图像分类新方法：局部张量奇异值分解
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本文档"Image classification using local tensor singular value decompositions"探讨了在图像分类领域中应用一种新颖的算法，即局部张量奇异值分解（tSVD）进行精确而成本效益高的分类。传统的线性分类器和神经网络方法虽然在图像分类方面取得了显著成果，但往往伴随着较高的存储成本和复杂的计算过程。tSVD作为一种非线性方法，其核心在于利用张量代数中的优化特性，能够对图像数据进行有效压缩并保持高精度。
作者们提出了一种创新的图像分类策略，它通过截断局部张量的奇异值分解来实现。这种方法在保持结果准确性的同时，显著降低了存储需求，这对于大数据时代尤为重要。局部tSVD的优势在于它能够适应图像数据的局部特征，并通过分解捕捉到图像的不同模式和结构。通过这种方式，它能准确地确定一张图片属于哪个类别，且避免了传统算法可能遇到的过拟合问题。
论文进一步阐述了tSVD在图像处理中的具体操作步骤，包括如何提取图像的局部特征、如何构建和分解张量，以及如何利用这些信息进行分类决策。与传统的基于全局特征或深度学习的方法相比，局部tSVD具有更高的灵活性和效率，因为它能更好地处理图像中的局部变化。
此外，文中还可能探讨了实验部分，展示了在各种标准图像数据集上的性能比较，包括CIFAR-10、MNIST等，证明了该方法在实际应用中的有效性。为了增强可扩展性和适应性，论文可能也讨论了如何处理不同尺度和维度的图像，以及如何在实时性和计算效率之间找到平衡。
这篇论文对于那些寻求在图像分类任务中实现高效、准确且资源友好的解决方案的研究者和技术人员来说，提供了一个有价值的新视角。它不仅介绍了tSVD在图像分类中的独特优势，还可能提出了未来研究的方向，如结合其他机器学习技术的集成方法或者针对特定领域的优化策略。
Image classification using local tensor singular
value decompositions
Elizabeth Newman
Department of Mathematics
Tufts University
Medford, Massachusetts 02155
Email: e.newman@tufts.edu
Misha Kilmer
Department of Mathematics
Tufts University
Medford, Massachusetts 02155
Email: misha.kilmer@tufts.edu
Lior Horesh
IBM TJ Watson Research Center
1101 Kitchawan Road
Yorktown Heights, NY
Email: lhoresh@us.ibm.com
Abstract—From linear classifiers to neural networks, image
classification has been a widely explored topic in mathematics,
and many algorithms have proven to be effective classifiers.
However, the most accurate classifiers typically have significantly
high storage costs, or require complicated procedures that may
be computationally expensive. We present a novel (nonlinear)
classification approach using truncation of local tensor singular
value decompositions (tSVD) that robustly offers accurate results,
while maintaining manageable storage costs. Our approach takes
advantage of the optimality of the representation under the tensor
algebra described to determine to which class an image belongs.
We extend our approach to a method that can determine specific
pairwise match scores, which could be useful in, for example,
object recognition problems where pose/position are different. We
demonstrate the promise of our new techniques on the MNIST
data set.
I. INTRODUCTION
Image classification is a well-explored problem in which an
image is identified as belonging to one of a known number
of classes. Researchers seek to extract particular features
from which to determine patterns comprising an image. Algo-
rithms to determine these essential features include statistical
methods such as centroid-based clustering, connectivity/graph-
based clustering, distribution-based clustering, and density-
based clustering [13], [14], [15], as well as learning algorithms
(linear discriminant analysis, support vector machines, neural
networks) [5].
Our approach differs significantly from techniques in the
literature in that it uses local tensor singular value decompo-
sitions (tSVD) to form the feature space of an image. Tensor
approaches are gaining increasing popularity for tasks such as
image recognition and dictionary learning and reconstruction
[3], [9], [7], [10]. These are favored over matrix-vector-based
approaches as it has been demonstrated that a tensor-based
approach enables retention of the original image structural
correlations that are lost by image vectorization. Tensor ap-
proaches for image classification appear to be in their infancy,
although some approaches based on the tensor HOSVD [11]
have been explored in the literature [6].
Here, we are motivated by the work in [3] which em-
ploys optimal low tubal-rank tensor factorizations through
use of the t-product [1] and by the work in [2] describing
tensor orthogonal projections. We present a new approach
for classification based on the tensor SVD from [1], called
the tSVD, which is elegant for its straightforward mathe-
matical interpretation and implementation, and which has the
advantage that it can be easily parallelized for great com-
putational advantage. State-of-the-art matrix decompositions
are asymptotically challenged in dealing with the demand to
process ever-growing datasets of larger and more complex
objects [16], so the importance of this dimension of this study
cannot be overstated. Our method is in direct contrast to deep
neural network based approaches which require many layers
of complexity and for which theoretical interpretation is not
readily available [17]. Our approach is also different from
the tensor approach in [6] because truncating the tSVD has
optimality properties that truncating the HOSVD does not
enjoy. We conclude this study with a demonstration on the
MNIST [4] dataset.
A. Notation and Preliminaries
In this paper, a tensor is a real-valued
1
third-order tensor,
or three-dimensional array of data, denoted by a capital script
letter. As depicted in Figure 1, A is an ×m×n tensor. Frontal
slices A
(k)
for k =1,...,n are × m matrices. Lateral slices
A
j
for j =1,...,m are × n matrices oriented along the
third dimension. Tubes a
ij
for i =1,..., and j =1,...,m
are n × 1 column vectors oriented along the third dimension
[2].
(a) Tensor A. (b) Frontal
slices A
(k)
.
(c) Lateral
slices
A
j
.
(d) Tubes a
ij
.
Fig. 1. Representations of third-order tensors.
To multiply a pair of tensors, we need to understand the
t-product, which requires the following tensor reshaping ma-
chinery. Given A∈R
×m×n
, the unfold function reshapes
1
We assume real-valued tensors because we are working with real-valued
image data. However, the subsequent notation and definitions can be extended
to complex-valued tensors [8].
2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
978-1-5386-1251-4/17/$31.00 ©2017 IEEE
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