arXiv subjects

Laura Balzano

Publications and source records attributed to Laura Balzano.

At least 19 recordsLinked to original sources

arXiv2026

In-span learning: adapting reduced-order models using their own predictions

Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data. Adaptive methods address this by updating the subspace online with external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace. By streaming the model's own predictions through an incremental singular-value decomposition with forgetting, we obtain a trajectory-informed spectral preconditioner, in which the subspace is unchanged but the basis is reweighted and realigned toward the modes visited by the dynamics. This enables the model to absorb future out-of-span corrections more effectively. We expose aspects of this mechanism on a three-dimensional spiral and confirm it on viscous Burgers and Fisher-KPP dynamics. We also discuss how in-span learning can be viewed as a dynamical-systems analogue of in-context learning. More broadly, in-span learning suggests a new principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.

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arXiv2026

Benign Landscape of Quadratic Programs with Orthogonality Constraints and Its Application to Heteroscedastic Probabilistic PCA

In this work, we study the optimization landscape of homogeneous quadratic programs with orthogonality constraints (QPOC) and apply the resulting theory to heteroscedastic probabilistic PCA (HePPCA). For QPOC, we establish a complete characterization of the benign optimization landscape by showing that every critical point is either a global maximizer or a strict saddle point. Our analysis builds on a closed-form characterization of the critical point set, from which we derive a necessary and sufficient condition for global optimality and show that every non-optimal critical point has a direction of positive curvature. As an application, we show that the population version of HePPCA is a special instance of QPOC and therefore has a benign optimization landscape; moreover, it satisfies local geodesic strong concavity near every global maximizer. We furthermore prove that, when the sample size is sufficiently large, the sample version of HePPCA inherits these favorable properties with high probability. Together with existing theory on avoiding strict saddle points, our results provide a theoretical justification for the observed local linear convergence of retraction-based optimization methods to global solutions for both QPOC and HePPCA. Finally, we present numerical experiments to corroborate our theoretical results.

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arXiv2026

A Concentration Inequality for the Covariance Matrix of an Arbitrary Subset of Random Vectors

Concentration inequalities for sample covariance matrices are fundamental tools in high-dimensional probability. Classical results typically assume that the selected random vectors are independent of the selection rule. In this paper, we study spectral concentration for sample covariance matrices formed from arbitrary, possibly data-dependent subsets of i.i.d. random vectors. Such data-dependent selection destroys the usual independence structure and makes standard covariance concentration bounds inapplicable. For i.i.d. Gaussian random vectors, we prove high-probability lower and upper bounds for the minimal and maximal eigenvalues of such selected covariance matrices. Compared with a direct union-bound argument, our results provide substantially sharper guarantees and allow much smaller subset proportions. We further discuss extensions from Gaussian to sub-Gaussian random vectors, and beyond independence to weakly dependent observations, with geometrically strong-mixing Gaussian sequences serving as a representative example of the latter. Finally, we apply the developed concentration inequalities to the K-subspace clustering problem under a low-rank Gaussian mixture model, where the optimal clusters are inherently data-dependent. Our results yield recovery guarantees showing that the clustering error of global minimizers decays polynomially with the signal-to-noise ratio.

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arXiv2026

The Effect of Training Task Diversity on In-Context Learning through the Lens of Low-Dimensional Subspaces

The transformer's emergent ability to perform in-context learning (ICL) has sparked a wide range of studies designed to understand its underlying mechanisms. Existing works often study how training task diversity, defined either as the number of ICL training task vectors or as the number of function classes from which the task vectors are drawn, shapes both the learning dynamics and generalization capabilities of ICL. While both definitions have uncovered many interesting phenomena, many observations under the latter definition remain theoretically unexplained. This paper presents a minimal analytical model under which these phenomena provably emerge from the properties of the training data. By modeling the training task vectors as a mixture of low-rank Gaussians, we show how training task diversity, defined by the number of non-overlapping columns between subspaces that parameterize the covariance matrices, improves both the generalization and optimization trajectory of ICL with linear attention. In particular, we show that our model can explain (i) why training with task diversity shortens the ICL plateau and (ii) why ICL appears to achieve out-of-distribution generalization. We conclude by empirically demonstrating how our results extend to nonlinear transformers and nonlinear function classes. Overall, our work presents a tractable framework to unify existing observations.

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arXiv2026

History-aware adaptive reduced-order models via incremental singular value decomposition

Reduced-order models (ROMs) can accelerate high-dimensional dynamical simulations, but their accuracy often deteriorates when online dynamics leave the regime represented by offline training data. We develop a projection-based adaptive ROM framework based on incremental singular value decomposition (iSVD), in which occasional full-order operator evaluations provide correction snapshots for online basis updates. The intrusive ROMs considered here are fully parameterized by the basis, so each update naturally propagates to reduced operators and hyper-reduction machinery. Through its evolving singular structure, iSVD retains an encoded history of the observed dynamics and is history-aware in this sense. We study the method on three nonlinear problems of increasing complexity: the one-dimensional viscous Burgers equation, the Sod shock tube, and a stiff one-dimensional ten-species rotating detonation engine (RDE). The Burgers problem is used to analyze the method and compare iSVD with alternative basis adaptation rules, showing that history-aware updates outperform instantaneous updates and that iSVD gives the strongest overall performance. The Sod and RDE cases demonstrate that these advantages persist in more challenging compressible-flow settings. For the RDE problem, the iSVD adaptive ROM improves upon the current state-of-the-art Direct adaptive ROM baseline in both predictive accuracy and computational efficiency. A cost analysis shows that the dominant online cost comes from interacting with the full-order model to obtain correction snapshots, while the iSVD update itself is negligible. These results identify iSVD as an effective mechanism for online learning of reduced subspaces and suggest a path toward ROMs that remain predictive over horizons several orders of magnitude longer than their initial training window.

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arXiv2026

Emergent Low-Rank Training Dynamics in MLPs with Smooth Activations

Recent empirical evidence has demonstrated that the training dynamics of large-scale deep neural networks occur within low-dimensional subspaces. While this has inspired new research into low-rank training, compression, and adaptation, theoretical justification for these dynamics in nonlinear networks remains limited. %compared to deep linear settings. To address this gap, this paper analyzes the learning dynamics of multi-layer perceptrons (MLPs) under gradient descent (GD). We demonstrate that the weight dynamics concentrate within invariant low-dimensional subspaces throughout training. Theoretically, we precisely characterize these invariant subspaces for two-layer networks with smooth nonlinear activations, providing insight into their emergence. Experimentally, we validate that this phenomenon extends beyond our theoretical assumptions. Leveraging these insights, we empirically show there exists a low-rank MLP parameterization that, when initialized within the appropriate subspaces, matches the classification performance of fully-parameterized counterparts on a variety of classification tasks.

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arXiv2025

ALPCAHUS: Subspace Clustering for Heteroscedastic Data

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. Various methods have been proposed to extend PCA to the union of subspace (UoS) setting for clustering data that comes from multiple subspaces like K-Subspaces (KSS). However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a heteroscedastic-based subspace clustering method, named ALPCAHUS, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace bases associated with the low-rank structure of the data. This clustering algorithm builds on K-Subspaces (KSS) principles by extending the recently proposed heteroscedastic PCA method, named LR-ALPCAH, for clusters with heteroscedastic noise in the UoS setting. Simulations and real-data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing clustering algorithms. Code available at https://github.com/javiersc1/ALPCAHUS.

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arXiv2025

MonarchAttention: Zero-Shot Conversion to Fast, Hardware-Aware Structured Attention

Transformers have achieved state-of-the-art performance across various tasks, but suffer from a notable quadratic complexity in sequence length due to the attention mechanism. In this work, we propose MonarchAttention -- a novel approach to sub-quadratic attention approximation via Monarch matrices, an expressive class of structured matrices. Based on the variational form of softmax, we describe an efficient optimization-based algorithm to compute an approximate projection of softmax attention onto the class of Monarch matrices with $\Theta(N\sqrt{N} d)$ computational complexity and $\Theta(Nd)$ memory/IO complexity. Unlike previous approaches, MonarchAttention is both (1) transferable, yielding minimal performance loss with no additional training, even when replacing every attention layer of the Transformer, and (2) hardware-efficient, utilizing the highest-throughput tensor core units on modern GPUs. With optimized kernels, MonarchAttention achieves substantial speed-ups in wall-time over FlashAttention-2: $1.4\times$ for shorter sequences $(N=256)$, $4.5\times$ for medium-length sequences $(N=4K)$, and $8.2\times$ for longer sequences $(N=16K)$. We demonstrate the quality of MonarchAttention on diverse tasks and architectures in vision and language problems, showing that it flexibly and accurately approximates softmax attention in a variety of contexts. Our code is available at https://github.com/cjyaras/monarch-attention.

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arXiv2025

Out-of-Distribution Generalization of In-Context Learning: A Low-Dimensional Subspace Perspective

The transformer's remarkable ability to perform in-context learning (ICL) has sparked a wide range of studies designed to understand its strengths and limitations. However, a theoretical understanding of when ICL can and cannot generalize beyond its pre-training data still remains unclear. This paper puts forth a minimal mathematical model that provably identifies when ICL can generalize out-of-distribution (OOD). By studying linear regression tasks parameterized with low-rank covariance matrices, we model distribution shifts as varying angles between subspaces and derive conditions under which a single-layer linear attention model interpolates across all angles. We show that if pre-training task vectors are drawn from a union of subspaces, transformers can generalize to all angle shifts--enabling ICL even in regions with zero probability mass in the training distribution. On the other hand, if the pre-training tasks are drawn from a single Gaussian, the test risk shows a non-negligible dependence on the angle, implying that ICL cannot generalize OOD. We empirically show that our results also hold for models such as GPT-2, and present experiments on how our results extend to nonlinear function classes.

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arXiv2025

ALPCAH: Subspace Learning for Sample-wise Heteroscedastic Data

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a subspace learning method, named ALPCAH, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace basis associated with the low-rank structure of the data. Our method makes no distributional assumptions of the low-rank component and does not assume that the noise variances are known. Further, this method uses a soft rank constraint that does not require subspace dimension to be known. Additionally, this paper develops a matrix factorized version of ALPCAH, named LR-ALPCAH, that is much faster and more memory efficient at the cost of requiring subspace dimension to be known or estimated. Simulations and real data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing algorithms. Code available at https://github.com/javiersc1/ALPCAH.

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arXiv2025

Truncated Matrix Completion - An Empirical Study

Low-rank Matrix Completion (LRMC) describes the problem where we wish to recover missing entries of partially observed low-rank matrix. Most existing matrix completion work deals with sampling procedures that are independent of the underlying data values. While this assumption allows the derivation of nice theoretical guarantees, it seldom holds in real-world applications. In this paper, we consider various settings where the sampling mask is dependent on the underlying data values, motivated by applications in sensing, sequential decision-making, and recommender systems. Through a series of experiments, we study and compare the performance of various LRMC algorithms that were originally successful for data-independent sampling patterns.

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arXiv2025

An Overview of Low-Rank Structures in the Training and Adaptation of Large Models

The substantial computational demands of modern large-scale deep learning present significant challenges for efficient training and deployment. Recent research has revealed a widespread phenomenon wherein deep networks inherently learn low-rank structures in their weights and representations during training. This tutorial paper provides a comprehensive review of advances in identifying and exploiting these low-rank structures, bridging mathematical foundations with practical applications. We present two complementary theoretical perspectives on the emergence of low-rankness: viewing it through the optimization dynamics of gradient descent throughout training, and understanding it as a result of implicit regularization effects at convergence. Practically, these theoretical perspectives provide a foundation for understanding the success of techniques such as Low-Rank Adaptation (LoRA) in fine-tuning, inspire new parameter-efficient low-rank training strategies, and explain the effectiveness of masked training approaches like dropout and masked self-supervised learning.

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arXiv2024

A Spectral Framework for Tracking Communities in Evolving Networks

Discovering and tracking communities in time-varying networks is an important task in network science, motivated by applications in fields ranging from neuroscience to sociology. In this work, we characterize the celebrated family of spectral methods for static clustering in terms of the low-rank approximation of high-dimensional node embeddings. From this perspective, it becomes natural to view the evolving community detection problem as one of subspace tracking on the Grassmann manifold. While the resulting optimization problem is nonconvex, we adopt a block majorize-minimize Riemannian optimization scheme to learn the Grassmann geodesic which best fits the data. Our framework generalizes any static spectral community detection approach and leads to algorithms achieving favorable performance on synthetic and real temporal networks, including those that are weighted, signed, directed, mixed-membership, multiview, hierarchical, cocommunity-structured, bipartite, or some combination thereof. We demonstrate how to specifically cast a wide variety of methods into our framework, and demonstrate greatly improved dynamic community detection results in all cases.

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arXiv2024

Symmetric Matrix Completion with ReLU Sampling

We study the problem of symmetric positive semi-definite low-rank matrix completion (MC) with deterministic entry-dependent sampling. In particular, we consider rectified linear unit (ReLU) sampling, where only positive entries are observed, as well as a generalization to threshold-based sampling. We first empirically demonstrate that the landscape of this MC problem is not globally benign: Gradient descent (GD) with random initialization will generally converge to stationary points that are not globally optimal. Nevertheless, we prove that when the matrix factor with a small rank satisfies mild assumptions, the nonconvex objective function is geodesically strongly convex on the quotient manifold in a neighborhood of a planted low-rank matrix. Moreover, we show that our assumptions are satisfied by a matrix factor with i.i.d. Gaussian entries. Finally, we develop a tailor-designed initialization for GD to solve our studied formulation, which empirically always achieves convergence to the global minima. We also conduct extensive experiments and compare MC methods, investigating convergence and completion performance with respect to initialization, noise level, dimension, and rank.

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arXiv2024

Compressible Dynamics in Deep Overparameterized Low-Rank Learning & Adaptation

While overparameterization in machine learning models offers great benefits in terms of optimization and generalization, it also leads to increased computational requirements as model sizes grow. In this work, we show that by leveraging the inherent low-dimensional structures of data and compressible dynamics within the model parameters, we can reap the benefits of overparameterization without the computational burdens. In practice, we demonstrate the effectiveness of this approach for deep low-rank matrix completion as well as fine-tuning language models. Our approach is grounded in theoretical findings for deep overparameterized low-rank matrix recovery, where we show that the learning dynamics of each weight matrix are confined to an invariant low-dimensional subspace. Consequently, we can construct and train compact, highly compressed factorizations possessing the same benefits as their overparameterized counterparts. In the context of deep matrix completion, our technique substantially improves training efficiency while retaining the advantages of overparameterization. For language model fine-tuning, we propose a method called "Deep LoRA", which improves the existing low-rank adaptation (LoRA) technique, leading to reduced overfitting and a simplified hyperparameter setup, while maintaining comparable efficiency. We validate the effectiveness of Deep LoRA on natural language tasks, particularly when fine-tuning with limited data. Our code is available at https://github.com/cjyaras/deep-lora-transformers.

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arXiv2024

Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms

We analyze inexact Riemannian gradient descent (RGD) where Riemannian gradients and retractions are inexactly (and cheaply) computed. Our focus is on understanding when inexact RGD converges and what is the complexity in the general nonconvex and constrained setting. We answer these questions in a general framework of tangential Block Majorization-Minimization (tBMM). We establish that tBMM converges to an $\epsilon$-stationary point within $O(\epsilon^{-2})$ iterations. Under a mild assumption, the results still hold when the subproblem is solved inexactly in each iteration provided the total optimality gap is bounded. Our general analysis applies to a wide range of classical algorithms with Riemannian constraints including inexact RGD and proximal gradient method on Stiefel manifolds. We numerically validate that tBMM shows improved performance over existing methods when applied to various problems, including nonnegative tensor decomposition with Riemannian constraints, regularized nonnegative matrix factorization, and low-rank matrix recovery problems.

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arXiv2023

Convergence and complexity of block majorization-minimization for constrained block-Riemannian optimization

Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex optimization that sequentially minimizes a majorizing surrogate of the objective function in each block coordinate while the other block coordinates are held fixed. We consider a family of BMM algorithms for minimizing smooth nonconvex objectives, where each parameter block is constrained within a subset of a Riemannian manifold. We establish that this algorithm converges asymptotically to the set of stationary points, and attains an $\epsilon$-stationary point within $\widetilde{O}(\epsilon^{-2})$ iterations. In particular, the assumptions for our complexity results are completely Euclidean when the underlying manifold is a product of Euclidean or Stiefel manifolds, although our analysis makes explicit use of the Riemannian geometry. Our general analysis applies to a wide range of algorithms with Riemannian constraints: Riemannian MM, block projected gradient descent, optimistic likelihood estimation, geodesically constrained subspace tracking, robust PCA, and Riemannian CP-dictionary-learning. We experimentally validate that our algorithm converges faster than standard Euclidean algorithms applied to the Riemannian setting.

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arXiv2023

Efficient Compression of Overparameterized Deep Models through Low-Dimensional Learning Dynamics

Overparameterized models have proven to be powerful tools for solving various machine learning tasks. However, overparameterization often leads to a substantial increase in computational and memory costs, which in turn requires extensive resources to train. In this work, we present a novel approach for compressing overparameterized models, developed through studying their learning dynamics. We observe that for many deep models, updates to the weight matrices occur within a low-dimensional invariant subspace. For deep linear models, we demonstrate that their principal components are fitted incrementally within a small subspace, and use these insights to propose a compression algorithm for deep linear networks that involve decreasing the width of their intermediate layers. We empirically evaluate the effectiveness of our compression technique on matrix recovery problems. Remarkably, by using an initialization that exploits the structure of the problem, we observe that our compressed network converges faster than the original network, consistently yielding smaller recovery errors. We substantiate this observation by developing a theory focused on deep matrix factorization. Finally, we empirically demonstrate how our compressed model has the potential to improve the utility of deep nonlinear models. Overall, our algorithm improves the training efficiency by more than 2x, without compromising generalization.

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