Prototypical Networks Protonets
Tags: #machine learning #meta learningEquation
$$c_{k}=\frac{1}{S_{k}}\sum_{(x_{i},y_{i}) \in S_{k}} f_{\phi}(x) \\ p_{\phi}(y=kx)=\frac{\exp(d(f_{\phi}(x), c_{k}))}{\sum_{k^{'}} \exp(d(f_{\phi}(x), c_{k^{'}})} \\\min J(\phi)=\log p_{\phi}(y=kx)$$Latex Code
c_{k}=\frac{1}{S_{k}}\sum_{(x_{i},y_{i}) \in S_{k}} f_{\phi}(x) \\ p_{\phi}(y=kx)=\frac{\exp(d(f_{\phi}(x), c_{k}))}{\sum_{k^{'}} \exp(d(f_{\phi}(x), c_{k^{'}})} \\\min J(\phi)=\log p_{\phi}(y=kx)
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Introduction
Equation
Latex Code
c_{k}=\frac{1}{S_{k}}\sum_{(x_{i},y_{i}) \in S_{k}} f_{\phi}(x) \\ p_{\phi}(y=kx)=\frac{\exp(d(f_{\phi}(x), c_{k}))}{\sum_{k^{'}} \exp(d(f_{\phi}(x), c_{k^{'}})} \\\min J(\phi)=\log p_{\phi}(y=kx)
Explanation
Prototypical networks compute an Mdimensional representation c_{k} or prototype, of each class through an embedding f_{\phi}(.) with parameters \phi. Each prototype is the mean vector of the embedded support points belonging to its class k. Prototypical networks then produce a distribution over classes for a query point x based on a softmax over distances to the prototypes in the embedding space as p(y=kx). Then the negative loglikelihood of J(\theta) is calculated over query set.
Related Documents
 See paper Prototypical Networks for Fewshot Learning for more detail.
Related Videos
Prototypical Networks as Mixture Density Estimation
Bregman divergences
d_{\phi}(z,z^{'})=\phi(z)  \phi(z^{'})(zz^{'})^{T} \nabla \phi(z^{'})
Mixture Density Estimation
p_{\phi}(y=kz)=\frac{\pi_{k} \exp(d(z, \mu (\theta_{k})))}{\sum_{k^{'}} \pi_{k^{'}} \exp(d(z, \mu (\theta_{k})))}
Explanation
The prototypi cal networks algorithm is equivalent to performing mixture density estimation on the support set with an exponential family density. A regular Bregman divergence d_{\phi} is defined as above. \phi is a differentiable, strictly convex function of the Legendre type. Examples of Bregman divergences include squared Euclidean distance and Mahalanobis distance.
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