在這篇論文我們提出了一個新穎的譜群聚法--基於距離的譜群聚法(distance-based spectral clustering)，相對傳統的譜群聚法(spectral clustering)來說，若未能預知適當的相似度量(similarity measure)則其群聚結果將易產生不正確的群聚結果。因本法無需假設輸入的資料須預知適當的相似度量及預知其群聚數，輸入資料的群聚結果完全是根據資料本身的性質來判定完成。再者，因為對基於距離的譜群聚法而言所需的是成對距離矩陣(pairwise distance matrix)，所以我們不需要對輸入的資料進行成對相似矩陣的轉化，基於此點，本方法能明顯地有別於傳統的譜群聚法。除此之外，本方法能考量各群聚的內部結構及不同群聚間的關係，也因此增強本方法對於不同群聚的區分能力進而能夠正確地萃取出各個群聚。 本方法成功地以Laplace operator 運用於成對距離矩陣而計算出對稱之高斯-拉普拉斯權重矩陣(Laplacian-of-Gaussian weighted matrix, i.e., LoG weighted matrix)，並參考對稱之高斯-拉普拉斯權重矩陣的最大特徵值所對應的特徵向量，根據此特徵向量中所有元素的升冪排列將成對距離矩陣中的元素進行重新排列後形成了對角化之區塊結構(diagonal block-wise structure)，我們可根據對角化產生之區塊結構進行自動的群聚萃取(automatic cluster extraction)，在實驗結果分析上證明其群聚結果有著相當的正確性，同時也顯示本群聚法可適用於不同形態或具有龐大雜訊的模擬資料組。 除此之外，我們也將基於距離的譜群聚法應用於影像處理(image processing)及生物資訊(bioinformatics) 的真實問題上，對於不同領域的資料型態之實驗結果中驗證了此群聚法的可靠性ヽ可行性及可適性。我們相信基於距離的譜群聚法是個值得注意的群聚法。 In this dissertation, we propose a novel spectral clustering method, the distance-based spectral clustering, which makes no assumption on regarding both the suitable similarity measure and the prior-knowledge of cluster number. The proposed method is a new version of traditionally spectral clustering method, of which a distance pairwise matrix can be directly employed without transformation as a pairwise similarity matrix in advance. Moreover, the inter-cluster structure and the intra-cluster pairwise relationships are maximized in the proposed method to increase the discrimination capability on extracting clusters. The Laplace operator is successfully applied by the proposed method to the pairwise distance matrix to produce the symmetric Laplacian of Gaussian (LoG) weighted matrix. According to the ascending order of the elements of the eigenvector corresponding to the first largest eigenvalue of the symmetric LoG weighted matrix, the pairwise distance matrix is reordered to exhibits the diagonal block-wise structure. Besides, the automatic cluster extraction is accomplished based on the diagonal block-wise structure. The experimental results of a number of various datasets show that the correctness of the extracted clusters and it is robust against noises even the high level of noises, besides, the experimental analysis demonstrates the outstanding performance of the proposed method. Moreover, we apply the distance-based spectral clustering to the real problems in different fields including image processing and bioinformatics. Experimental results of these different types of data demonstrate its reliability, feasibility, and adaptability. We believe that the proposed distance-based spectral clustering is a remarkable clustering method.
