Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker, convolutional, block diagonal, sum, or product structure. In this paper, we propose a simple but general framework for large-scale linear algebra problems in machine learning, named CoLA (Compositional Linear Algebra). By combining a linear operator abstraction with compositional dispatch rules, CoLA automatically constructs memory and runtime efficient numerical algorithms. Moreover, CoLA provides memory efficient automatic differentiation, low precision computation, and GPU acceleration in both JAX and PyTorch, while also accommodating new objects, operations, and rules in downstream packages via multiple dispatch. CoLA can accelerate many algebraic operations, while making it easy to prototype matrix structures and algorithms, providing an appealing drop-in tool for virtually any computational effort that requires linear algebra. We showcase its efficacy across a broad range of applications, including partial differential equations, Gaussian processes, equivariant model construction, and unsupervised learning.

This paper addresses aircraft delays, emphasizing their impact on safety and financial losses. To mitigate these issues, an innovative machine learning (ML)-enhanced landing scheduling methodology is proposed, aiming to improve automation and safety. Analyzing flight arrival delay scenarios reveals strong multimodal distributions and clusters in arrival flight time durations. A multi-stage conditional ML predictor enhances separation time prediction based on flight events. ML predictions are then integrated as safety constraints in a time-constrained traveling salesman problem formulation, solved using mixed-integer linear programming (MILP). Historical flight recordings and model predictions address uncertainties between successive flights, ensuring reliability. The proposed method is validated using real-world data from the Atlanta Air Route Traffic Control Center (ARTCC ZTL). Case studies demonstrate an average 17.2% reduction in total landing time compared to the First-Come-First-Served (FCFS) rule. Unlike FCFS, the proposed methodology considers uncertainties, instilling confidence in scheduling. The study concludes with remarks and outlines future research directions.

A system of coupled oscillators on an arbitrary graph is locally driven by the tendency to mutual synchronization between nearby oscillators, but can and often exhibit nonlinear behavior on the whole graph. Understanding such nonlinear behavior has been a key challenge in predicting whether all oscillators in such a system will eventually synchronize. In this paper, we demonstrate that, surprisingly, such nonlinear behavior of coupled oscillators can be effectively linearized in certain latent dynamic spaces. The key insight is that there is a small number of `latent dynamics filters', each with a specific association with synchronizing and non-synchronizing dynamics on subgraphs so that any observed dynamics on subgraphs can be approximated by a suitable linear combination of such elementary dynamic patterns. Taking an ensemble of subgraph-level predictions provides an interpretable predictor for whether the system on the whole graph reaches global synchronization. We propose algorithms based on supervised matrix factorization to learn such latent dynamics filters. We demonstrate that our method performs competitively in synchronization prediction tasks against baselines and black-box classification algorithms, despite its simple and interpretable architecture.

Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from a significant computational burden. Thus, a meaningful line of work consists at proposing alternatives to reduce this burden while still enjoying its properties. In this thesis, we focus on alternatives which use projections on subspaces. The main such alternative is the Sliced-Wasserstein distance, which we first propose to extend to Riemannian manifolds in order to use it in Machine Learning applications for which using such spaces has been shown to be beneficial in the recent years. We also study sliced distances between positive measures in the so-called unbalanced OT problem. Back to the original Euclidean Sliced-Wasserstein distance between probability measures, we study the dynamic of gradient flows when endowing the space with this distance in place of the usual Wasserstein distance. Then, we investigate the use of the Busemann function, a generalization of the inner product in metric spaces, in the space of probability measures. Finally, we extend the subspace detour approach to incomparable spaces using the Gromov-Wasserstein distance.

Stochastic gradient-based optimization methods such as SGD and Adam (Kingma & Ba, 2014) are the main workhorse behind modern machine learning. Such methods sequentially apply stochastic gradient steps to update the trained model and their performance crucially depends on the choice of a learning rate sequence, or schedule, used throughout this process to determine the magnitude of the sequential updates. All in all, effectively tuning the learning rate schedule is widely considered a tedious task requiring extensive, sometimes prohibitive, hyper-parameter search, resulting in a significant excess of engineering time and compute resources usage in ML training. A prominent approach to address this issue gave rise to a plethora of adaptive optimization methods (most notably Duchi et al., 2011 and Kingma & Ba, 2014), where the learning rate parameter is automatically tuned during the optimization process based on previously received stochastic gradients. In some important applications these methods provide superior convergence performance, while their theoretical guarantees match the state-of-the-art in the stochastic convex and (smooth) non-convex optimization settings (Li & Orabona, 2019; Ward et al., 2020; Attia & Koren, 2023). However, despite the adaptivity incorporated into these methods, auxiliary learning rate schedules are often still required to actually attain their optimal performance (e.g., Loshchilov & Hutter, 2016), and the nuisance of laborious and extensive manual tuning still remain relevant for these methods as well.

Learning and forecasting stochastic time series is essential in various scientific fields. However, despite the proposals of nonlinear filters and deep-learning methods, it remains challenging to capture nonlinear dynamics from a few noisy samples and predict future trajectories with uncertainty estimates while maintaining computational efficiency. Here, we propose a fast algorithm to learn and forecast nonlinear dynamics from noisy time series data. A key feature of the proposed model is kernel functions applied to projected lines, enabling fast and efficient capture of nonlinearities in the latent dynamics. Through empirical case studies and benchmarking, the model demonstrates its effectiveness in learning and forecasting complex nonlinear dynamics, offering a valuable tool for researchers and practitioners in time series analysis.

This work addresses the problem of motion planning for a group of nonholonomic robots under Visible Light Communication (VLC) connectivity requirements. In particular, we consider an inspection task performed by a Robot Chain Control System (RCCS), where a leader must visit relevant regions of an environment while the remaining robots operate as relays, maintaining the connectivity between the leader and a base station. We leverage Mixed-Integer Linear Programming (MILP) to design a trajectory planner that can coordinate the RCCS, minimizing time and control effort while also handling the issues of directed Line-Of-Sight (LOS), connectivity over directed networks, and the nonlinearity of the robots' dynamics. The efficacy of the proposal is demonstrated with realistic simulations in the Gazebo environment using the Turtlebot3 robot platform.

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model builds on the random geometric graphs (Gilbert, 1961), one of the basic models of random graphs for spatial networks, in the same way that the well-studied stochastic block model builds on the Erd\H{o}s-R\'{en}yi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancements in community detection. To analyze the geometric block model, we first provide new connectivity results for random annulus graphs which are generalizations of random geometric graphs. The connectivity properties of geometric graphs have been studied since their introduction, and analyzing them has been more difficult than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge formation. We then use the connectivity results of random annulus graphs to provide necessary and sufficient conditions for efficient recovery of communities for the geometric block model. We show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. For this we consider the following two regimes of graph density. In the regime where the average degree of the graph grows logarithmically with the number of vertices, we show that our algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model in the logarithmic degree regime. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.

A random $m\times n$ matrix $S$ is an oblivious subspace embedding (OSE) with parameters $\epsilon>0$, $\delta\in(0,1/3)$ and $d\leq m\leq n$, if for any $d$-dimensional subspace $W\subseteq R^n$, $P\big(\,\forall_{x\in W}\ (1+\epsilon)^{-1}\|x\|\leq\|Sx\|\leq (1+\epsilon)\|x\|\,\big)\geq 1-\delta.$ It is known that the embedding dimension of an OSE must satisfy $m\geq d$, and for any $\theta > 0$, a Gaussian embedding matrix with $m\geq (1+\theta) d$ is an OSE with $\epsilon = O_\theta(1)$. However, such optimal embedding dimension is not known for other embeddings. Of particular interest are sparse OSEs, having $s\ll m$ non-zeros per column, with applications to problems such as least squares regression and low-rank approximation. We show that, given any $\theta > 0$, an $m\times n$ random matrix $S$ with $m\geq (1+\theta)d$ consisting of randomly sparsified $\pm1/\sqrt s$ entries and having $s= O(\log^4(d))$ non-zeros per column, is an oblivious subspace embedding with $\epsilon = O_{\theta}(1)$. Our result addresses the main open question posed by Nelson and Nguyen (FOCS 2013), who conjectured that sparse OSEs can achieve $m=O(d)$ embedding dimension, and it improves on $m=O(d\log(d))$ shown by Cohen (SODA 2016). We use this to construct the first oblivious subspace embedding with $O(d)$ embedding dimension that can be applied faster than current matrix multiplication time, and to obtain an optimal single-pass algorithm for least squares regression. We further extend our results to construct even sparser non-oblivious embeddings, leading to the first subspace embedding with low distortion $\epsilon=o(1)$ and optimal embedding dimension $m=O(d/\epsilon^2)$ that can be applied in current matrix multiplication time.

Automated synthesis of correct-by-construction controllers for autonomous systems is crucial for their deployment in safety-critical scenarios. Such autonomous systems are naturally modeled as stochastic dynamical models. The general problem is to compute a controller that provably satisfies a given task, represented as a probabilistic temporal logic specification. However, factors such as stochastic uncertainty, imprecisely known parameters, and hybrid features make this problem challenging. We have developed an abstraction framework that can be used to solve this problem under various modeling assumptions. Our approach is based on a robust finite-state abstraction of the stochastic dynamical model in the form of a Markov decision process with intervals of probabilities (iMDP). We use state-of-the-art verification techniques to compute an optimal policy on the iMDP with guarantees for satisfying the given specification. We then show that, by construction, we can refine this policy into a feedback controller for which these guarantees carry over to the dynamical model. In this short paper, we survey our recent research in this area and highlight two challenges (related to scalability and dealing with nonlinear dynamics) that we aim to address with our ongoing research.