Efficient exploration remains one of the most important open problems in reinforcement learning. Discovering novel states or transitions requires policies that efficiently direct the agent away from the regions of the state space that are already well explored. We introduce Novel Exploration via Orthogonality (NEO), an approach that automatically uncovers not only which regions of the environment are novel but also how to reach them by leveraging Laplacian representations. NEO uses the eigenvectors of a modified graph Laplacian to induce gradient flows from states that are frequently visited (less novel) to states that are seldom visited (more novel). We show that NEO's modified Laplacian yields eigenvectors whose extreme values align with the most novel regions of the state space. We provide bounds for the eigenvalues of the modified Laplacian; and we show that the smoothest eigenvectors with real eigenvalues below certain thresholds provide guaranteed gradients to novel states for both undirected and directed graphs. In an empirical evaluation in online, incremental settings, NEO outperformed related state-of-theart approaches, including eigen-options and cover options, in a large collection of undirected and directed environments with varying connectivity structures.

Computation in recurrent networks of neurons has been hypothesized to occur at the level of low-dimensional latent dynamics, both in artificial systems and in the brain. This hypothesis seems at odds with evidence from large-scale neuronal recordings in mice showing that neuronal population activity is high-dimensional. To demonstrate that low-dimensional latent dynamics and high-dimensional activity can be two sides of the same coin, we present an analytically solvable recurrent neural network (RNN) model whose dynamics can be exactly reduced to a lowdimensional dynamical system, but generates an activity manifold that has a high linear embedding dimension. This raises the question: Do low-dimensional latents explain the high-dimensional activity observed in mouse visual cortex? Spectral theory tells us that the covariance eigenspectrum alone does not allow us to recover the dimensionality of the latents, which can be low or high, when neurons are nonlinear. To address this indeterminacy, we develop Neural Cross-Encoder (NCE), an interpretable, nonlinear latent variable modeling method for neuronal recordings, and find that high-dimensional neuronal responses to drifting gratings and spontaneous activity in visual cortex can be reduced to low-dimensional latents, while the responses to natural images cannot. We conclude that the high-dimensional activity measured in certain conditions, such as in the absence of a stimulus, is explained by low-dimensional latents that are nonlinearly processed by individual neurons.

Spectral decomposition of linear operators plays a central role in many areas of machine learning and scientific computing. Recent work has explored training neural networks to approximate eigenfunctions of such operators, enabling scalable approaches to representation learning, dynamical systems, and partial differential equations (PDEs). In this paper, we revisit a classical optimization framework from the computational physics literature known as the orbital minimization method (OMM), originally proposed in the 1990s for solving eigenvalue problems in computational chemistry. We provide a simple linear-algebraic proof of the consistency of the OMM objective, and reveal connections between this method and several ideas that have appeared independently across different domains. Our primary goal is to justify its broader applicability in modern learning pipelines. We adapt this framework to train neural networks to decompose positive semidefinite operators, and demonstrate its practical advantages across a range of benchmark tasks. Our results highlight how revisiting classical numerical methods through the lens of modern theory and computation can provide not only a principled approach for deploying neural networks in numerical simulation, but also effective and scalable tools for machine learning.

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting afterstudy inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d.

Vertex hunting (VH) is the task of estimating a simplex from noisy data points and has many applications in areas such as network and text analysis. We introduce a new variant, semi-supervised vertex hunting (SSVH), in which partial information is available in the form of barycentric coordinates for some data points, known only up to an unknown transformation. To address this problem, we develop a method that leverages properties of orthogonal projection matrices, drawing on novel insights from linear algebra. We establish theoretical error bounds for our method and demonstrate that it achieves a faster convergence rate than existing unsupervised VH algorithms. Finally, we apply SSVH to two practical settings-- semi-supervised network mixed membership estimation and semi-supervised topic modeling--resulting in efficient and scalable algorithms.

We study the problem of preconditioning in sequential prediction. From the theoretical lens of linear dynamical systems, we show that convolving the target sequence corresponds to applying a polynomial to the hidden transition matrix. Building on this insight, we propose a universal preconditioning method that convolves the target with coefficients from orthogonal polynomials such as Chebyshev or Legendre. We prove that this approach reduces regret for two distinct prediction algorithms and yields the first ever sublinear and hidden-dimension-independent regret bounds (up to logarithmic factors) that hold for systems with marginally stable and asymmetric transition matrices. Finally, extensive synthetic and realworld experiments show that this simple preconditioning strategy improves the performance of a diverse range of algorithms, including recurrent neural networks, and generalizes to signals beyond linear dynamical systems.

AResults883 In Table 4, we show a summary of the results of AlgoTuner for each of the four frontier models tests.884 Speedup percentage is calculated as the percentage of tasks for which AlgoTuner gets at least a 1.1 speedup. Speedup is calculated as the ratio between the reference solve function's time and the LM-generated solve function's time. The LM receives an initial message, consisting of general instructions on how to888 use the system (see C.1), Numba (Lam et al., 2015), Dask (Rocklin, 2015), and Cython (Behnel et al.,889 2011) (for a full list see Appendix E). Additionally, the LM is given the task's description, which890 includes input and output descriptions and examples, as well as the task's solveand is_solution891 functions.

Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, realworld matrices are often observed with noise, arising from sampling, sketching, and quantization. The spectral-norm robustness of low-rank inverse approximations remains poorly understood. We systematically study the spectral-norm error ( A 1)p A 1p for an n n symmetric matrix A, where A 1p denotes the best rank-papproximation of A 1, and A = A+E is a noisy observation. Under mild assumptions on the noise, we derive sharp non-asymptotic perturbation bounds that reveal how the error scales with the eigengap, spectral decay, and noise alignment with low-curvature directions of A. Our analysis introduces a novel application of contour integral techniques to the non-entire function f(z) = 1/z, yielding bounds that improve over naive adaptations of classical full-inverse bounds by up to a factor of n. Empirically, our bounds closely track the true perturbation error across a variety of real-world and synthetic matrices, while estimates based on classical results tend to significantly overpredict. These findings offer practical, spectrum-aware guarantees for low-rank inverse approximations in noisy computational environments.

Recurrent neural networks (RNNs) trained on neuroscience-inspired tasks offer powerful models of brain computation. However, typical training paradigms rely on open-loop, supervised settings, whereas real-world learning unfolds in closed-loop environments. Here, we develop a mathematical theory describing the learning dynamics of linear RNNs trained in closed-loop contexts. We first demonstrate that two otherwise identical RNNs, trained in either closed-or open-loop modes, follow markedly different learning trajectories. To probe this divergence, we analytically characterize the closed-loop case, revealing distinct stages aligned with the evolution of the training loss. Specifically, we show that the learning dynamics of closed-loop RNNs, in contrast to open-loop ones, are governed by an interplay between two competing objectives: short-term policy improvement and long-term stability of the agent-environment interaction. Finally, we apply our framework to a realistic motor control task, highlighting its broader applicability. Taken together, our results underscore the importance of modeling closed-loop dynamics in a biologically plausible setting.

One of the chronic problems of deep-learning models is shortcut learning. In a case where the majority of training data are dominated by a certain feature, neural networks prefer to learn such a feature even if the feature is not generalizable outside the training set. Based on the framework of Neural Tangent Kernel (NTK), we analyzed the case of linear neural networks to derive some important properties of shortcut learning. We defined a "feature" of a neural network as an eigenfunction of NTK. Then, we found that shortcut features correspond to features with larger eigenvalues when the shortcuts stem from the imbalanced number of samples in the clustered distribution. We also showed that the features with larger eigenvalues still have a large influence on the neural network output even after training, due to data variances in the clusters. Such a preference for certain features remains even when a margin of a neural network output is controlled, which shows that the max-margin bias is not the only major reason for shortcut learning. These properties of linear neural networks are empirically extended for more complex neural networks as a two-layer fully-connected ReLU network and a ResNet-18.