强化结构保持非负矩阵分解：解决高维数据降维与噪声问题
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非负矩阵分解（Nonnegative Matrix Factorization, NMF）作为一种线性降维方法，在机器学习和数据挖掘等领域中表现出广泛的应用。然而，它存在两个关键局限性。首先，NMF局限于欧氏空间中的语义分解，无法揭示高维数据分布的内在几何结构。这限制了其在处理复杂数据集时的表现，特别是在需要保留数据的原始结构和模式的情况下。在许多实际问题中，如图像识别或物体分析，这种对数据几何特征的忽视可能导致降维后的信息丢失。
其次，NMF对噪声数据敏感，尤其是在真实世界的数据集中，噪声往往难以避免。这些噪声可能来源于测量误差、异常值或数据采样过程中的干扰。对于噪声数据的处理不力，可能导致NMF的结果被误导，降低模型的稳健性和准确性。
为了克服这些问题，研究者们提出了Robust Structure Preserving Nonnegative Matrix Factorization（鲁棒结构保持非负矩阵分解）。这种方法旨在通过增强NMF对数据几何结构的保真度，并引入噪声抗性机制，提高模型的稳健性。具体来说，该方法可能包括以下几个关键步骤：
1. **结构保真性**：通过引入约束或者优化策略，使得分解后的因子矩阵能够更好地近似原数据矩阵的奇异值分解，从而更好地捕捉数据的低维几何特征。
2. **噪声抑制**：可能采用诸如迭代硬阈值（Iterative Hard Thresholding）或者稀疏编码等技术，去除或减弱噪声对分解结果的影响。这有助于减少噪声对数据表示的扭曲。
3. **迭代算法**：优化算法可能是交替最小化、块主成分分析（Block Principal Component Analysis, BPCA）或其他非负优化方法的改进版本，以确保在每个迭代步骤中都保持非负性和结构的稳定性。
4. **评估和选择**：可能引入新的评估指标，比如鲁棒性分数或者结构相似度，来衡量降维后的数据是否保留了原始数据的结构，并据此调整模型参数。
5. **应用领域**：尽管本文提到的Robust Structure Preserving NMF主要针对的是机器学习和数据挖掘，但其方法论可以广泛应用于诸如人脸识别（FACE RECOGNITION）、对象识别、图像分割和物体部分识别（PARTS）等领域，以提升这些应用的性能和鲁棒性。
总结来说，Robust Structure Preserving Nonnegative Matrix Factorization是一种重要的改进，它在保持数据结构的同时提高了对噪声的抵抗能力，为解决高维数据中的复杂问题提供了有效的解决方案。通过结合先进的优化策略和鲁棒性处理，NMF得以在实际应用中发挥更大的潜力。
Mathematical Problems in Engineering 3
feature samples manifold structure are also considered.
Although the graph-based NMF algorithm incorporates the
prior knowledge of manifold structure into NMF framework,
overemphasizing local structure preservation sometimes will
degrade its performance. A structure preserving nonnega-
tive matrix factorization (SPNMF) is proposed in [10], in
which the intrinsic manifold is approximated with local
and distant graphs. Experiment results demonstrate SPNMF
outperforms GNMF and its variants.
All the abovementioned NMF variants are based on the
premisethattheinputdataarenoise-free;itisimpractical
to real-world data samples which are oen corrupted by
noise. For boosting NMF robustness to outliers,
2,1
-norm
based NMF reconstruction function is applied in [41, 42]. In
[43],
2,1
-norm and
1
-norm are used to formulate the graph
embedding and data reconstruction function, which endows
the algorithm with robustness to unreliable graphs and noisy
labels.
Among the existing methods, SPNMF is closely relevant
to our method and performs better on some specic datasets.
ere are still some dierences between our proposed algo-
rithm and SPNMF. Firstly, the Laplacian matrix of distant
repulsion graph from SPNMF needs to be updated in each
iteration, which is time consuming. Additionally, to learn a
better parts-based representation, the basis vector of SPNMF
is required to be as orthogonal as possible. Finally, SPNMF is
sensitive to noisy data points.
3. The Proposed Method
Our goal is to perform matrix factorization on the hidden
semanticsubspace.Tothisend,weimposethelocalanity
and distant repulsion structure preservation on the new
robust NMF framework. In this section, we will rst explain
the structure preservation and robust data reconstruction
terms of RSPNMF, followed by its optimization algorithm.
e convergence of the proposed algorithm is proved nally.
3.1. Problem Formulation. Despite the diverse motivations of
various dimensionality reduction algorithms, most of them
can be explained within a graph embedding framework.
Let ={,}be an undirected weighted graph, where
vertex set corresponds to a dataset and ∈R
𝑁×𝑁
+
is
an anity matrix whose elements measure the similarity of
each pair of vertices. In graph embedding algorithm, graph
characterizes the prior knowledge of geometric structure
from data distribution.
If we use the
1
-nearest neighboring graph to character-
ize the local structure of data distribution, the weight matrix
=[
𝑖𝑗
]
𝑁×𝑁
canbedenedasfollows:
𝑖𝑗
=
−‖𝑥
𝑖
−𝑥
𝑗
‖
2
/𝜎
2
, if
𝑖
∈
𝑘
1
𝑗
or
𝑗
∈
𝑘
1
𝑖
,
0, otherwise,
(4)
where is the bandwidth parameter and
𝑘
1
(
𝑖
)denotes the
set of
1
nearest neighbors of
𝑖
.
Local invariance assumption which encourages neigh-
boring data pairs in the original space to be still close in the
low-dimensional embedding subspace can be formulated as
min
𝑉
1
2
𝑁
𝑖,𝑗=1
V
𝑖
−V
𝑗
2
𝑖𝑗
,
(5)
where V
𝑖
is the corresponding low-dimensional representa-
tion of
𝑖
.
By dening the diagonal matrix and Laplacian matrix
of graph are as
=−,
𝑖𝑖
=
𝑖 =𝑗
𝑖𝑗
, ∀,
(6)
we can rewrite (5) as
min
𝑉
1
2
𝑁
𝑖,𝑗=1
V
𝑖
−V
𝑗
2
𝑖𝑗
=min
𝑉
𝑁
𝑖=1
V
𝑇
𝑖
V
𝑖
𝑖𝑖
−
𝑁
𝑖,𝑗=1
V
𝑇
𝑖
V
𝑗
𝑖𝑗
=min
𝑉
Tr
𝑇
−Tr
𝑇
=min
𝑉
Tr
𝑇
,
(7)
where Tr()denotes the trace of a matrix.
Equation (7) essentially holds the smoothness of
dimensionality reduction process. It is based on two vital
assumptions; rstly, the neighboring data samples in high-
dimensional space are semantic similarity, and secondly
the preservation of anity structure plays great role for
low-dimensional representation.
Local invariance assumption essentially exploits the
favorite relationship among similar data samples under unsu-
pervised condition; however, it ignores unfavorite relation-
ship between divergent data pairs. In this paper, we conjecture
that the distant data pairs are always semantic dierence. A
new distant neighboring graph
𝐶
={,
𝐶
}which is used
to describe the repulsion relationship between dissimilar data
pairs is also constructed. e corresponding weight matrix
𝐶
=[
𝑐
𝑖𝑗
]
𝑁×𝑁
is dened as follows:
𝑐
𝑖𝑗
=
−‖𝑥
𝑖
−𝑥
𝑗
‖
2
/𝜎
2
, if
𝑖
∈
𝑘
2
𝑗
or
𝑗
∈
𝑘
2
𝑖
,
1, otherwise,
(8)
where
𝑘
2
(
𝑖
)denotes the remotest
2
data samples of
𝑖
in
givensample dataset.
From (8), we can we can nd that the more the distance
between
𝑖
and
𝑗
, the smaller the value of
𝑐
𝑖𝑗
.Ifwewant
the corresponding low-dimensional representation V
𝑖
and V
𝑗
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