Conditional ordinal random fields for structured ordinal-valued label prediction

Abstract

Predicting labels of structured data such as sequences or images is a very important problem in statistical machine learning and data mining. The conditional random field (CRF) is perhaps one of the most successful approaches for structured label prediction via conditional probabilistic modeling. In such models, it is traditionally assumed that each label is a random variable from a nominal category set (e.g., class categories) where all categories are symmetric and unrelated from one another. In this paper we consider a different situation of ordinal-valued labels where each label category bears a particular meaning of preference or order. This setup fits many interesting problems/datasets for which one is interested in predicting labels that represent certain degrees of intensity or relevance. We propose a fairly intuitive and principled CRF-like model that can effectively deal with the ordinal-scale labels within an underlying correlation structure. Unlike standard log-linear CRFs, learning the proposed model incurs non-convex optimization. However, the new model can be learned accurately using efficient gradient search. We demonstrate the improved prediction performance achieved by the proposed model on several intriguing sequence/image label prediction tasks.

Appendix

The gradients $\frac{\partial z_k(r,c)}{\partial \mu }$, for $k=0,1$ and $j=1,\dots ,R-2$, in (16) are summarized as follows:

$$\begin{aligned} \frac{\partial z_k(r,c)}{\partial \mathbf{a}}&= -\frac{1}{\sigma _0^2} {\varvec{\phi }}(\mathbf{x}_r), \end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial z_k(r,c)}{\partial \sigma _0}&= -\frac{2\big (b_{c-k} - \mathbf{a}^{\top }{\varvec{\phi }}(\mathbf{x}_r)\big )}{\sigma _0^3}, \end{aligned}$$

(21)

$$\begin{aligned} \frac{\partial z_0(r,c)}{\partial b_1}&= \left\{ \begin{array}{ll} 0 \quad \quad \mathrm{if} \quad c=R \\ \frac{1}{\sigma _0^2} \quad \mathrm{otherwise} \end{array} \right. , \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial z_1(r,c)}{\partial b_1}&= \left\{ \begin{array}{ll} 0 \quad \quad \mathrm{if} \quad c=1 \\ \frac{1}{\sigma _0^2} \quad \mathrm{otherwise} \end{array} \right. , \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial z_0(r,c)}{\partial \delta _j}&= \left\{ \begin{array}{ll} 0 \quad \quad \mathrm{if} \quad c \in \{1,\dots ,j,R\} \\ \frac{2\delta _j}{\sigma _0^2} \quad \mathrm{otherwise} \end{array} \right. , \end{aligned}$$

(24)

$$\begin{aligned} \frac{\partial z_1(r,c)}{\partial \delta _j}&= \left\{ \begin{array}{ll} 0 \quad \quad \mathrm{if} \quad c \in \{1,\dots ,j+1\} \\ \frac{2\delta _j}{\sigma _0^2} \quad \mathrm{otherwise} \end{array} \right. . \end{aligned}$$

(25)