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Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. In this paper, we show that there exists a certain "sparsifer": a subset of elements, of size $O(|S^{opt}| \cdot 1/\varepsilon)$, where $S^{opt}$ denotes the optimal solution, that is guaranteed to contain a $3/2 + \varepsilon$ approximation, while guaranteeing certain robustness properties. We call such a small subset a Density Constrained Subset (DCS), which is inspired by the Edge-Degree Constrained Subgraph (EDCS) [Bernstein and Stein, 2015], originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the density-based decomposition. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models.

Representational spaces learned via language modeling are fundamental to Natural Language Processing (NLP), however there has been limited understanding regarding how and when during training various types of linguistic information emerge and interact. Leveraging a novel information theoretic probing suite, which enables direct comparisons of not just task performance, but their representational subspaces, we analyze nine tasks covering syntax, semantics and reasoning, across 2M pre-training steps and five seeds. We identify critical learning phases across tasks and time, during which subspaces emerge, share information, and later disentangle to specialize. Across these phases, syntactic knowledge is acquired rapidly after 0.5% of full training. Continued performance improvements primarily stem from the acquisition of open-domain knowledge, while semantics and reasoning tasks benefit from later boosts to long-range contextualization and higher specialization. Measuring cross-task similarity further reveals that linguistically related tasks share information throughout training, and do so more during the critical phase of learning than before or after. Our findings have implications for model interpretability, multi-task learning, and learning from limited data.

Precise relative navigation is a critical enabler for distributed satellites to achieve new mission objectives impossible for a monolithic spacecraft. Carrier phase differential GPS (CDGPS) with integer ambiguity resolution (IAR) is a promising means of achieving cm-level accuracy for high-precision Rendezvous, Proximity-Operations and Docking (RPOD), In-Space Servicing, Assembly and Manufacturing (ISAM) as well as satellite formation flying and swarming. However, IAR is sensitive to received GPS signal noise, especially under severe multi-path or high thermal noise. This paper proposes a sensor-fusion approach to achieve IAR under such conditions in two coupling stages. A loose coupling stage fuses through an Extended Kalman Filter the CDGPS measurements with on-board sensor measurements such as range from cross-links, and vision-based bearing angles. A second tight-coupling stage augments the cost function of the integer weighted least-squares minimization with a soft constraint function using noise-weighted observed-minus-computed residuals from these external sensor measurements. Integer acceptance tests are empirically modified to reflect added constraints. Partial IAR is applied to graduate integer fixing. These proposed techniques are packaged into flight-capable software, with ground truths simulated by the Stanford Space Rendezvous Laboratory's S3 library using state-of-the-art force modelling with relevant sources of errors, and validated in two scenarios: (1) a high multi-path scenario involving rendezvous and docking in low Earth orbit, and (2) a high thermal noise scenario relying only on GPS side-lobe signals during proximity operations in geostationary orbit. This study demonstrates successful IAR in both cases, using the proposed sensor-fusion approach, thus demonstrating potential for high-precision state estimation under adverse signal-to-noise conditions.

Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann-Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular kernels. For a class of kernel functions that includes the finite Hilbert transformation in 1D and the principal part of the Maxwell volume integral operator used for DDA in dimensions 2 and 3, we show that the method, which does not fit into known frameworks of projection methods, can nevertheless be considered as a finite section method for an infinite block Toeplitz matrix. The symbol of this matrix is given by a Fourier series that does not converge absolutely. We use Ewald's method to obtain an exponentially fast convergent series representation of this symbol and show that it is a bounded function, thereby allowing to describe the spectrum and the numerical range of the matrix. It turns out that this numerical range includes the numerical range of the integral operator, but that it is in some cases strictly larger. In these cases the discretization method does not provide a spectrally correct approximation, and while it is stable for a large range of the spectral parameter $\lambda$, there are values of $\lambda$ for which the singular integral equation is well posed, but the discretization method is unstable.

Online gradient descent (OGD) is well known to be doubly optimal under strong convexity or monotonicity assumptions: (1) in the single-agent setting, it achieves an optimal regret of $\Theta(\log T)$ for strongly convex cost functions; and (2) in the multi-agent setting of strongly monotone games, with each agent employing OGD, we obtain last-iterate convergence of the joint action to a unique Nash equilibrium at an optimal rate of $\Theta(\frac{1}{T})$. While these finite-time guarantees highlight its merits, OGD has the drawback that it requires knowing the strong convexity/monotonicity parameters. In this paper, we design a fully adaptive OGD algorithm, \textsf{AdaOGD}, that does not require a priori knowledge of these parameters. In the single-agent setting, our algorithm achieves $O(\log^2(T))$ regret under strong convexity, which is optimal up to a log factor. Further, if each agent employs \textsf{AdaOGD} in strongly monotone games, the joint action converges in a last-iterate sense to a unique Nash equilibrium at a rate of $O(\frac{\log^3 T}{T})$, again optimal up to log factors. We illustrate our algorithms in a learning version of the classical newsvendor problem, where due to lost sales, only (noisy) gradient feedback can be observed. Our results immediately yield the first feasible and near-optimal algorithm for both the single-retailer and multi-retailer settings. We also extend our results to the more general setting of exp-concave cost functions and games, using the online Newton step (ONS) algorithm.

One challenge in spoken language translation is that plenty of spoken content is long-form, but short units are necessary for obtaining high-quality translations. To address this mismatch, we adapt large language models (LLM) to split long ASR transcripts into segments that can be independently translated so as to maximize the overall translation quality. To combat the tendency of hallucination by LLMs, we incorporate finite-state constraints during decoding to eliminate invalid outputs. We discover that LLMs are adaptable to transcripts containing ASR errors through prompt-tuning or fine-tuning. In comparison to a state-of-the-art automatic punctuation baseline, our best LLM improves the average BLEU for English-German, English-Spanish, and English-Arabic TED talk translation in 9 test sets by 2.9 points, just by improving segmentation.

Implicit models such as Deep Equilibrium Models (DEQs) have garnered significant attention in the community for their ability to train infinite layer models with elegant solution-finding procedures and constant memory footprint. However, despite several attempts, these methods are heavily constrained by model inefficiency and optimization instability. Furthermore, fair benchmarking across relevant methods for vision tasks is missing. In this work, we revisit the line of implicit models and trace them back to the original weight-tied models. Surprisingly, we observe that weight-tied models are more effective, stable, as well as efficient on vision tasks, compared to the DEQ variants. Through the lens of these simple-yet-clean weight-tied models, we further study the fundamental limits in the model capacity of such models and propose the use of distinct sparse masks to improve the model capacity. Finally, for practitioners, we offer design guidelines regarding the depth, width, and sparsity selection for weight-tied models, and demonstrate the generalizability of our insights to other learning paradigms.

On the journey to enable robots to interact with the real world where humans, animals, and unpredictable elements are acting as independent agents; it is crucial for robots to have the capability to detect dynamic objects. In this paper, we argue that the detection of dynamic objects can be solved by computing the spatiotemporal normals of a point cloud. In our experiments, we demonstrate that this simple method can be used robustly for LiDAR and depth cameras with performances similar to the state of the art while offering a significantly simpler method.

Brain tumors are increasingly prevalent, characterized by the uncontrolled spread of aberrant tissues in the brain, with almost 700,000 new cases diagnosed globally each year. Magnetic Resonance Imaging (MRI) is commonly used for the diagnosis of brain tumors and accurate classification is a critical clinical procedure. In this study, we propose an efficient solution for classifying brain tumors from MRI images using custom transfer learning networks. While several researchers have employed various pre-trained architectures such as RESNET-50, ALEXNET, VGG-16, and VGG-19, these methods often suffer from high computational complexity. To address this issue, we present a custom and lightweight model using a Convolutional Neural Network-based pre-trained architecture with reduced complexity. Specifically, we employ the VGG-19 architecture with additional hidden layers, which reduces the complexity of the base architecture but improves computational efficiency. The objective is to achieve high classification accuracy using a novel approach. Finally, the result demonstrates a classification accuracy of 96.42%.

Graphs are important data representations for describing objects and their relationships, which appear in a wide diversity of real-world scenarios. As one of a critical problem in this area, graph generation considers learning the distributions of given graphs and generating more novel graphs. Owing to their wide range of applications, generative models for graphs, which have a rich history, however, are traditionally hand-crafted and only capable of modeling a few statistical properties of graphs. Recent advances in deep generative models for graph generation is an important step towards improving the fidelity of generated graphs and paves the way for new kinds of applications. This article provides an extensive overview of the literature in the field of deep generative models for graph generation. Firstly, the formal definition of deep generative models for the graph generation and the preliminary knowledge are provided. Secondly, taxonomies of deep generative models for both unconditional and conditional graph generation are proposed respectively; the existing works of each are compared and analyzed. After that, an overview of the evaluation metrics in this specific domain is provided. Finally, the applications that deep graph generation enables are summarized and five promising future research directions are highlighted.