Estimating the geometric median of a dataset is a robust counterpart to mean estimation, and is a fundamental problem in computational geometry.

Generalization is a key challenge in machine learning, specifically in reasoning tasks, where models are expected to solve problems more complex than those encountered during training. Existing approaches typically train reasoning models in an end-to-end fashion, directly mapping input instances to solutions. While this allows models to learn useful heuristics from data, it often results in limited generalization beyond the training distribution. In this work, we propose a novel approach to reasoning generalization by learning energy landscapes over the solution spaces of smaller, more tractable subproblems. At test time, we construct a global energy landscape for a given problem by combining the energy functions of multiple subproblems. This compositional approach enables the incorporation of additional constraints during inference, allowing the construction of energy landscapes for problems of increasing difficulty. To improve the sample quality from this newly constructed energy landscape, we introduce Parallel Energy Minimization (PEM). We evaluate our approach on a wide set of reasoning problems. Our method outperforms existing state-of-the-art methods, demonstrating its ability to generalize to larger and more complex problems.

Invariant object recognition-the ability to identify objects despite changes in appearance-is a hallmark of visual processing in the brain, yet its understanding remains a central challenge in systems neuroscience. Artificial neural networks trained to predict neural responses to visual stimuli ("digital twins") could provide a powerful framework for studying such complex computations in silico. However, while current models accurately capture single-neuron responses within individual visual areas, their ability to reproduce how populations of neurons represent object identity, and how these representations transform across the cortical hierarchy, remains largely unexplored. Here we examine key functional signatures observed experimentally and find that current models account for hierarchical changes in basic single-neuron properties, such as receptive field size, but fail to capture more complex population-level phenomena, particularly invariant object representations. To address this gap, we introduce a biologically inspired hierarchical readout scheme that mirrors cortical anatomy, modeling each visual area as a projection from a distinct depth within a shared core network. This approach significantly improves the prediction of population-level representational transformations, outperforming standard models that use only the final layer, as well as alternatives with modified architecture, regularization, and loss function. Our results suggest that incorporating anatomical information provides a strong inductive bias in digital twin models, enabling them to better capture general principles of brain function.

We establish a comprehensive finite-sample and asymptotic theory for stochastic gradient descent (SGD) with constant learning rates. First, we propose a novel linear approximation technique to provide a quenched central limit theorem (CLT) for SGD iterates with refined tail properties, showing that regardless of the chosen initialization, the fluctuations of the algorithm around its target point converge to a multivariate normal distribution. Our conditions are substantially milder than those required in the classical CLTs for SGD, yet offering a stronger convergence result. Furthermore, we derive the first Berry-Esseen bound - the Gaussian approximation error - for the constant learning-rate SGD, which is sharp compared to the decaying learning-rate schemes in the literature. Beyond the moment convergence, we also provide the Nagaev-type inequality for the SGD tail probabilities by adopting the autoregressive approximation techniques, which entails non-asymptotic largedeviation guarantees. These results are verified via numerical simulations, paving the way for theoretically grounded uncertainty quantification, especially with non-asymptotic validity.

The stable periodic patterns present in the time series data serve as the foundation for long-term forecasting. However, existing models suffer from limitations such as continuous and chaotic input partitioning, as well as weak inductive biases, which restrict their ability to capture such recurring structures. In this paper, we propose MoFo, which interprets periodicity as both the correlation of periodaligned time steps and the trend of period-offset time steps. We first design periodstructured patches--2D tensors generated through discrete sampling--where each row contains only period-aligned time steps, enabling direct modeling of periodic correlations. Period-offset time steps within a period are aligned in columns.

We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.

Transformer architectures can solve unseen tasks based on input-output pairs in a given prompt due to in-context learning (ICL). Existing theoretical studies on ICL have mainly focused on linear regression tasks, often with i.i.d.

In this work, we introduce TreeGen, a novel generative framework modeling distributions over hierarchies. We extend Bayesian Flow Networks (BFNs) to enable transitions between probabilistic and discrete hierarchies parametrized via categorical distributions. Our proposed scheduler provides smooth and consistent entropy decay across varying numbers of categories. We empirically evaluate TreeGen on the jet-clustering task in high-energy physics, demonstrating that it consistently generates valid trees that adhere to physical constraints and closely align with ground-truth log-likelihoods. Finally, by comparing TreeGen's samples to the exact posterior distribution and performing likelihood maximization via rejection sampling, we demonstrate that TreeGen outperforms various baselines.

Goal-conditioned policy learning for robotic manipulation presents significant challenges in maintaining performance across diverse objectives and environments. We introduce Hyper-GoalNet, a framework that generates task-specific policy network parameters from goal specifications using hypernetworks. Unlike conventional methods that simply condition fixed networks on goal-state pairs, our approach separates goal interpretation from state processing - the former determines network parameters while the latter applies these parameters to current observations. To enhance representation quality for effective policy generation, we implement two complementary constraints on the latent space: (1) a forward dynamics model that promotes state transition predictability, and (2) a distance-based constraint ensuring monotonic progression toward goal states. We evaluate our method on a comprehensive suite of manipulation tasks with varying environmental randomization. Results demonstrate significant performance improvements over state-of-the-art methods, particularly in high-variability conditions.

The rise of smart manufacturing under Industry 4.0 introduces mass customization and dynamic production, demanding more advanced and flexible scheduling techniques. The flexible job-shop scheduling problem (FJSP) has attracted significant attention due to its complex constraints and strong alignment with real-world production scenarios. Current deep reinforcement learning (DRL)-based approaches to FJSP predominantly employ constructive methods. While effective, they often fall short of reaching (near-)optimal solutions. In contrast, improvement-based methods iteratively explore the neighborhood of initial solutions and are more effective in approaching optimality.