這項工作探索了極高維度圖案的卷積距離，並提出了一種新穎的基於卷積相似性的分群方法，並應用在非線性降維上。歐幾里得距離有一項涉及內積。將一個三維圖案通過另一個圖案進行卷積的操作會獲得許多卷積內積，每個內積對應於由任意三維位移引起的排列。卷積內積的最大值表示測量了最大相似性數值的最佳圖案匹配。提出了一種基於最大相似性的準則來分群極高維度圖案的新方法，還有基於最大相似性的準則的最近相鄰關係。分群後，以各群的局部中心點為代表點，並透過最近相鄰關係定義次級代表點。在此基礎上透過PCA(principal components analysis)建構支援空間，將資料降維到中間k維空間，計算代表點之間的混合距離，混合距離構成了目標空間中輸入影像的非線性約束，這指向了一個個體的LNE(Local nonlinear embedding)問題，單個LNE問題可以透過Levenberg-Marquardt算法解決。

This work explores convolution distances of extremely high dimensional patterns and proposes a novel approach for clustering based on convolution similarity with application to do nonlinear dimensionality reduction. The Euclidean distance has one term involving an inner product. The operation of convoluting a 3-dimensional pattern through the other pattern attains many convolutional inner products, each corresponding an alignment caused by an arbitrary 3-dimensional shift. The largest value of convolutional inner products indicates the best pattern match, measuring the quantity of maximal similarity. A novel approach for clustering extremely high-dimensional patterns and the nearest neighboring relations are proposed based on the criterion of maximal similarity. After clustering, the local center point of each cluster is preserved as the representative, and the secondary representative is defined through the nearest neighboring relations. On this basis, the support space is constructed through principal components analysis. The data is reduced to the intermediate k-dimensional space, and the mixing distance between representatives is calculated. The mixing distance constitutes the nonlinear constraint of the input image in the target space. This specifies an individual locally nonlinear embedding problem (LNE). A single LNE problem can be solved by the Levenberg-Marquardt algorithm.

1. Introduction 1
2. Methods 4
2.1 Neighboring relations of centroids in neural modules for NDR mapping 4
2.2 Convolution distance 6
2.3 Clustering based on convolution distances 9
2.4 Second-level clustering and posterior processes 12
2.5 K nearest neighbors and locally nonlinear embedding 13
3. Numerical Experiments 15
3.1 Experiment 1 15
3.2 Experiment 2 18
3.3 Experiment 3 19
3.4 Improvement method 21
4. Conclusions 23
Reference 24

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