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add similarities and finish parts of matandmeth
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lukas-heiligenbrunner 2024-12-30 14:06:47 +01:00
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@ -283,21 +283,40 @@ The softmax function has high similarities with the Boltzmann distribution and w
=== Cross Entropy Loss === Cross Entropy Loss
#todo[Maybe remove this section] #todo[Maybe remove this section]
Cross Entropy Loss is a well established loss function in machine learning. 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. @crel #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. 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) &= -sum_(x in cal(X)) p(x) log q(x)\
H(p,q) &= -(p log(q) + (1-p) log(1-q))\ 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)) 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 === 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 === 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 == Alternative Methods
There are several alternative methods to few-shot learning which are not used in this bachelor thesis. There are several alternative methods to few-shot learning which are not used in this bachelor thesis.
#todo[Do it!]