運用切片逆迴歸法可以找出有效的維度縮減方向來探索高維度資料的內在結構。在本論文中，我們針對非線性維度縮減問題，提出利用幾何測地線距離逼近法的一個混合型切片逆迴歸法，我們稱此方法為等軸距切片逆迴歸法。所提的方法中，第一步是先計算兩兩資料點等軸距距離，然後根據群集分析(例如:階層式群集分析)或排序方法(例如:秩二橢圓排序法)在這個距離矩陣上的分群結果，當成切片的依據，使得傳統的切片逆迴歸演算法可以被應用。 我們將說明等軸距切片逆迴歸法可以重新找到非線性流形資料 (例如瑞士捲資料) 內隱的維度和幾何結構。進一步，我們將應用所找到的特徵向量在分類問題上。 說明的例子會有一般的實際資料及微陣列基因表現資料。所提的方法也會和其它現存的幾個維度縮減方法相比較。 Sliced inverse regression (SIR) was introduced to find an effective linear dimension-reduction direction to explore the intrinsic structure of high dimensional data. In this study, we present isometric SIR for nonlinear dimension reduction - a hybrid of the SIR method using the geodesic distance approximation. First, the proposed method computes the isometric distance between data points; the resulting distance matrix is then sliced according to hierarchical clustering results with rank-two ellipse seriation, and the classical SIR algorithm is applied. We show that the isometric SIR can recover the embedded dimensionality and geometric structure of a nonlinear manifold dataset (e.g., the Swiss-roll). We illustrate how isometric SIR features can further be used for the classification problems. Finally, we report and discuss this novel method in comparison to several existing dimension-reduction techniques.
