With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed [D. Li, K. Yang, W. Wong, Advances in Neural Information Processing Systems (2016) 1099-1107] to learn an adaptive piecewise constant approximation defined on a binary sequential partition of the underlying domain, where the star discrepancy is adopted to measure the uniformity of particle distribution. However, the calculation of the star discrepancy is NP-hard and it does not satisfy the reflection invariance and rotation invariance either. To this end, we use the mixture discrepancy and the comparison of moments as a replacement of the star discrepancy, leading to the density estimation via mixture discrepancy based sequential partition (DSP-mix) and density estimation via moments based sequential partition (MSP), respectively. Both DSP-mix and MSP are computationally tractable and exhibit the reflection and rotation invariance. Numerical experiments in reconstructing the $d$-D mixture of Gaussians and Betas with $d=2, 3, \dots, 6$ demonstrate that DSP-mix and MSP both run approximately ten times faster than DSP while maintaining the same accuracy.

The timely identification of mental disorders in adolescents is a global public health challenge.Single factor is difficult to detect the abnormality due to its complex and subtle nature. Additionally, the generalized multimodal Computer-Aided Screening (CAS) systems with interactive robots for adolescent mental disorders are not available. Here, we design an android application with mini-games and chat recording deployed in a portable robot to screen 3,783 middle school students and construct the multimodal screening dataset, including facial images, physiological signs, voice recordings, and textual transcripts.We develop a model called GAME (Generalized Model with Attention and Multimodal EmbraceNet) with novel attention mechanism that integrates cross-modal features into the model. GAME evaluates adolescent mental conditions with high accuracy (73.34%-92.77%) and F1-Score (71.32%-91.06%).We find each modality contributes dynamically to the mental disorders screening and comorbidities among various mental disorders, indicating the feasibility of explainable model. This study provides a system capable of acquiring multimodal information and constructs a generalized multimodal integration algorithm with novel attention mechanisms for the early screening of adolescent mental disorders.