# One-class SVM formula

Recently I have been studying one-class SVM and am a little bit confused about the offset $$\rho$$. The common optimization problem is to find a function $$f(x)= w^\top x-\rho$$ by solving $$\begin{array}{l} \min\limits_{w, \rho} \frac{1}{2}\| w\|^2 + \frac{1}{n\nu}\sum\limits_{i=1}^n\xi_i -\rho\\ \text{s.t. } \ \ w^\top x \geq \rho - \xi_i,\ \ i \in[n]\\ \qquad \xi_i\geq 0, \end{array}$$ whose dual problem is $$\begin{array}{l} \min\limits_{\alpha_i} \frac{1}{2}\sum\limits_i\sum\limits_j\alpha_i\alpha_j x_i^\top x_j\\ \text{s.t. } \ \ \sum\limits_{i=1}^n\alpha_i=1\\ \qquad 0\leq\alpha_i\leq \frac{1}{n\nu}, i\in[n] \end{array}$$

The above dual problem can only give us $$w$$ but the $$\rho$$ has to be determined by KKT condition, which is involved with finding out those $$\alpha_i$$ between $$0$$ and $$\frac{1}{n\nu}$$ (both exclusively). However, my consideration is due to numerical results of those $$\alpha_i$$'s, what kind of value can be treated as $$0$$ or $$\frac{1}{n\nu}$$? Different person may take different threshold or even sometimes we cannot find those $$0<\alpha_i<\frac{1}{n\nu}$$ due to an improper threshold, so it may cause the offset estimation unstable or troublesome, although maybe we can take the average over those indces $$i$$ whose corresponding $$\alpha_i$$ between $$0$$ and $$\frac{1}{n\nu}$$ to get an relative nice offset for former case.

Based on this argument, if we can directly set $$\rho = 1$$? If it is, then we needn't to estimate the $$\rho$$ any more, which may be safe. If not (or in general not), what will happen?

• Welcome to DataScienceSE. In case nobody answers, I'd suggest asking on stats.stackexchange.com. Theoretical questions often get good answers there. Jun 13 at 23:29
• Thank you @Erwan and your editing as well as suggestion. Jun 14 at 0:32