A.1 Behavioral Task A male nonhuman primate (NHP, Macaca mulatta), Monkey N (age 7 at the beginning of the dataset, age 11 at the end), was trained to perform a trial-based, two degree-of-freedom (DOF) dexterous finger movement task, shown in Figure 1. During all sessions, Monkey N sat in a primate chair (Crist Instruments, Hagerstown, MA) in a shielded chamber, with his arms fixed at his sides and flexed 90 degrees at the elbow, resting on a table. The left hand was positioned securely in a manipulandum, which used bend sensors (FS-L-0073-103-ST, Spectra Symbol, Salt Lake City, UT) to measure the flexion of two finger groups, index (IDX) and middle-ring-small (MRS). At the beginning of each experimental session (and as needed throughout a session), these flexion sensors were calibrated such that a reading of 1 indicated full flexion of a finger group and a reading of 0 indicated full extension. These readings were used to update the positions of the corresponding finger groups of a virtual hand presented on a screen in front of Monkey N. Bend sensor values were sampled at 1000 Hz. Updates to the virtual hand were limited to the refresh rate of the monitor (120 Hz). The task itself involved trial-based target acquisitions. At the beginning of each trial, two color-coded spherical targets, one for each DOF, were placed on the screen, covering 15% of the full arc of motion (see Figure 1A). Monkey N then acquired the targets by moving his fingers to the correct positions and holding his position for 750 ms.

We study potential presence of statistical-computational gaps (SCG) in symmetric binary perceptrons (SBP) via a parametric utilization of \emph{fully lifted random duality theory} (fl-RDT) [96]. A structural change from decreasingly to arbitrarily ordered $c$-sequence (a key fl-RDT parametric component) is observed on the second lifting level and associated with \emph{satisfiability} ($α_c$) -- \emph{algorithmic} ($α_a$) constraints density threshold change thereby suggesting a potential existence of a nonzero computational gap $SCG=α_c-α_a$. The second level estimate is shown to match the theoretical $α_c$ whereas the $r\rightarrow \infty$ level one is proposed to correspond to $α_a$. For example, for the canonical SBP ($κ=1$ margin) we obtain $α_c\approx 1.8159$ on the second and $α_a\approx 1.6021$ (with converging tendency towards $\sim 1.59$ range) on the seventh level. Our propositions remarkably well concur with recent literature: (i) in [20] local entropy replica approach predicts $α_{LE}\approx 1.58$ as the onset of clustering defragmentation (presumed driving force behind locally improving algorithms failures); (ii) in $α\rightarrow 0$ regime we obtain on the third lifting level $κ\approx 1.2385\sqrt{\frac{α_a}{-\log\left ( α_a \right ) }}$ which qualitatively matches overlap gap property (OGP) based predictions of [43] and identically matches local entropy based predictions of [24]; (iii) $c$-sequence ordering change phenomenology mirrors the one observed in asymmetric binary perceptron (ABP) in [98] and the negative Hopfield model in [100]; and (iv) as in [98,100], we here design a CLuP based algorithm whose practical performance closely matches proposed theoretical predictions.

In this paper, we provide an in-depth study of Stochastic Backpropagation (SBP) when training deep neural networks for standard image classification and object detection tasks. During backward propagation, SBP calculates gradients by using only a subset of feature maps to save GPU memory and computational cost. We interpret SBP as an efficient way to implement stochastic gradient decent by performing backpropagation dropout, which leads to significant memory saving and training run-time reduction, with a minimal impact on the overall model accuracy. We offer best practices to apply SBP for training image recognition models, which can be adopted in learning a wide range of deep neural networks. Experiments on image classification and object detection show that SBP can save up to 40% of GPU memory with less than 1% accuracy degradation.

Given two boundary distributions, the Schr\"odinger Bridge (SB) problem seeks the ``most likely`` random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for generative modeling and distribution matching. While these methods perform well in Euclidean domains, they are not directly applicable to topological domains such as graphs and simplicial complexes, which are crucial for data defined over network entities, such as node signals and edge flows. In this work, we propose the Topological Schr\"odinger Bridge problem (TSBP) for matching signal distributions on a topological domain. We set the reference process to follow some linear tractable topology-aware stochastic dynamics such as topological heat diffusion. For the case of Gaussian boundary distributions, we derive a closed-form topological SB (TSB) in terms of its time-marginal and stochastic differential. In the general case, leveraging the well-known result, we show that the optimal process follows the forward-backward topological dynamics governed by some unknowns. Building on these results, we develop TSB-based models for matching topological signals by parameterizing the unknowns in the optimal process as (topological) neural networks and learning them through likelihood training. We validate the theoretical results and demonstrate the practical applications of TSB-based models on both synthetic and real-world networks, emphasizing the role of topology. Additionally, we discuss the connections of TSB-based models to other emerging models, and outline future directions for topological signal matching.

Large language models (LLMs) demonstrate remarkable performance on many NLP tasks, yet often exhibit order dependence: simply reordering semantically identical tokens (e.g., answer choices in multiple-choice questions) can lead to inconsistent predictions. Recent work proposes Set-Based Prompting (SBP) as a way to remove order information from designated token subsets, thereby mitigating positional biases. However, applying SBP on base models induces an out-of-distribution input format, which can degrade in-distribution performance. We introduce a fine-tuning strategy that integrates SBP into the training process, "pulling" these set-formatted prompts closer to the model's training manifold. We show that SBP can be incorporated into a model via fine-tuning. Our experiments on in-distribution (MMLU) and out-of-distribution (CSQA, ARC Challenge) multiple-choice tasks show that SBP fine-tuning significantly improves accuracy and robustness to answer-order permutations, all while preserving broader language modeling capabilities. We discuss the broader implications of order-invariant modeling and outline future directions for building fairer, more consistent LLMs.

The application of reinforcement learning (RL) to dynamic resource allocation in optical networks has been the focus of intense research activity in recent years, with almost 100 peer-reviewed papers. We present a review of progress in the field, and identify significant gaps in benchmarking practices and reproducibility. To determine the strongest benchmark algorithms, we systematically evaluate several heuristics across diverse network topologies. We find that path count and sort criteria for path selection significantly affect the benchmark performance. We meticulously recreate the problems from five landmark papers and apply the improved benchmarks. Our comparisons demonstrate that simple heuristics consistently match or outperform the published RL solutions, often with an order of magnitude lower blocking probability. Furthermore, we present empirical lower bounds on network blocking using a novel defragmentation-based method, revealing that potential improvements over the benchmark heuristics are limited to 19--36\% increased traffic load for the same blocking performance in our examples. We make our simulation framework and results publicly available to promote reproducible research and standardized evaluation https://doi.org/10.5281/zenodo.12594495.