--We consider the phase retrieval problem, which involves recovering a rank-one positive semidefinite matrix from rank-one measurements. A recently proposed algorithm based on Bures-Wasserstein gradient descent (BWGD) exhibits superlinear convergence, but it is unstable, and existing theory can only prove local linear convergence for higher rank matrix recovery. We resolve this gap by revealing that BWGD implements Newton's method with a nonsmooth and nonconvex objective. We develop a smoothing framework that regularizes the objective, enabling a stable method with rigorous superlinear convergence guarantees. Experiments on synthetic data demonstrate this superior stability while maintaining fast convergence. Phase retrieval--the problem of recovering a real or complex signal from magnitude-only measurements--is a fundamental problem in signal processing. Its applications range from X-ray crystallography to astronomical imaging, where measurement systems capture a form of intensity [Harrison, 1993, Fienup, 1982, Fienup and Dainty, 1987, Miao et al., 1998]. The seemingly simple constraint of measuring magnitudes transforms what would be a linear problem into a challenging nonlinear and nonconvex optimization problem. More critically, the direct optimization formulation using least squares yields a nonconvex objective function, making it difficult to solve effectively. The phase retrieval problem is a specific instance of a broader class of low-rank matrix sensing problems that arise throughout signal processing [Recht et al., 2010].

To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear algebra, such as matrices, as morphisms in the category of matrices, $\mathbf{Mat_{k}}$. This framework is further extended by generalizing the results to arbitrary monoidal semiadditive categories. To enrich this perspective and accommodate higher-rank matrices (tensors), we define semiadditive 2-categories, where matrices $T_{ij}$ are represented as 1-morphisms, and tensors with four indices $T_{ijkl}$ as 2-morphisms. This formalization provides an index-free, typed linear algebra framework that includes matrices and tensors with up to four indices. Furthermore, we extend the framework to monoidal semiadditive 2-categories and demonstrate detailed operations and vectorization within the 2-category of 2Vec introduced by Kapranov and Voevodsky.

We describe a thermodynamic-inspired computing paradigm based on oscillatory neural networks (ONNs). While ONNs have been widely studied as Ising machines for tackling complex combinatorial optimization problems, this work investigates their feasibility in solving linear algebra problems, specifically the inverse matrix. Grounded in thermodynamic principles, we analytically demonstrate that the linear approximation of the coupled Kuramoto oscillator model leads to the inverse matrix solution. Numerical simulations validate the theoretical framework, and we examine the parameter regimes that computation has the highest accuracy.

We propose a novel framework for learning hidden graph structures from data using geometric analysis and nonlinear dynamics. Our approach: (1) Defines discrete Sobolev spaces on graphs for scalar/vector fields, establishing key functional properties; (2) Introduces gauge-equivalent nonlinear Schrödinger and Landau--Lifshitz dynamics with provable stable stationary solutions smoothly dependent on input data and graph weights; (3) Develops a stochastic gradient algorithm over graph moduli spaces with sparsity regularization. Theoretically, we guarantee: topological correctness (homology recovery), metric convergence (Gromov--Hausdorff), and efficient search space utilization. Our dynamics-based model achieves stronger generalization bounds than standard neural networks, with complexity dependent on the data manifold's topology.

We propose an efficient federated dual decomposition algorithm for calculating the Wasserstein barycenter of several distributions, including choosing the support of the solution. The algorithm does not access local data and uses only highly aggregated information. It also does not require repeated solutions to mass transportation problems. Because of the absence of any matrix-vector operations, the algorithm exhibits a very low complexity of each iteration and significant scalability. We illustrate its virtues and compare it to the state-of-the-art methods on several examples of mixture models.

We study the problem of learning a structured approximation (low-rank, sparse, banded, etc.) to an unknown matrix $A$ given access to matrix-vector product (matvec) queries of the form $x \rightarrow Ax$ and $x \rightarrow A^Tx$. This problem is of central importance to algorithms across scientific computing and machine learning, with applications to fast multiplication and inversion for structured matrices, building preconditioners for first-order optimization, and as a model for differential operator learning. Prior work focuses on obtaining query complexity upper and lower bounds for learning specific structured matrix families that commonly arise in applications. We initiate the study of the problem in greater generality, aiming to understand the query complexity of learning approximations from general matrix families. Our main result focuses on finding a near-optimal approximation to $A$ from any finite-sized family of matrices, $\mathcal{F}$. Standard results from matrix sketching show that $O(\log|\mathcal{F}|)$ matvec queries suffice in this setting. This bound can also be achieved, and is optimal, for vector-matrix-vector queries of the form $x,y\rightarrow x^TAy$, which have been widely studied in work on rank-$1$ matrix sensing. Surprisingly, we show that, in the matvec model, it is possible to obtain a nearly quadratic improvement in complexity, to $\tilde{O}(\sqrt{\log|\mathcal{F}|})$. Further, we prove that this bound is tight up to log-log factors.Via covering number arguments, our result extends to well-studied infinite families. As an example, we establish that a near-optimal approximation from any \emph{linear matrix family} of dimension $q$ can be learned with $\tilde{O}(\sqrt{q})$ matvec queries, improving on an $O(q)$ bound achievable via sketching techniques and vector-matrix-vector queries.

The original version of this story appeared in Quanta Magazine. In 1994, an earthquake of a proof shook up the mathematical world. The mathematician Andrew Wiles had finally settled Fermat's Last Theorem, a central problem in number theory that had remained open for over three centuries. The proof didn't just enthral mathematicians--it made the front page of The New York Times. But to accomplish it, Wiles (with help from the mathematician Richard Taylor) first had to prove a more subtle intermediate statement--one with implications that extended beyond Fermat's puzzle.

We develop a criterion to certify whether causal effects are identifiable in linear structural equation models with latent variables. Linear structural equation models correspond to directed graphs whose nodes represent the random variables of interest and whose edges are weighted with linear coefficients that correspond to direct causal effects. In contrast to previous identification methods, we do not restrict ourselves to settings where the latent variables constitute independent latent factors (i.e., to source nodes in the graphical representation of the model). Our novel latent-subgraph criterion is a purely graphical condition that is sufficient for identifiability of causal effects by rational formulas in the covariance matrix. To check the latent-subgraph criterion, we provide a sound and complete algorithm that operates by solving an integer linear program. While it targets effects involving observed variables, our new criterion is also useful for identifying effects between latent variables, as it allows one to transform the given model into a simpler measurement model for which other existing tools become applicable.

Many signals in the real world that evolve in time can be modeled as a stochastic process with the signal randomly jumping from one state to another as time proceeds. When the signal can only exhibit a finite number of possible states, one can interpret the evolution of the signal as a random walk on a graph with vertices representing the states of the signal and edge weights giving way to the transition probabilities from one state to another. In particular, one arrives at a Markov chain representation of the signal that can be estimated using only the signal data. However, many realistic signals can take on a continuum of values, and so the goal of this work is to present a framework for modeling continuous-space stochastic signals and to identify metastable and coherent sets via clustering techniques. We present a data-driven method to learn the discrete-time transition probabilities of stochastic signals evolving in continuous space, which can be regarded as a generalization of the discrete space case considered in [25, 22]. The underlying theory is developed by evoking the concept of a graphon, which can be defined as the limit of sequences of dense networks that grow without bound [35, 34, 21, 18]. As recently shown in [43], graphons provide a well-developed framework for extending the concepts of random walks on finite graphs to stochastic processes evolving in continuous space. For example, random walks on graphs can be used to measure the centrality of vertices, and these concepts can also be extended to graphons [4]. Our goal is to identify transition probabilities, clusters, and the graphon itself from random walk data.

The sparse identification of nonlinear dynamics (SINDy) approach can discover the governing equations of dynamical systems based on measurement data, where the dynamical model is identified as the sparse linear combination of the given basis functions. A major challenge in SINDy is the design of a library, which is a set of candidate basis functions, as the appropriate library is not trivial for many dynamical systems. To overcome this difficulty, this study proposes SINDy with library optimization mechanism (SINDy-LOM), which is a combination of the sparse regression technique and the novel learning strategy of the library. In the proposed approach, the basis functions are parametrized. The SINDy-LOM approach involves a two-layer optimization architecture: the inner-layer, in which the data-driven model is extracted as the sparse linear combination of the candidate basis functions, and the outer-layer, in which the basis functions are optimized from the viewpoint of the recursive long-term (RLT) prediction accuracy; thus, the library design is reformulated as the optimization of the parametrized basis functions. The resulting SINDy-LOM model has good interpretability and usability, as the proposed approach yields the parsimonious model. The library optimization mechanism significantly reduces user burden. The RLT perspective improves the reliability of the resulting model compared with the traditional SINDy approach that can only ensure the one-step-ahead prediction accuracy. The validity of the proposed approach is demonstrated by applying it to a diesel engine airpath system, which is a well-known complex industrial system.