arXiv subjects

Laura Balzano

Publications and source records attributed to Laura Balzano.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Semidefinite Relaxation for Sums of Heterogeneous Quadratic Forms on the Stiefel Manifold

We study the maximization of sums of heterogeneous quadratic forms over the Stiefel manifold, a nonconvex problem that arises in several modern signal processing and machine learning applications such as heteroscedastic probabilistic principal component analysis (HPPCA). In this work, we derive a novel semidefinite program (SDP) relaxation of the original problem and study a few of its theoretical properties. We prove a global optimality certificate for the original nonconvex problem via a dual certificate, which leads to a simple feasibility problem to certify global optimality of a candidate solution on the Stiefel manifold. In addition, our relaxation reduces to an assignment linear program for jointly diagonalizable problems and is therefore known to be tight in that case. We generalize this result to show that it is also tight for close-to jointly diagonalizable problems, and we show that the HPPCA problem has this characteristic. Numerical results validate our global optimality certificate and sufficient conditions for when the SDP is tight in various problem settings.

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arXiv2025

Streaming Heteroscedastic Probabilistic PCA with Missing Data

Streaming principal component analysis (PCA) is an integral tool in large-scale machine learning for rapidly estimating low-dimensional subspaces from very high-dimensional data arriving at a high rate. However, modern datasets increasingly combine data from a variety of sources, and thus may exhibit heterogeneous quality across samples. Standard streaming PCA algorithms do not account for non-uniform noise, so their subspace estimates can quickly degrade. While the recently proposed Heteroscedastic Probabilistic PCA Technique (HePPCAT) addresses this heterogeneity, it was not designed to handle streaming data, which may exhibit non-stationary behavior. Moreover, HePPCAT does not allow for missing entries in the data, which can be common in streaming data. This paper proposes the Streaming HeteroscedASTic Algorithm for PCA (SHASTA-PCA) to bridge this divide. SHASTA-PCA employs a stochastic alternating expectation maximization approach that jointly learns the low-rank latent factors and the unknown noise variances from streaming data that may have missing entries and heteroscedastic noise, all while maintaining a low memory and computational footprint. Numerical experiments demonstrate the superior subspace estimation of our method compared to state-of-the-art streaming PCA algorithms in the heteroscedastic setting. Finally, we illustrate SHASTA-PCA applied to highly heterogeneous real data from astronomy.

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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.

cs.SI↗

arXiv2024

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 $ε$-stationary point within $\widetilde{O}(ε^{-2})$ 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.

math.OC↗

arXiv2024

Online Bilevel Optimization: Regret Analysis of Online Alternating Gradient Methods

This paper introduces \textit{online bilevel optimization} in which a sequence of time-varying bilevel problems is revealed one after the other. We extend the known regret bounds for online single-level algorithms to the bilevel setting. Specifically, we provide new notions of \textit{bilevel regret}, develop an online alternating time-averaged gradient method that is capable of leveraging smoothness, and give regret bounds in terms of the path-length of the inner and outer minimizer sequences.

math.OC↗

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.

cs.LG↗

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 $ε$-stationary point within $O(ε^{-2})$ 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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arXiv2024

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.

cs.LG↗

arXiv2024

Learning physics-based reduced-order models from data using nonlinear manifolds

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

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arXiv2023

ALPCAH: Sample-wise Heteroscedastic PCA with Tail Singular Value Regularization

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH

stat.ML↗

arXiv2023

Matrix Completion over Finite Fields: Bounds and Belief Propagation Algorithms

We consider the low rank matrix completion problem over finite fields. This problem has been extensively studied in the domain of real/complex numbers, however, to the best of authors' knowledge, there exists merely one efficient algorithm to tackle the problem in the binary field, due to Saunderson et al. [1]. In this paper, we improve upon the theoretical guarantees for the algorithm provided in [1]. Furthermore, we formulate a new graphical model for the matrix completion problem over the finite field of size $q$, $\Bbb{F}_q$, and present a message passing (MP) based approach to solve this problem. The proposed algorithm is the first one for the considered matrix completion problem over finite fields of arbitrary size. Our proposed method has a significantly lower computational complexity, reducing it from $O(n^{2r+3})$ in [1] down to $O(n^2)$ (where, the underlying matrix has dimension $n \times n$ and $r$ denotes its rank), while also improving the performance.

cs.IT↗

arXiv2023

Learning latent representations in high-dimensional state spaces using polynomial manifold constructions

We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem's intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques.

math.NA↗

arXiv2023

The Law of Parsimony in Gradient Descent for Learning Deep Linear Networks

Over the past few years, an extensively studied phenomenon in training deep networks is the implicit bias of gradient descent towards parsimonious solutions. In this work, we investigate this phenomenon by narrowing our focus to deep linear networks. Through our analysis, we reveal a surprising "law of parsimony" in the learning dynamics when the data possesses low-dimensional structures. Specifically, we show that the evolution of gradient descent starting from orthogonal initialization only affects a minimal portion of singular vector spaces across all weight matrices. In other words, the learning process happens only within a small invariant subspace of each weight matrix, despite the fact that all weight parameters are updated throughout training. This simplicity in learning dynamics could have significant implications for both efficient training and a better understanding of deep networks. First, the analysis enables us to considerably improve training efficiency by taking advantage of the low-dimensional structure in learning dynamics. We can construct smaller, equivalent deep linear networks without sacrificing the benefits associated with the wider counterparts. Second, it allows us to better understand deep representation learning by elucidating the linear progressive separation and concentration of representations from shallow to deep layers. We also conduct numerical experiments to support our theoretical results. The code for our experiments can be found at https://github.com/cjyaras/lawofparsimony.

cs.LG↗

arXiv2023

Dynamic Subspace Estimation with Grassmannian Geodesics

Dynamic subspace estimation, or subspace tracking, is a fundamental problem in statistical signal processing and machine learning. This paper considers a geodesic model for time-varying subspaces. The natural objective function for this model is non-convex. We propose a novel algorithm for minimizing this objective and estimating the parameters of the model from data with Grassmannian-constrained optimization. We show that with this algorithm, the objective is monotonically non-increasing. We demonstrate the performance of this model and our algorithm on synthetic data, video data, and dynamic fMRI data.

eess.SP↗

arXiv2023

Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold

When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon for normalized features. Based on an unconstrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddles with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization.

cs.LG↗

arXiv2023

HeMPPCAT: Mixtures of Probabilistic Principal Component Analysers for Data with Heteroscedastic Noise

Mixtures of probabilistic principal component analysis (MPPCA) is a well-known mixture model extension of principal component analysis (PCA). Similar to PCA, MPPCA assumes the data samples in each mixture contain homoscedastic noise. However, datasets with heterogeneous noise across samples are becoming increasingly common, as larger datasets are generated by collecting samples from several sources with varying noise profiles. The performance of MPPCA is suboptimal for data with heteroscedastic noise across samples. This paper proposes a heteroscedastic mixtures of probabilistic PCA technique (HeMPPCAT) that uses a generalized expectation-maximization (GEM) algorithm to jointly estimate the unknown underlying factors, means, and noise variances under a heteroscedastic noise setting. Simulation results illustrate the improved factor estimates and clustering accuracies of HeMPPCAT compared to MPPCA.

stat.ME↗

arXiv2022

Mode Reduction for Markov Jump Systems

Switched systems are capable of modeling processes with underlying dynamics that may change abruptly over time. To achieve accurate modeling in practice, one may need a large number of modes, but this may in turn increase the model complexity drastically. Existing work on reducing system complexity mainly considers state space reduction, yet reducing the number of modes is less studied. In this work, we consider Markov jump linear systems (MJSs), a special class of switched systems where the active mode switches according to a Markov chain, and several issues associated with its mode complexity. Specifically, inspired by clustering techniques from unsupervised learning, we are able to construct a reduced MJS with fewer modes that approximates well the original MJS under various metrics. Furthermore, both theoretically and empirically, we show how one can use the reduced MJS to analyze stability and design controllers with significant reduction in computational cost while achieving guaranteed accuracy.

eess.SY↗

arXiv2022

Optimally Weighted PCA for High-Dimensional Heteroscedastic Data

Modern data are increasingly both high-dimensional and heteroscedastic. This paper considers the challenge of estimating underlying principal components from high-dimensional data with noise that is heteroscedastic across samples, i.e., some samples are noisier than others. Such heteroscedasticity naturally arises, e.g., when combining data from diverse sources or sensors. A natural way to account for this heteroscedasticity is to give noisier blocks of samples less weight in PCA by using the leading eigenvectors of a weighted sample covariance matrix. We consider the problem of choosing weights to optimally recover the underlying components. In general, one cannot know these optimal weights since they depend on the underlying components we seek to estimate. However, we show that under some natural statistical assumptions the optimal weights converge to a simple function of the signal and noise variances for high-dimensional data. Surprisingly, the optimal weights are not the inverse noise variance weights commonly used in practice. We demonstrate the theoretical results through numerical simulations and comparisons with existing weighting schemes. Finally, we briefly discuss how estimated signal and noise variances can be used when the true variances are unknown, and we illustrate the optimal weights on real data from astronomy.

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