From 0b5a0647e257aa0247d57dcb5d31624ff0e6722d Mon Sep 17 00:00:00 2001
From: lukas-heiligenbrunner <lukas.heiligenbrunner@gmail.com>
Date: Mon, 30 Dec 2024 14:06:47 +0100
Subject: [PATCH] add similarities and finish parts of matandmeth

---
 materialandmethods.typ | 27 +++++++++++++++++++++++----
 1 file changed, 23 insertions(+), 4 deletions(-)

diff --git a/materialandmethods.typ b/materialandmethods.typ
index 822f99c..755eb32 100644
--- a/materialandmethods.typ
+++ b/materialandmethods.typ
@@ -283,21 +283,40 @@ The softmax function has high similarities with the Boltzmann distribution and w
 === Cross Entropy Loss
 #todo[Maybe remove this section]
 Cross Entropy Loss is a well established loss function in machine learning.
-Equation~\eqref{eq:crelformal}\cite{crossentropy} shows the formal general definition of the Cross Entropy Loss.
-And equation~\eqref{eq:crelbinary} is the special case of the general Cross Entropy Loss for binary classification tasks.
+@crel #cite(<crossentropy>) shows the formal general definition of the Cross Entropy Loss.
+And @crel is the special case of the general Cross Entropy Loss for binary classification tasks.
 
 $
 H(p,q) &= -sum_(x in cal(X)) p(x) log q(x)\
 H(p,q) &= -(p log(q) + (1-p) log(1-q))\
 cal(L)(p,q) &= -1/N sum_(i=1)^(cal(B)) (p_i log(q_i) + (1-p_i) log(1-q_i))
-$
+$ <crel>
+#todo[Check how multiline equation refs work]
 
-Equation~$cal(L)(p,q)$~\eqref{eq:crelbinarybatch}\cite{handsonaiI} is the Binary Cross Entropy Loss for a batch of size $cal(B)$ and used for model training in this Practical Work.
+Equation~$cal(L)(p,q)$ @crel #cite(<handsonaiI>) is the Binary Cross Entropy Loss for a batch of size $cal(B)$ and used for model training in this Practical Work.
 
 === Cosine Similarity
+To measure the distance between two vectors some common distance measures are used.
+One popular of them is the Cosine Similarity (@cosinesimilarity).
+It measures the cosine of the angle between two vectors.
+The Cosine Similarity is especially useful when the magnitude of the vectors is not important.
 
+$
+  cos(theta) &:= (A dot B) / (||A|| dot ||B||)\
+  &= (sum_(i=1)^n  A_i B_i)/ (sqrt(sum_(i=1)^n A_i^2) dot sqrt(sum_(i=1)^n B_i^2))
+$ <cosinesimilarity>
+
+#todo[Source?]
 === Euclidean Distance
+The euclidean distance (@euclideannorm) is a simpler method to measure the distance between two points in a vector space.
+It just calculates the square root of the sum of the squared differences of the coordinates.
+the euclidean distance can also be represented as the L2 norm (euclidean norm) of the difference of the two vectors.
 
+$
+  cal(d)(A,B) = ||A-B|| := sqrt(sum_(i=1)^n (A_i - B_i)^2)
+$ <euclideannorm>
+#todo[Source?]
 == Alternative Methods
 
 There are several alternative methods to few-shot learning which are not used in this bachelor thesis.
+#todo[Do it!]