Toshinari Morimoto

Yen-Shiu Chin, Zhi-Yu Jou, Toshinari Morimoto et al.

The inference of conditional distributions is a fundamental problem in statistics, essential for prediction, uncertainty quantification, and probabilistic modeling. A wide range of methodologies have been developed for this task. This article reviews and compares several representative approaches spanning classical nonparametric methods and modern generative models. We begin with the single-index method of Hall and Yao (2005), which estimates the conditional distribution through a dimension-reducing index and nonparametric smoothing of the resulting one-dimensional cumulative conditional distribution function. We then examine the basis-expansion approaches, including FlexCode (Izbicki and Lee, 2017) and DeepCDE (Dalmasso et al., 2020), which convert conditional density estimation into a set of nonparametric regression problems. In addition, we discuss two recent generative simulation-based methods that leverage modern deep generative architectures: the generative conditional distribution sampler (Zhou et al., 2023) and the conditional denoising diffusion probabilistic model (Fu et al., 2024; Yang et al., 2025). A systematic numerical comparison of these approaches is provided using a unified evaluation framework that ensures fairness and reproducibility. The performance metrics used for the estimated conditional distribution include the mean-squared errors of conditional mean and standard deviation, as well as the Wasserstein distance. We also discuss their flexibility and computational costs, highlighting the distinct advantages and limitations of each approach.

Toshinari Morimoto, Su-Yun Huang

In this paper, we propose a dimension reduction method specifically designed for tensor-structured feature data in deep neural networks. The method is implemented as a hidden layer, called the TensorProjection layer, which transforms input tensors into output tensors with reduced dimensions through mode-wise projections. The projection directions are treated as model parameters of the layer and are optimized during model training. Our method can serve as an alternative to pooling layers for summarizing image data, or to convolutional layers as a technique for reducing the number of channels. We conduct experiments on tasks such as medical image classification and segmentation, integrating the TensorProjection layer into commonly used baseline architectures to evaluate its effectiveness. Numerical experiments indicate that the proposed method can outperform traditional downsampling methods, such as pooling layers, in our tasks, suggesting it as a promising alternative for feature summarization.