We adapt the generic knowledge from existing studies that are applicable to different datasets. Generic knowledge is expressed as probabilities. The generic knowledge is categorized into three types: expression-dependent single AU probabilities, expression-dependent joint AU probabilities, and expression-independent joint AU probabilities. 1) For expression-dependent single AU probabilities, two sources are considered. According to FACS, given an expression, AUs can be grouped into primary (P) and secondary (S) categories. The primary AUs are the most expressive AUs with respective to the expression, and the secondary AUs may co-occur with primary AUs providing additional supports for the expression.

Our proposed approach can be applied to other AUs as well. In Tab.6, LP-SM also considers apex frames on CK+, and The comparison to LP-SM is consistent. In Tab.8, we apply FMPN-FER and DeepEmotion to our pre-processed We will consider a pre-trained VGGFace model in our further work. R2 2.1 The novelty compared to prior work. Facial expression can be a group of AUs.

In this paper, we establish Multilayer Correlation Clustering, a novel generalization of Correlation Clustering (Bansal et al., FOCS '02) to the multilayer setting. In this model, we are given a series of inputs of Correlation Clustering (called layers) over the common set $V$. The goal is then to find a clustering of $V$ that minimizes the $\ell_p$-norm ($p\geq 1$) of the disagreements vector, which is defined as the vector (with dimension equal to the number of layers), each element of which represents the disagreements of the clustering on the corresponding layer. For this generalization, we first design an $O(L\log n)$-approximation algorithm, where $L$ is the number of layers, based on the well-known region growing technique. We then study an important special case of our problem, namely the problem with the probability constraint. For this case, we first give an $(\alpha+2)$-approximation algorithm, where $\alpha$ is any possible approximation ratio for the single-layer counterpart. For instance, we can take $\alpha=2.5$ in general (Ailon et al., JACM '08) and $\alpha=1.73+\epsilon$ for the unweighted case (Cohen-Addad et al., FOCS '23). Furthermore, we design a $4$-approximation algorithm, which improves the above approximation ratio of $\alpha+2=4.5$ for the general probability-constraint case. Computational experiments using real-world datasets demonstrate the effectiveness of our proposed algorithms.

This paper studies the problem of estimation for general finite mixture models, with a particular focus on the elliptical mixture models (EMMs). Instead of using the widely adopted Kullback-Leibler divergence, we provide a stable solution to the EMMs that is robust to initialisations and attains superior local optimum by adaptively optimising along a manifold of an approximate Wasserstein distance. More specifically, we first summarise computable and identifiable EMMs, in order to identify the optimisation problem. Due to a probability constraint, solving this problem is cumbersome and unstable, especially under the Wasserstein distance. We thus resort to an efficient optimisation on a statistical manifold defined under an approximate Wasserstein distance, which allows for explicit metrics and operations. This is shown to significantly stabilise and improve the EMM estimations. We also propose an adaptive method to further accelerate the convergence. Experimental results demonstrate excellent performances of the proposed solver.

When a planner must decide whether it has enough evidence to make a decision based on probability, it faces the sample size problem. Current planners using probabilities need not deal with this problem because they do not generate their probabilities from observations. This paper presents an event based language in which the planner's probabilities are calculated from the binomial random variable generated by the observed ratio of one type of event to another. Such probabilities are subject to error, so the planner must introspect about their validity. Inferences about the probability of these events can be made using statistics. Inferences about the validity of the approximations can be made using interval estimation. Interval estimation allows the planner to avoid making choices that are only weakly supported by the planner's evidence.