# Relevance vector machine

In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification.[1] The RVM has an identical functional form to the support vector machine, but provides probabilistic classification.

It is actually equivalent to a Gaussian process model with covariance function:

${\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\sum _{j=1}^{N}{\frac {1}{\alpha _{j}}}\varphi (\mathbf {x} ,\mathbf {x} _{j})\varphi (\mathbf {x} ',\mathbf {x} _{j})}$

where ${\displaystyle \varphi }$ is the kernel function (usually Gaussian), ${\displaystyle \alpha _{j}}$ are the variances of the prior on the weight vector ${\displaystyle w\sim N(0,\alpha ^{-1}I)}$, and ${\displaystyle \mathbf {x} _{1},\ldots ,\mathbf {x} _{N}}$ are the input vectors of the training set.[2]

Compared to that of support vector machines (SVM), the Bayesian formulation of the RVM avoids the set of free parameters of the SVM (that usually require cross-validation-based post-optimizations). However RVMs use an expectation maximization (EM)-like learning method and are therefore at risk of local minima. This is unlike the standard sequential minimal optimization (SMO)-based algorithms employed by SVMs, which are guaranteed to find a global optimum (of the convex problem).

The relevance vector machine was patented in the United States by Microsoft (patent expired September 4, 2019).[3]

## References

1. ^ Tipping, Michael E. (2001). "Sparse Bayesian Learning and the Relevance Vector Machine". Journal of Machine Learning Research. 1: 211–244.
2. ^ Candela, Joaquin Quiñonero (2004). "Sparse Probabilistic Linear Models and the RVM". Learning with Uncertainty - Gaussian Processes and Relevance Vector Machines (PDF) (Ph.D.). Technical University of Denmark. Retrieved April 22, 2016.
3. ^ US 6633857, Michael E. Tipping, "Relevance vector machine"