CAJ | 학술논문

局部切空间排列（LTSA）算法是一种有效的流形学习算法, 能较好地学习出高维数据的低维嵌入坐标。数据点的切空间在LTSA算法中起着重要的作用, 其局部几何特征多是在样本点的切空间内表示。但是在实际中, LTSA算法是把数据点邻域的样本协方差矩阵的主元所张成的空间当做数据点的切空间, 导致了在非均匀采样或样本邻域均值点与样本自身偏离程度较大时, 原算法的误差增大, 甚至失效。为此, 提出一种更严谨的数据点切空间的计算方法, 即数据点的邻域矩阵按照数据点本身进行中心化。通过数学推导, 证明了在一阶泰勒展开的近似下, 提出的计算方法所得到的空间即为数据点自身的切空间。在此基础上, 提出了一种改进的局部切空间排列算法, 并通过实验结果体现了该方法的有效性和稳定性。与已有经典算法相比, 提出的计算方法没有增加任何计算复杂度。
국부절공간배렬（LTSA）산법시일충유효적류형학습산법, 능교호지학습출고유수거적저유감입좌표。수거점적절공간재LTSA산법중기착중요적작용, 기국부궤하특정다시재양본점적절공간내표시。단시재실제중, LTSA산법시파수거점린역적양본협방차구진적주원소장성적공간당주수거점적절공간, 도치료재비균균채양혹양본린역균치점여양본자신편리정도교대시, 원산법적오차증대, 심지실효。위차, 제출일충경엄근적수거점절공간적계산방법, 즉수거점적린역구진안조수거점본신진행중심화。통과수학추도, 증명료재일계태륵전개적근사하, 제출적계산방법소득도적공간즉위수거점자신적절공간。재차기출상, 제출료일충개진적국부절공간배렬산법, 병통과실험결과체현료해방법적유효성화은정성。여이유경전산법상비, 제출적계산방법몰유증가임하계산복잡도。
As one of the classical manifold learning algorithms, LTSA algorithm can yield low-dimensional embedding coordinates from high-dimensional space effectively. Tangent space plays a central role in LTSA algorithm by projecting each neighborhood into the tangent space to obtain the local coordinates. However, in practice, LTSA algorithm takes the space which spanned by principal components of the sample covariance matrix of the neighborhood as the tangent space of the point. This paper presented a more rigorous method to calculate tangent space, that the neighborhood matrix of data points was centralized in accordance with the data point itself. By mathematical deduction, it proved that, under the approximation of first order Taylor, the space attained by our method is even the tangent space of data points itself. Based on this method, it proposed an improved local tangent space alignment algorithm. The effectiveness and stability of this algorithm are further confirmed by some experiments. Moreover, the proposed algorithm has no increase in the computational complexity.