Kernel化张量机：一种结构保留的监督学习方法

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"Kernelized Support Tensor Machines (KSTM) 是一种新型的机器学习方法，它在监督张量学习的背景下，结合了张量分解理论和核方法，旨在保留数据的结构信息并挖掘张量数据中的非线性判别关系，以提升学习任务的性能。通过引入核化的张量分解技术，KSTM能够在核空间中近似张量数据，从而探索复杂的数据内部非线性关联。此外，还设计了双结构保持核来学习张量数据之间的非线性边界。通过联合优化，KSTM中得到的核在判别分析中表现出更好的泛化能力。实验结果显示，在实际的神经影像数据集上，KSTM优于现有的先进技术。" 在机器学习领域，张量（Tensor）作为一种多维数组，广泛用于表示复杂的数据结构，例如图像、视频和高维特征。传统的机器学习模型可能无法有效地处理这些数据的结构特性，因此，KSTM的提出是为了克服这一挑战。张量分解是处理高维数据的一种有效手段，它将高阶张量分解为一组低阶张量的组合，从而揭示隐藏的模式和结构。核方法是机器学习中的一个重要工具，它允许我们将数据映射到一个高维的特征空间，在这个空间中，原本非线性可分的问题变得线性可分。KSTM引入了核化的张量分解，即将原始的张量数据通过核函数映射到高维空间，使得在原始空间中难以捕捉的非线性关系变得可以被探索。 KSTM的关键创新在于设计了双结构保持核，这允许模型不仅在单个张量的维度之间学习非线性边界，而且在张量与张量之间建立联系，进一步增强了模型的分类或预测能力。通过这种方式，KSTM能够更好地捕获数据的复杂结构，提高模型的泛化性能。实验部分，KSTM在实际的神经影像数据集上进行了验证，这些数据集通常包含大量的多模态信息，如MRI和fMRI扫描，对于理解大脑功能和疾病诊断具有重要意义。KSTM在这些任务上的优越表现证明了其在处理复杂结构的张量数据时的有效性和优势。 Kernelized Support Tensor Machines 是一种强大的工具，它利用张量分解和核方法的结合，提高了对复杂非线性结构数据的建模能力，尤其适用于那些需要处理高维结构数据的领域，如医学影像分析、社交网络分析等。

Kernelized Support Tensor Machines

Lifang He

Chun-Ta Lu

Guixiang Ma

Shen Wang

Linlin Shen

Philip S. Yu

1 3

Ann B. Ragin

Abstract

In the context of supervised tensor learning, pre-

serving the structural information and exploit-

ing the discriminative nonlinear relationships of

tensor data are crucial for improving the perfor-

mance of learning tasks. Based on tensor fac-

torization theory and kernel methods, we pro-

pose a novel Kernelized Support Tensor Ma-

chine (KSTM) which integrates kernelized ten-

sor factorization with maximum-margin crite-

rion. Speciﬁcally, the kernelized factorization

technique is introduced to approximate the ten-

sor data in kernel space such that the complex

nonlinear relationships within tensor data can be

explored. Further, dual structural preserving ker-

nels are devised to learn the nonlinear bound-

ary between tensor data. As a result of joint

optimization, the kernels obtained in KSTM ex-

hibit better generalization power to discrimina-

tive analysis. The experimental results on real-

world neuroimaging datasets show the superior-

ity of KSTM over the state-of-the-art techniques.

1. Introduction

In many real-world applications, data samples intrinsically

come in the form of two-dimensional (matrices) or multi-

dimensional arrays (tensors). In medical neuroimaging, for

instance, a functional magnetic resonance imaging (fMRI)

sample is naturally a third-order tensor consisting of 3D

voxels. There has been extensive work in supervised tensor

learning (STL) recently. For example, (Tao et al., 2007)

proposed a STL framework that extends the standard lin-

ear support vector machine (SVM) learning framework to

tensor patterns by constructing multilinear models. Under

Department of Computer Science, University of Illinois at

Chicago, Chicago, IL, USA

Institute for Computer Vision, Shen-

zhen University, Shenzhen, China

Institute for Data Science,

Tsinghua University, Beijing, China

Department of Radiology,

Northwestern University, Chicago, IL, USA. Correspondence to:

Linlin Shen <llshen@szu.edu.cn>.

Proceedings of the 34

International Conference on Machine

by the author(s).

this learning framework, several tensor-based linear mod-

els (Zhou et al., 2013; Hao et al., 2013) have been devel-

oped. These methods assume, explicitly or implicitly, that

data are linearly separable in the input space. However,

in practice this assumption is often violated and the linear

decision boundaries do not adequately separate the classes.

In order to apply kernel methods for tensor data, several

works (Signoretto et al., 2011; 2012; Zhao et al., 2013a)

have been presented to convert the input tensors into vec-

tors (or matrices), which are then used to construct kernels.

This kind of conversion, though, will destroy the structure

information of the tensor data. Moreover, the dimension-

ality of the resulting vector typically becomes very high,

which leads to the curse of dimensionality and small sam-

ple size problems (Lu et al., 2008; Yan et al., 2007).

Recently, (Hao et al., 2013; He et al., 2014; Ma et al.,

2016) employed CANDECOMP/PARAFAC (CP) factor-

ization (Kolda & Bader, 2009) on the input tensor to foster

the use of kernel methods for STL problems. However, as

indicated in (Rubinov et al., 2009; Luo et al., 2011), the

underlying structure of real data is often nonlinear. Al-

though the CP factorization provides a good approximation

to the original tensor data, it only concerned with multilin-

ear formulas. Thus, it is difﬁcult to model complex non-

linear relationships within the tensor data. Most recently,

(He et al., 2017) extended CP factorization to the nonlinear

case through the exploitation of the representer theorem,

and then used kernelized CP (KCP) factorization to facil-

itate kernel learning. To the best of our knowledge, there

is no existing work that tackles factorization and prediction

as a joint optimization problem over the kernel methods.

This paper focuses on developing kernelized tensor factor-

ization with kernel maximum-margin constraint, referred

as Kernelized Support Tensor Machine (KSTM). KSTM

includes two principal ingredients. First, inspired by (Sig-

noretto et al., 2013), we introduce a general formulation

of kernelized tensor factorization in the tensor product re-

producing kernel Hilbert space, namely kernelized Tucker

model, which provides a new perspective on understand-

ing the KCP process. Second, we apply kernel trick to

embed the compact representations extracted by KCP into

the dual structure-preserving kernels (He et al., 2014) in

conjunction with a maximum-margin method to solve the

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