关于GTM中Φ : Y → X含义及数据空间高维坐标的技术咨询

Hey there, let's break down these two concepts from your paragraph step by step to make them easier to grasp!

First, let's tackle "high dimensional coordinates in the data space":
In the context of dimensionality reduction methods like SOM and GTM, the "data space" refers to the original high-dimensional space where your raw data lives. For example, if you're working with 200x200 pixel images, each image is represented as a 40,000-dimensional vector (one value per pixel)—that 40,000-dimensional space is your data space. The "high dimensional coordinates" are simply the positions of your actual data points (and the prototype points wj mentioned) within this space. Each coordinate corresponds to a feature or dimension of your data.

Now, onto Φ : Y → X:
Let's unpack the notation first:

• Y is the low-dimensional latent space mentioned in the text (for GTM, this is the real plane with a probability distribution peaked at regular grid positions).
• X is the high-dimensional data space we just talked about.
• Φ is a parameterized generalized linear function that acts as a bridge between these two spaces.

In plain terms: this function takes any point from the low-dimensional latent space Y and maps it to a corresponding point in the high-dimensional data space X. For GTM specifically, the prototype points wj (which are the high-dimensional counterparts to the grid positions in Y) aren't directly assigned like they are in SOM—instead, each prototype is the output of applying Φ to a grid point from the latent space Y. This gives GTM more flexibility than SOM, since the mapping between latent space and data space is governed by a learnable function rather than fixed grid assignments.

To tie it back to the original text, here's the relevant snippet for reference:

These points are associated to high dimensional coordinates in the data space, the parameters of the mapping, called prototypes wj, which are directly assigned to grid positions by means of the index in case of SOM, or which are images of a parameterized generalized linear function Φ : Y → X in case of GTM.

备注：内容来源于stack exchange，提问作者tahasozgen