when combined with low-rank learning. Our results show that SLTrain adds minimal extra parameters and memory costs compared to pretraining with low-rank parameterization, yet achieves substantially better performance, which is comparable to full-rank training. Remarkably, when combined with quantization and per-layer updates, SLTrain can reduce memory requirements by up to 73% when pretraining the LLaMA 7B model.