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We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets -- movie ratings, hyperspectral images, and high-dimensional functions -- suggest that the proposed algorithms are more efficient and have better reconstruction ability than other candidates.

SoCs are now designed with their own AI accelerator segment to accommodate the ever-increasing demand of Deep Learning (DL) applications. With powerful MAC engines for matrix multiplications, these accelerators show high computing performance. However, because of limited memory resources (i.e., bandwidth and capacity), they fail to achieve optimum system performance during large batch training and inference. In this work, we propose a memory system with high on-chip capacity and bandwidth to shift the gear of AI accelerators from memory-bound to achieving system-level peak performance. We develop the memory system with DTCO-enabled customized SOT-MRAM as large on-chip memory through STCO and detailed characterization of the DL workloads. %We evaluate our workload-aware memory system on the CV and NLP benchmarks and observe significant PPA improvement compared to an SRAM-based in both inference and training modes. Our workload-aware memory system achieves 8X energy and 9X latency improvement on Computer Vision (CV) benchmarks in training and 8X energy and 4.5X latency improvement on Natural Language Processing (NLP) benchmarks in training while consuming only around 50% of SRAM area at iso-capacity.

Document-based Question-Answering (QA) tasks are crucial for precise information retrieval. While some existing work focus on evaluating large language model's performance on retrieving and answering questions from documents, assessing the LLMs' performance on QA types that require exact answer selection from predefined options and numerical extraction is yet to be fully assessed. In this paper, we specifically focus on this underexplored context and conduct empirical analysis of LLMs (GPT-4 and GPT 3.5) on question types, including single-choice, yes-no, multiple-choice, and number extraction questions from documents. We use the Cogtale dataset for evaluation, which provide human expert-tagged responses, offering a robust benchmark for precision and factual grounding. We found that LLMs, particularly GPT-4, can precisely answer many single-choice and yes-no questions given relevant context, demonstrating their efficacy in information retrieval tasks. However, their performance diminishes when confronted with multiple-choice and number extraction formats, lowering the overall performance of the model on this task, indicating that these models may not be reliable for the task. This limits the applications of LLMs on applications demanding precise information extraction from documents, such as meta-analysis tasks. However, these findings hinge on the assumption that the retrievers furnish pertinent context necessary for accurate responses, emphasizing the need for further research on the efficacy of retriever mechanisms in enhancing question-answering performance. Our work offers a framework for ongoing dataset evaluation, ensuring that LLM applications for information retrieval and document analysis continue to meet evolving standards.

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

Despite making significant progress in multi-modal tasks, current Multi-modal Large Language Models (MLLMs) encounter the significant challenge of hallucination, which may lead to harmful consequences. Therefore, evaluating MLLMs' hallucinations is becoming increasingly important in model improvement and practical application deployment. Previous works are limited in high evaluation costs (e.g., relying on humans or advanced LLMs) and insufficient evaluation dimensions (e.g., types of hallucination and task). In this paper, we propose an LLM-free multi-dimensional benchmark AMBER, which can be used to evaluate both generative task and discriminative task including object existence, object attribute and object relation hallucination. Based on AMBER, we design a low-cost and efficient evaluation pipeline. Additionally, we conduct a comprehensive evaluation and detailed analysis of mainstream MLLMs including GPT-4V(ision), and also give guideline suggestions for mitigating hallucinations. The data and code of AMBER are available at //github.com/junyangwang0410/AMBER.

We present a methodology for using unlabeled data to design semi supervised learning (SSL) methods that improve the prediction performance of supervised learning for regression tasks. The main idea is to design different mechanisms for integrating the unlabeled data, and include in each of them a mixing parameter $\alpha$, controlling the weight given to the unlabeled data. Focusing on Generalized Linear Models (GLM) and linear interpolators classes of models, we analyze the characteristics of different mixing mechanisms, and prove that in all cases, it is invariably beneficial to integrate the unlabeled data with some nonzero mixing ratio $\alpha>0$, in terms of predictive performance. Moreover, we provide a rigorous framework to estimate the best mixing ratio $\alpha^*$ where mixed SSL delivers the best predictive performance, while using the labeled and unlabeled data on hand. The effectiveness of our methodology in delivering substantial improvement compared to the standard supervised models, in a variety of settings, is demonstrated empirically through extensive simulation, in a manner that supports the theoretical analysis. We also demonstrate the applicability of our methodology (with some intuitive modifications) to improve more complex models, such as deep neural networks, in real-world regression tasks.

For years, SIMD/vector units have enhanced the capabilities of modern CPUs in High-Performance Computing (HPC) and mobile technology. Typical commercially-available SIMD units process up to 8 double-precision elements with one instruction. The optimal vector width and its impact on CPU throughput due to memory latency and bandwidth remain challenging research areas. This study examines the behavior of four computational kernels on a RISC-V core connected to a customizable vector unit, capable of operating up to 256 double precision elements per instruction. The four codes have been purposefully selected to represent non-dense workloads: SpMV, BFS, PageRank, FFT. The experimental setup allows us to measure their performance while varying the vector length, the memory latency, and bandwidth. Our results not only show that larger vector lengths allow for better tolerance of limitations in the memory subsystem but also offer hope to code developers beyond dense linear algebra.

We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of the numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. Starting from a simple 1D illustration, we demonstrate numerically that the resulting MLMC estimator deteriorates the estimation of high-frequency components of the discretized expectation field compared to a Monte Carlo (MC) estimator. By adapting mathematical tools initially developed for multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. This analysis provides a spectral decomposition of the variance into contributions associated with each scale component of the discretized field. We then propose improved MLMC estimators using a filtering mechanism similar to the smoothing process of multigrid methods. The filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. These improvements are quantified for the specific class of simulators considered in our spectral analysis. The resulting filtered MLMC (F-MLMC) estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the proposed F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

We present the new Orthogonal Polynomials Approximation Algorithm (OPAA), a parallelizable algorithm that solves two problems from a functional analytic approach: first, it finds a smooth functional estimate of a density function, whether it is normalized or not; second, the algorithm provides an estimate of the normalizing weight. In the context of Bayesian inference, OPAA provides an estimate of the posterior function as well as the normalizing weight, which is also known as the evidence. A core component of OPAA is a special transform of the square root of the joint distribution into a special functional space of our construct. Through this transform, the evidence is equated with the $L^2$ norm of the transformed function, squared. Hence, the evidence can be estimated by the sum of squares of the transform coefficients. The computations can be parallelized and completed in one pass. To compute the transform coefficients, OPAA proposes a new computational scheme leveraging Gauss--Hermite quadrature in higher dimensions. Not only does it avoid the potential high variance problem associated with random sampling methods, it also enables one to speed up the computation by parallelization, and significantly reduces the complexity by a vector decomposition.

Recently pre-trained language representation models such as BERT have shown great success when fine-tuned on downstream tasks including information retrieval (IR). However, pre-training objectives tailored for ad-hoc retrieval have not been well explored. In this paper, we propose Pre-training with Representative wOrds Prediction (PROP) for ad-hoc retrieval. PROP is inspired by the classical statistical language model for IR, specifically the query likelihood model, which assumes that the query is generated as the piece of text representative of the "ideal" document. Based on this idea, we construct the representative words prediction (ROP) task for pre-training. Given an input document, we sample a pair of word sets according to the document language model, where the set with higher likelihood is deemed as more representative of the document. We then pre-train the Transformer model to predict the pairwise preference between the two word sets, jointly with the Masked Language Model (MLM) objective. By further fine-tuning on a variety of representative downstream ad-hoc retrieval tasks, PROP achieves significant improvements over baselines without pre-training or with other pre-training methods. We also show that PROP can achieve exciting performance under both the zero- and low-resource IR settings. The code and pre-trained models are available at //github.com/Albert-Ma/PROP.