text: Probability distributions produced by the cross-entropy loss for ordinal classification problems can possess undesired properties. We propose a straightforward technique to constrain discrete ordinal probability distributions to be unimodal via a combination of the Poisson probability mass function and the softmax nonlinearity. We evaluate this approach on two large ordinal image datasets and obtain promising results.
  @cite_2: Threshold models are one of the most common approaches for ordinal regression, based on projecting patterns to the real line and dividing this real line in consecutive intervals, one interval for each class. However, finding such one-dimensional projection can be too harsh an imposition for some datasets. This paper proposes a multidimensional latent space representation with the purpose of relaxing this projection, where the different classes are arranged based on concentric hyperspheres, each class containing the previous classes in the ordinal scale. The proposal is implemented through a neural network model, each dimension being a linear combination of a common set of basis functions. The model is compared to a nominal neural network, a neural network based on the proportional odds model and to other state-of-the-art ordinal regression methods for a total of 12 datasets. The proposed latent space shows an improvement on the two performance metrics considered, and the model based on the three-dimensional latent space obtains competitive performance when compared to the other methods.
summary: There are other ordinal techniques but which do not impose unimodal constraints. The proportional odds model (POM) and its neural network extensions (POMNN, CHNN @cite_2 ) do not suffer from the monotonicity issue due to the utilization of monotonically increasing biases in the calculation of probabilities. The stick-breaking approach by , which is a reformulation of the multinomial logit (softmax), could also be used in the ordinal case as it technically imposes an ordering on classes.