Poisson Loss¶
Description¶
The Poisson Loss is a cost function used to model count data, typically when observations follow a Poisson distribution. Unlike Mean Squared Error (MSE), it accounts for the intrinsic variance of count data. It is used in models like Poisson regression and in denoising benchmarks for scRNA-seq data. This loss evaluates the quality of reconstruction by comparing observed and predicted values while respecting the discrete and heteroscedastic nature of the data.
Formulas¶
The Poisson Loss between an observed value \(y\) and a prediction \(\hat{y}\) is given by :
Where :
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\(y\): observed value (non-negative integer)
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\(\hat{y}\): predicted value (strictly positive real number)
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\(\log(y!)\) is constant with respect to \(\hat{y}\) and is often ignored during optimization.
The loss is minimized when \(\hat{y} \approx y\), and it penalizes errors more heavily for larger values of \(y\), which aligns with the increasing variance of the Poisson distribution.
Value Range :¶
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The Poisson Loss is non-negative : \(\mathcal{L} \geq 0\)
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It tends to infinity as \(\hat{y} \to 0\) or when \(\hat{y}\) is far from \(y\)