Masahiro Yamamoto

Piermarco Cannarsa, Jacques Tort, Masahiro Yamamoto

This paper studies unique continuation for weakly degenerate parabolic equations in one space dimension. A new Carleman estimate of local type is obtained to deduce that all solutions that vanish on the degeneracy set, together with their conormal derivative, are identically equal to zero. An approximate controllability result for weakly degenerate parabolic equations under Dirichlet boundary condition is deduced.

Jie Yu, Yikan Liu, Masahiro Yamamoto

In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with both partial boundary and interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.

Masahiro Yamamoto

Since the invention of computers, communication through natural language (actual human language) has been a dream technology. However, natural language is extremely difficult to mathematically formulate, making it difficult to realize as an algorithm without considering programming. While there have been numerous technological developments, one cannot say that any results allowing free utilization have been achieved thus far. In the case of language learning in humans, for instance when learning one's mother tongue or foreign language, one must admit that this process is similar to the adage "practice makes perfect" in principle, even though the learning method is significant up to a point. Deep learning has played a central role in contemporary AI technology in recent years. When applied to natural language processing (NLP), this produced unprecedented results. Achievements exceeding the initial predictions have been reported from the results of learning vast amounts of textual data using deep learning. For instance, four arithmetic operations could be performed without explicit learning, thereby enabling the explanation of complex images and the generation of images from corresponding explanatory texts. It is an accurate example of the learner embodying the concept of "practice makes perfect" by using vast amounts of textual data. This report provides a technological explanation of how cutting-edge NLP has made it possible to realize the "practice makes perfect" principle. Additionally, examples of how this can be applied to business are provided. We reported in June 2022 in Japanese on the NLP movement from late 2021 to early 2022. We would like to summarize this as a memorandum since this is just the initial movement leading to the current large language models (LLMs).

Takashi Isozaki, Masahiro Yamamoto, Atsushi Noda

The problem of explaining the results produced by machine learning methods continues to attract attention. Neural network (NN) models, along with gradient boosting machines, are expected to be utilized even in tabular data with high prediction accuracy. This study addresses the related issues of pseudo-correlation, causality, and combinatorial reasons for tabular data in NN predictors. We propose a causal explanation method, CENNET, and a new explanation power index using entropy for the method. CENNET provides causal explanations for predictions by NNs and uses structural causal models (SCMs) effectively combined with the NNs although SCMs are usually not used as predictive models on their own in terms of predictive accuracy. We show that CEN-NET provides such explanations through comparative experiments with existing methods on both synthetic and quasi-real data in classification tasks.

Daijun Jiang, Yikan Liu, Masahiro Yamamoto

In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.