如何在二维图上描绘多维向量?
我在多维空间(可能有几千个维度)中有一组向量。在这个空间中,我可以计算两个向量之间的距离(如果重要的话,作为它们之间角度的余弦)。我想要的是可视化这些保持距离的向量。也就是说,如果在多维空间中向量 a
比向量 c
更接近向量 b
,那么它在 2- 上也一定更接近向量 b
。维度图。有没有什么图可以清楚的描绘出来?
I have a set of vectors in multidimensional space (may be several thousands of dimensions). In this space, I can calculate distance between 2 vectors (as a cosine of the angle between them, if it matters). What I want is to visualize these vectors keeping the distance. That is, if vector a
is closer to vector b
than to vector c
in multidimensional space, it also must be closer to it on 2-dimensional plot. Is there any kind of diagram that can clearly depict it?
我不这么认为。想象任何四面体的二维图片。无法在二维中以彼此相等的距离描绘四个顶点。因此,您将很难尝试在二维中描述三个以上的 n 维向量并保持它们的相互距离。
(但现在我想不出严格的证明。)
更新:
好吧,第二个想法,也许很愚蠢:如果您尝试找到更紧密关联的对象/文本的簇,然后计算每个簇的中心或平均向量。然后你就可以减少问题空间。首先找到保持其相对距离的簇的二维组成。然后插入主向量,仅考虑它们在簇内的相对距离以及它们到两个或三个最近的簇中心的距离。
这种方法适用于大量向量。但它并不准确,因为总会有一些相似的向量最终出现在遥远的地方。
I don't think so. Imagine any twodimensional picture of a tetrahedron. There is no way of depicting the four vertices in two dimensions with equal distances from each other. So you will have a hard time trying to depict more than three n-dimensional vectors in 2 dimensions conserving their mutual distances.
(But right now I can't think of a rigorous proof.)
Update:
Ok, second idea, maybe it's dumb: If you try and find clusters of closer associated objects/texts, then calculate the center or mean vector of each cluster. Then you can reduce the problem space. At first find a 2D composition of the clusters that preserves their relative distances. Then insert the primary vectors, only accounting for their relative distances within a cluster and their distance to the center of to two or three closest clusters.
This approach will be ok for a large number of vectors. But it will not be accurate in that there always will be somewhat similar vectors ending up at distant places.