# Label propagation equation: what are the terms?

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• Master1022
In summary, the conversation discusses the variables and equations in a paper on label propagation. It is determined that ##f## and ##y## are functions of ##v##, where ##v## designates a vertex, ##S## is an operator, and ##\alpha## is a positive scalar. It is also noted that ##f(v)## can be written as a vector over a given set of vertices ##V##, but ##y## cannot be a scalar.
Master1022
TL;DR Summary
What terms in the equation (from the linked paper) are vectors or scalars?
Hi,

This is a simple question that I just wanted to clarify. I was reading the following paper on label propagation: HERE and I can't understand whether the terms are vectors or scalars in one of the equations - specifically, equation (2.15) shown in the image below.

My understanding:
- ##f## is a vector
- ## S ## is a matrix
- ## \alpha ## is a scalar
- I am not too sure about ##y##: could be a vector or a scalar.
- ##\nu##: I am not too sure, but I think it could be referring to a specific node? That is, ## f(\nu) ## could be the value of the vector ## f## at node ## \nu ##.
- ## y ##: I am not sure, but I think it is a vector (see reasoning below).

Case is ## y ## is a scalar:
- That would make sense mathematically, but does that mean that we are using the same scalar ## y ## the equation for all nodes. That is, it doesn't matter what node ## \nu ## we are considering, we will always have the same ## y ## scalar in the equation? However, there is another equation above (shown below) which uses y as follows. This suggests that ##y## is a vector because then we have matrix-vector multiplication:

Case if ## y ## is a vector:
- It could be a vector (as suggested by image above), but then we are adding a vector ## (1 - \alpha) y ## to a scalar ## \alpha S f ## is a vector, and we are extracted the value at a certain node ## \nu ##, so it is a scalar. Therefore, it seems unlikely that ## y ## is a vector unless my interpretation of ## \nu ## is incorrect.Apologies if this is sparse with information. I didn't want to rewrite the paper in this post and I am unsure of some of the definitions of variables in there. Any help would be greatly appreciated.

##f## and ##y## are functions of ##v##, where ##v## designates a vertex, ##S## an operator, ##\alpha## a positive scalar.

For a given set of vertices ##V##, it is possible to write ##f## and ##y## as a vector over the set of vertices, and ##S## as a matrix.

Note that ##y## can't be a scalar otherwise eq. (2.15) would represent the sum of disparate elements.

Master1022
DrClaude said:
##f## and ##y## are functions of ##v##, where ##v## designates a vertex, ##S## an operator, ##\alpha## a positive scalar.

For a given set of vertices ##V##, it is possible to write ##f## and ##y## as a vector over the set of vertices, and ##S## as a matrix.

Note that ##y## can't be a scalar otherwise eq. (2.15) would represent the sum of disparate elements.
Many thanks for the response!

So would ## f( \nu ) ## be a vector instead of just referring to the entry of vector ## f## corresponding to ## \nu ##?

Master1022 said:
So would ## f( \nu ) ## be a vector instead of just referring to the entry of vector ## f## corresponding to ## \nu ##?

##f(v)## is a function, but if you have a discrete set ##V## of vertices ##v##, then ##f(v)## over ##V## can be written as a vector.

## 1. What is the label propagation equation?

The label propagation equation is a mathematical formula used in the field of machine learning for semi-supervised learning tasks. It is used to predict the labels of unlabeled data points based on the labels of their neighboring data points.

## 2. What are the terms used in the label propagation equation?

The terms used in the label propagation equation include the weight matrix, the label matrix, and the initial label matrix. The weight matrix represents the relationship between data points, the label matrix contains the known labels of data points, and the initial label matrix contains the predicted labels for the unlabeled data points.

## 3. How does the label propagation equation work?

The label propagation equation works by iteratively updating the initial label matrix based on the label matrix and weight matrix. This process continues until the predicted labels converge and the algorithm reaches a stable solution.

## 4. What is the purpose of the label propagation equation?

The main purpose of the label propagation equation is to improve the accuracy of label predictions for unlabeled data points in semi-supervised learning tasks. It can also be used for data clustering and dimensionality reduction.

## 5. Are there any limitations to the label propagation equation?

Yes, there are some limitations to the label propagation equation. It assumes that the data points are connected in a continuous manner and that the labels of neighboring data points are similar. It may also struggle with data sets that have a high level of noise or outliers.

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