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Source author record

Bamdev Mishra

Bamdev Mishra appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

ResearcherUnclaimed source record
Machine Learningmath.OCArtificial IntelligenceComputation and LanguageComputer VisionNumerical AnalysisInformation Theorymath.ITCryptography and SecurityMathematical SoftwareSystems and Control

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Published work

19 published item(s)

preprint2026arXiv

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.

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preprint2026arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Differentially private Riemannian optimization

In this paper, we study the differentially private empirical risk minimization problem where the parameter is constrained to a Riemannian manifold. We introduce a framework of differentially private Riemannian optimization by adding noise to the Riemannian gradient on the tangent space. The noise follows a Gaussian distribution intrinsically defined with respect to the Riemannian metric. We adapt the Gaussian mechanism from the Euclidean space to the tangent space compatible to such generalized Gaussian distribution. We show that this strategy presents a simple analysis as compared to directly adding noise on the manifold. We further show privacy guarantees of the proposed differentially private Riemannian (stochastic) gradient descent using an extension of the moments accountant technique. Additionally, we prove utility guarantees under geodesic (strongly) convex, general nonconvex objectives as well as under the Riemannian Polyak-Łojasiewicz condition. We show the efficacy of the proposed framework in several applications.

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preprint2022arXiv

Riemannian accelerated gradient methods via extrapolation

In this paper, we propose a simple acceleration scheme for Riemannian gradient methods by extrapolating iterates on manifolds. We show when the iterates are generated from Riemannian gradient descent method, the accelerated scheme achieves the optimal convergence rate asymptotically and is computationally more favorable than the recently proposed Riemannian Nesterov accelerated gradient methods. Our experiments verify the practical benefit of the novel acceleration strategy.

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preprint2020arXiv

A Simple Approach to Learning Unsupervised Multilingual Embeddings

Recent progress on unsupervised learning of cross-lingual embeddings in bilingual setting has given impetus to learning a shared embedding space for several languages without any supervision. A popular framework to solve the latter problem is to jointly solve the following two sub-problems: 1) learning unsupervised word alignment between several pairs of languages, and 2) learning how to map the monolingual embeddings of every language to a shared multilingual space. In contrast, we propose a simple, two-stage framework in which we decouple the above two sub-problems and solve them separately using existing techniques. The proposed approach obtains surprisingly good performance in various tasks such as bilingual lexicon induction, cross-lingual word similarity, multilingual document classification, and multilingual dependency parsing. When distant languages are involved, the proposed solution illustrates robustness and outperforms existing unsupervised multilingual word embedding approaches. Overall, our experimental results encourage development of multi-stage models for such challenging problems.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Geometry-aware Domain Adaptation for Unsupervised Alignment of Word Embeddings

We propose a novel manifold based geometric approach for learning unsupervised alignment of word embeddings between the source and the target languages. Our approach formulates the alignment learning problem as a domain adaptation problem over the manifold of doubly stochastic matrices. This viewpoint arises from the aim to align the second order information of the two language spaces. The rich geometry of the doubly stochastic manifold allows to employ efficient Riemannian conjugate gradient algorithm for the proposed formulation. Empirically, the proposed approach outperforms state-of-the-art optimal transport based approach on the bilingual lexicon induction task across several language pairs. The performance improvement is more significant for distant language pairs.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Learning Geometric Word Meta-Embeddings

We propose a geometric framework for learning meta-embeddings of words from different embedding sources. Our framework transforms the embeddings into a common latent space, where, for example, simple averaging of different embeddings (of a given word) is more amenable. The proposed latent space arises from two particular geometric transformations - the orthogonal rotations and the Mahalanobis metric scaling. Empirical results on several word similarity and word analogy benchmarks illustrate the efficacy of the proposed framework.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A Riemannian gossip approach to decentralized matrix completion

In this paper, we propose novel gossip algorithms for the low-rank decentralized matrix completion problem. The proposed approach is on the Riemannian Grassmann manifold that allows local matrix completion by different agents while achieving asymptotic consensus on the global low-rank factors. The resulting approach is scalable and parallelizable. Our numerical experiments show the good performance of the proposed algorithms on various benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

A Sparse and Low-Rank Optimization Framework for Index Coding via Riemannian Optimization

Side information provides a pivotal role for message delivery in many communication scenarios to accommodate increasingly large data sets, e.g., caching networks. Although index coding provides a fundamental modeling framework to exploit the benefits of side information, the index coding problem itself still remains open and only a few instances have been solved. In this paper, we propose a novel sparse and low- rank optimization modeling framework for the index coding problem to characterize the tradeoff between the amount of side information and the achievable data rate. Specifically, sparsity of the model measures the amount of side information, while low- rankness represents the achievable data rate. The resulting sparse and low-rank optimization problem has non-convex sparsity inducing objective and non-convex rank constraint. To address the coupled challenges in objective and constraint, we propose a novel Riemannian optimization framework by exploiting the quotient manifold geometry of fixed-rank matrices, accompanied by a smooth sparsity inducing surrogate. Simulation results demonstrate the appealing sparsity and low-rankness tradeoff in the proposed model, thereby revealing the tradeoff between the amount of side information and the achievable data rate in the index coding problem.

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preprint2016arXiv

Low-rank tensor completion: a Riemannian manifold preconditioning approach

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Riemannian preconditioning

This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.

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preprint2016arXiv

Riemannian preconditioning for tensor completion

We propose a novel Riemannian preconditioning approach for the tensor completion problem with rank constraint. A Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop a preconditioned nonlinear conjugate gradient algorithm for the problem. To this end, concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithm robustly outperforms state-of-the-art algorithms across different problem instances encompassing various synthetic and real-world datasets.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Scaled stochastic gradient descent for low-rank matrix completion

The paper looks at a scaled variant of the stochastic gradient descent algorithm for the matrix completion problem. Specifically, we propose a novel matrix-scaling of the partial derivatives that acts as an efficient preconditioning for the standard stochastic gradient descent algorithm. This proposed matrix-scaling provides a trade-off between local and global second order information. It also resolves the issue of scale invariance that exists in matrix factorization models. The overall computational complexity is linear with the number of known entries, thereby extending to a large-scale setup. Numerical comparisons show that the proposed algorithm competes favorably with state-of-the-art algorithms on various different benchmarks.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Topological Interference Management with User Admission Control via Riemannian Optimization

Topological interference management (TIM) provides a promising way to manage interference only based on the network connectivity information. Previous works on the TIM problem mainly focus on using the index coding approach and graph theory to establish conditions of network topologies to achieve the feasibility of topological interference management. In this paper, we propose a novel user admission control approach via sparse and low-rank optimization to maximize the number of admitted users for achieving the feasibility of topological interference management. To assist efficient algorithms design for the formulated rank-constrained (i.e., degrees-of-freedom (DoF) allocation) l0-norm maximization (i.e., user capacity maximization) problem, we propose a regularized smoothed l1- norm minimization approach to induce sparsity pattern, thereby guiding the user selection. We further develop a Riemannian trust-region algorithm to solve the resulting rank-constrained smooth optimization problem via exploiting the quotient manifold of fixed-rank matrices. Simulation results demonstrate the effectiveness and near-optimal performance of the proposed Riemannian algorithm to maximize the number of admitted users for topological interference management.

0 citations0 reviewsNo code linkedOpen access
preprint2015arXiv

Heterogeneous Tensor Decomposition for Clustering via Manifold Optimization

Tensors or multiarray data are generalizations of matrices. Tensor clustering has become a very important research topic due to the intrinsically rich structures in real-world multiarray datasets. Subspace clustering based on vectorizing multiarray data has been extensively researched. However, vectorization of tensorial data does not exploit complete structure information. In this paper, we propose a subspace clustering algorithm without adopting any vectorization process. Our approach is based on a novel heterogeneous Tucker decomposition model. In contrast to existing techniques, we propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model. All but the last mode have closed-form updates. Updating the last mode reduces to optimizing over the so-called multinomial manifold, for which we investigate second order Riemannian geometry and propose a trust-region algorithm. Numerical experiments show that our proposed algorithm compete effectively with state-of-the-art clustering algorithms that are based on tensor factorization.

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preprint2015arXiv

Sparse plus low-rank autoregressive identification in neuroimaging time series

This paper considers the problem of identifying multivariate autoregressive (AR) sparse plus low-rank graphical models. Based on the corresponding problem formulation recently presented, we use the alternating direction method of multipliers (ADMM) to efficiently solve it and scale it to sizes encountered in neuroimaging applications. We apply this decomposition on synthetic and real neuroimaging datasets with a specific focus on the information encoded in the low-rank structure of our model. In particular, we illustrate that this information captures the spatio-temporal structure of the original data, generalizing classical component analysis approaches.

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preprint2015arXiv

Symmetry-invariant optimization in deep networks

Recent works have highlighted scale invariance or symmetry that is present in the weight space of a typical deep network and the adverse effect that it has on the Euclidean gradient based stochastic gradient descent optimization. In this work, we show that these and other commonly used deep networks, such as those which use a max-pooling and sub-sampling layer, possess more complex forms of symmetry arising from scaling based reparameterization of the network weights. We then propose two symmetry-invariant gradient based weight updates for stochastic gradient descent based learning. Our empirical evidence based on the MNIST dataset shows that these updates improve the test performance without sacrificing the computational efficiency of the weight updates. We also show the results of training with one of the proposed weight updates on an image segmentation problem.

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preprint2015arXiv

Understanding symmetries in deep networks

Recent works have highlighted scale invariance or symmetry present in the weight space of a typical deep network and the adverse effect it has on the Euclidean gradient based stochastic gradient descent optimization. In this work, we show that a commonly used deep network, which uses convolution, batch normalization, reLU, max-pooling, and sub-sampling pipeline, possess more complex forms of symmetry arising from scaling-based reparameterization of the network weights. We propose to tackle the issue of the weight space symmetry by constraining the filters to lie on the unit-norm manifold. Consequently, training the network boils down to using stochastic gradient descent updates on the unit-norm manifold. Our empirical evidence based on the MNIST dataset shows that the proposed updates improve the test performance beyond what is achieved with batch normalization and without sacrificing the computational efficiency of the weight updates.

0 citations0 reviewsNo code linkedOpen access
preprint2013arXiv

Manopt, a Matlab toolbox for optimization on manifolds

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. We aim particularly at reaching practitioners outside our field.

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