arXiv subjects

Chuanfu Xiao

Publications and source records attributed to Chuanfu Xiao.

10 recordsLinked to original sources

arXiv2025

A Novel Block-Alternating Iterative Algorithm for Retrieving Top-$k$ Elements from Factorized Tensors

Tensors, especially higher-order tensors, are typically represented in low-rank formats to preserve the main information of the high-dimensional data while saving memory space. In practice, only a small fraction elements in high-dimensional data are of interest, such as the $k$ largest or smallest elements. Thus, retrieving the $k$ largest/smallest elements from a low-rank tensor is a fundamental and important task in a wide variety of applications. In this paper, we first model the top-$k$ elements retrieval problem to a continuous constrained optimization problem. To address the equivalent optimization problem, we develop a block-alternating iterative algorithm that decomposes the original problem into a sequence of small-scale subproblems. Leveraging the separable summation structure of the objective function, a heuristic algorithm is proposed to solve these subproblems in an alternating manner. Numerical experiments with tensors from synthetic and real-world applications demonstrate that the proposed algorithm outperforms existing methods in terms of accuracy and stability.

math.NA↗

arXiv2025

HaTT: Hadamard avoiding TT recompression

The Hadamard product of tensor train (TT) tensors is a fundamental nonlinear operation in scientific computing and data analysis. However, due to its tendency to significantly increase TT ranks, the Hadamard product poses a major computational challenge in TT tensor-based algorithms. To address this, it is crucial to develop recompression algorithms that mitigate the effects of this rank increase. Existing recompression algorithms require an explicit representation of the Hadamard product, resulting in high computational and storage costs. In this work, we propose a Hadamard avoiding TT recompression (HaTT) algorithm, which reduces both computational complexity and storage requirements. By leveraging the structure of the Hadamard product in TT tensors and exploiting its Hadamard product-free property, the HaTT algorithm achieves significantly lower complexity compared to existing TT recompression methods. This is confirmed through both complexity analysis and numerical experiments. Furthermore, the HaTT algorithm is applied to solve the Allen--Cahn equation, achieving substantial speedup over existing TT recompression algorithms without sacrificing accuracy.

math.NA↗

arXiv2025

AnTKV: Anchor Token-Aware Sub-Bit Vector Quantization for KV Cache in Large Language Models

Quantization has emerged as an effective and lightweight solution to reduce the memory footprint of the KV cache in Large Language Models. Nevertheless, minimizing the accuracy degradation caused by ultra-low-bit KV cache quantization remains a significant challenge. While scalar quantization is constrained by 1-bit bound, vector quantization exploits intra-vector correlations and enables sub-bit regimes, making it more suitable for ultra-low-bit quantization. To further mitigate quantization-induced degradation, we reveal that the degradation is highly uneven across tokens in attention quality. To investigate this unevenness, we introduce anchor score to measure each token's sensitivity to quantization. Our analysis and experiments show that preserving a small subset (1\%) of tokens with the highest Anchor Score significantly mitigates accuracy loss under aggressive quantization. We propose AnTKV, a dual-stage framework that leverages anchor token-aware vector quantization to compress the KV cache. It combines offline token-aware centroids learning and online anchor token selection to balance compression and accuracy. To enable efficient deployment, we design an online anchor token selection kernel compatible with FlashAttention. It allows LLaMA3-8B to scale to 840K tokens on a single 80GB A100, while delivering up to $3.5\times$ higher decoding throughput over the FP16 baseline. Experiments demonstrate that AnTKV matches or surpasses prior methods at 4-bit, and significantly reduce perplexity under ultra-low-bit quantization, achieving 6.32 at 1-bit on Mistral-7B, compared to 7.25 for CQ and 15.36 for KVQuant.

cs.CL↗

arXiv2025

Provable Low-Rank Tensor-Train Approximations in the Inverse of Large-Scale Structured Matrices

This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned with is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this paper, we provide a computationally verifiable sufficient condition such that the inverse matrix can be well approximated in a low-rank TT format. It not only answers what kind of structured matrix whose inverse has the low-rank TT representation but also motivates us to develop an efficient TT-based method to compute the inverse matrix. Furthermore, we prove that the inverse matrix indeed has the low-rank tensor format for a class of large-scale structured matrices induced by differential operators involved in several PDEs, such as the Poisson, Boltzmann, and Fokker-Planck equations. Thus, the proposed algorithm is suitable for solving these PDEs with massive degrees of freedom. Numerical results on the Poisson, Boltzmann, and Fokker-Planck equations validate the correctness of our theory and the advantages of our methodology.

math.NA↗

arXiv2024

APTT: An accuracy-preserved tensor-train method for the Boltzmann-BGK equation

Solving the Boltzmann-BGK equation with traditional numerical methods suffers from high computational and memory costs due to the curse of dimensionality. In this paper, we propose a novel accuracy-preserved tensor-train (APTT) method to efficiently solve the Boltzmann-BGK equation. A second-order finite difference scheme is applied to discretize the Boltzmann-BGK equation, resulting in a tensor algebraic system at each time step. Based on the low-rank TT representation, the tensor algebraic system is then approximated as a TT-based low-rank system, which is efficiently solved using the TT-modified alternating least-squares (TT-MALS) solver. Thanks to the low-rank TT representation, the APTT method can significantly reduce the computational and memory costs compared to traditional numerical methods. Theoretical analysis demonstrates that the APTT method maintains the same convergence rate as that of the finite difference scheme. The convergence rate and efficiency of the APTT method are validated by several benchmark test cases.

math.NA↗

arXiv2024

HOSCF: Efficient decoupling algorithms for finding the best rank-one approximation of higher-order tensors

Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, it is necessary to develop decoupling algorithms for computing the best rank-one approximation of higher-order tensors at large scales. In this paper, we first build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration frequently used in computational chemistry. The convergence theory of the HOSCF algorithm and an estimation of the convergence speed are further presented. In addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF. Numerical experiments show that the proposed algorithms can efficiently converge to the best rank-one approximation of both synthetic and real-world tensors and can scale with high parallel scalability on a modern parallel computer.

math.NA↗

arXiv2022

Tensor-Based Sketching Method for the Low-Rank Approximation of Data Streams

Low-rank approximation in data streams is a fundamental and significant task in computing science, machine learning and statistics. Multiple streaming algorithms have emerged over years and most of them are inspired by randomized algorithms, more specifically, sketching methods. However, many algorithms are not able to leverage information of data streams and consequently suffer from low accuracy. Existing data-driven methods improve accuracy but the training cost is expensive in practice. In this paper, from a subspace perspective, we propose a tensor-based sketching method for low-rank approximation of data streams. The proposed algorithm fully exploits the structure of data streams and obtains quasi-optimal sketching matrices by performing tensor decomposition on training data. A series of experiments are carried out and show that the proposed tensor-based method can be more accurate and much faster than the previous work.

cs.DS↗

arXiv2021

A rank-adaptive higher-order orthogonal iteration algorithm for truncated Tucker decomposition

We propose a novel rank-adaptive higher-order orthogonal iteration (HOOI) algorithm to compute the truncated Tucker decomposition of higher-order tensors with a given error tolerance, and prove that the method is locally optimal and monotonically convergent. A series of numerical experiments related to both synthetic and real-world tensors are carried out to show that the proposed rank-adaptive HOOI algorithm is advantageous in terms of both accuracy and efficiency. Some further analysis on the HOOI algorithm and the classical alternating least squares method are presented to further understand why rank adaptivity can be introduced into the HOOI algorithm and how it works.

math.NA↗

arXiv2021

Efficient Alternating Least Squares Algorithms for Low Multilinear Rank Approximation of Tensors

The low multilinear rank approximation, also known as the truncated Tucker decomposition, has been extensively utilized in many applications that involve higher-order tensors. Popular methods for low multilinear rank approximation usually rely directly on matrix SVD, therefore often suffer from the notorious intermediate data explosion issue and are not easy to parallelize, especially when the input tensor is large. In this paper, we propose a new class of truncated HOSVD algorithms based on alternating least squares (ALS) for efficiently computing the low multilinear rank approximation of tensors. The proposed ALS-based approaches are able to eliminate the redundant computations of the singular vectors of intermediate matrices and are therefore free of data explosion. Also, the new methods are more flexible with adjustable convergence tolerance and are intrinsically parallelizable on high-performance computers. Theoretical analysis reveals that the ALS iteration in the proposed algorithms is q-linear convergent with a relatively wide convergence region. Numerical experiments with large-scale tensors from both synthetic and real-world applications demonstrate that ALS-based methods can substantially reduce the total cost of the original ones and are highly scalable for parallel computing.

math.NA↗

arXiv2020

a-Tucker: Input-Adaptive and Matricization-Free Tucker Decomposition for Dense Tensors on CPUs and GPUs

Tucker decomposition is one of the most popular models for analyzing and compressing large-scale tensorial data. Existing Tucker decomposition algorithms usually rely on a single solver to compute the factor matrices and core tensor, and are not flexible enough to adapt with the diversities of the input data and the hardware. Moreover, to exploit highly efficient GEMM kernels, most Tucker decomposition implementations make use of explicit matricizations, which could introduce extra costs in terms of data conversion and memory usage. In this paper, we present a-Tucker, a new framework for input-adaptive and matricization-free Tucker decomposition of dense tensors. A mode-wise flexible Tucker decomposition algorithm is proposed to enable the switch of different solvers for the factor matrices and core tensor, and a machine-learning adaptive solver selector is applied to automatically cope with the variations of both the input data and the hardware. To further improve the performance and enhance the memory efficiency, we implement a-Tucker in a fully matricization-free manner without any conversion between tensors and matrices. Experiments with a variety of synthetic and real-world tensors show that a-Tucker can substantially outperform existing works on both CPUs and GPUs.

cs.DC↗