MathMistake Checker: A Comprehensive Demonstration for Step-by-Step Math Problem Mistake Finding by Prompt-Guided LLMs

We propose a novel system, MathMistake Checker, designed to automate step-by-step mistake finding in mathematical problems with lengthy answers through a two-stage process. The system aims to simplify grading, increase efficiency, and enhance learning experiences from a pedagogical perspective. It integrates advanced technologies, including computer vision and the chain-of-thought capabilities of the latest large language models (LLMs). Our system supports open-ended grading without reference answers and promotes personalized learning by providing targeted feedback. We demonstrate its effectiveness across various types of math problems, such as calculation and word problems.

Worth Their Weight: Randomized and Regularized Block Kaczmarz Algorithms without Preprocessing

Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares, the randomized block Kaczmarz method (RBK) is an appealing example of such an algorithm, but its convergence is only understood under sampling distributions that require potentially prohibitively expensive preprocessing steps. To address this limitation, we analyze RBK when the data is sampled uniformly, showing that its iterates converge in a Monte Carlo sense to a $\textit{weighted}$ least-squares solution. Unfortunately, for general problems the condition number of the weight matrix and the variance of the iterates can become arbitrarily large. We resolve these issues by incorporating regularization into the RBK iterations. Numerical experiments, including examples arising from natural gradient optimization, suggest that the regularized algorithm, ReBlocK, outperforms minibatch stochastic gradient descent for realistic problems that exhibit fast singular value decay.

Constructing Cell-type Taxonomy by Optimal Transport with Relaxed Marginal Constraints

The rapid emergence of single-cell data has facilitated the study of many different biological conditions at the cellular level. Cluster analysis has been widely applied to identify cell types, capturing the essential patterns of the original data in a much more concise form. One challenge in the cluster analysis of cells is matching clusters extracted from datasets of different origins or conditions. Many existing algorithms cannot recognize new cell types present in only one of the two samples when establishing a correspondence between clusters obtained from two samples. Additionally, when there are more than two samples, it is advantageous to align clusters across all samples simultaneously rather than performing pairwise alignment. Our approach aims to construct a taxonomy for cell clusters across all samples to better annotate these clusters and effectively extract features for downstream analysis. A new system for constructing cell-type taxonomy has been developed by combining the technique of Optimal Transport with Relaxed Marginal Constraints (OT-RMC) and the simultaneous alignment of clusters across multiple samples. OT-RMC allows us to address challenges that arise when the proportions of clusters vary substantially between samples or when some clusters do not appear in all the samples. Experiments on more than twenty datasets demonstrate that the taxonomy constructed by this new system can yield highly accurate annotation of cell types. Additionally, sample-level features extracted based on the taxonomy result in accurate classification of samples.

RAG4ITOps: A Supervised Fine-Tunable and Comprehensive RAG Framework for IT Operations and Maintenance

With the ever-increasing demands on Question Answering (QA) systems for IT operations and maintenance, an efficient and supervised fine-tunable framework is necessary to ensure the data security, private deployment and continuous upgrading. Although Large Language Models (LLMs) have notably improved the open-domain QA's performance, how to efficiently handle enterprise-exclusive corpora and build domain-specific QA systems are still less-studied for industrial applications. In this paper, we propose a general and comprehensive framework based on Retrieval Augmented Generation (RAG) and facilitate the whole business process of establishing QA systems for IT operations and maintenance. In accordance with the prevailing RAG method, our proposed framework, named with RAG4ITOps, composes of two major stages: (1) Models Fine-tuning \& Data Vectorization, and (2) Online QA System Process. At the Stage 1, we leverage a contrastive learning method with two negative sampling strategies to fine-tune the embedding model, and design the instruction templates to fine-tune the LLM with a Retrieval Augmented Fine-Tuning method. At the Stage 2, an efficient process of QA system is built for serving. We collect enterprise-exclusive corpora from the domain of cloud computing, and the extensive experiments show that our method achieves superior results than counterparts on two kinds of QA tasks. Our experiment also provide a case for applying the RAG4ITOps to real-world enterprise-level applications.

Costal Cartilage Segmentation with Topology Guided Deformable Mamba: Method and Benchmark

Costal cartilage segmentation is crucial to various medical applications, necessitating precise and reliable techniques due to its complex anatomy and the importance of accurate diagnosis and surgical planning. We propose a novel deep learning-based approach called topology-guided deformable Mamba (TGDM) for costal cartilage segmentation. The TGDM is tailored to capture the intricate long-range costal cartilage relationships. Our method leverages a deformable model that integrates topological priors to enhance the adaptability and accuracy of the segmentation process. Furthermore, we developed a comprehensive benchmark that contains 165 cases for costal cartilage segmentation. This benchmark sets a new standard for evaluating costal cartilage segmentation techniques and provides a valuable resource for future research. Extensive experiments conducted on both in-domain benchmarks and out-of domain test sets demonstrate the superiority of our approach over existing methods, showing significant improvements in segmentation precision and robustness.

Canonical Variates in Wasserstein Metric Space

In this paper, we address the classification of instances each characterized not by a singular point, but by a distribution on a vector space. We employ the Wasserstein metric to measure distances between distributions, which are then used by distance-based classification algorithms such as k-nearest neighbors, k-means, and pseudo-mixture modeling. Central to our investigation is dimension reduction within the Wasserstein metric space to enhance classification accuracy. We introduce a novel approach grounded in the principle of maximizing Fisher's ratio, defined as the quotient of between-class variation to within-class variation. The directions in which this ratio is maximized are termed discriminant coordinates or canonical variates axes. In practice, we define both between-class and within-class variations as the average squared distances between pairs of instances, with the pairs either belonging to the same class or to different classes. This ratio optimization is achieved through an iterative algorithm, which alternates between optimal transport and maximization steps within the vector space. We conduct empirical studies to assess the algorithm's convergence and, through experimental validation, demonstrate that our dimension reduction technique substantially enhances classification performance. Moreover, our method outperforms well-established algorithms that operate on vector representations derived from distributional data. It also exhibits robustness against variations in the distributional representations of data clouds.

The ESPRIT algorithm under high noise: Optimal error scaling and noisy super-resolution

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size $\mathcal{O}(1)$, but these analyses only show that the error $\epsilon$ decays at a slow rate $\epsilon=\mathcal{\tilde{O}}(n^{-1/2})$ with respect to the cutoff frequency $n$. In this work, we prove that under certain assumptions of bias and high noise, the ESPRIT algorithm can attain a significantly improved error scaling $\epsilon = \mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest.

CSST Strong Lensing Preparation: a Framework for Detecting Strong Lenses in the Multi-color Imaging Survey by the China Survey Space Telescope (CSST)

Strong gravitational lensing is a powerful tool for investigating dark matter and dark energy properties. With the advent of large-scale sky surveys, we can discover strong lensing systems on an unprecedented scale, which requires efficient tools to extract them from billions of astronomical objects. The existing mainstream lens-finding tools are based on machine learning algorithms and applied to cut-out-centered galaxies. However, according to the design and survey strategy of optical surveys by CSST, preparing cutouts with multiple bands requires considerable efforts. To overcome these challenges, we have developed a framework based on a hierarchical visual Transformer with a sliding window technique to search for strong lensing systems within entire images. Moreover, given that multi-color images of strong lensing systems can provide insights into their physical characteristics, our framework is specifically crafted to identify strong lensing systems in images with any number of channels. As evaluated using CSST mock data based on an Semi-Analytic Model named CosmoDC2, our framework achieves precision and recall rates of 0.98 and 0.90, respectively. To evaluate the effectiveness of our method in real observations, we have applied it to a subset of images from the DESI Legacy Imaging Surveys and media images from Euclid Early Release Observations. 61 new strong lensing system candidates are discovered by our method. However, we also identified false positives arising primarily from the simplified galaxy morphology assumptions within the simulation. This underscores the practical limitations of our approach while simultaneously highlighting potential avenues for future improvements.

Energy Allocation for Multi-User Cooperative Molecular Communication Systems in the Internet of Bio-Nano Things

Cooperative molecular communication (MC) is a promising technology for facilitating communication between nanomachines in the Internet of Bio-Nano Things (IoBNT) field. However, the performance of IoBNT is limited by the availability of energy for cooperative MC. This paper presents a novel transmitter design scheme that utilizes molecule movement between reservoirs, creating concentration differences through the consumption of free energy, and encoding information on molecule types. The performance of the transmitter is primarily influenced by energy costs, which directly impact the overall IoBNT system performance. To address this, the paper focuses on optimizing energy allocation in cooperative MC for enhanced transmitter performance. Theoretical analysis is conducted for two transmitters. For scenarios with more than two users, a genetic algorithm is employed in the energy allocation to minimize the total bit error rate (BER). Finally, numerical results show the effectiveness of the proposed energy allocation strategies in the considered cooperative MC system.

A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions

Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.