We propose a novel approach to nonlinear functional regression, called the Mapping-to-Parameter function model, which addresses complex and nonlinear functional regression problems in parameter space by employing any supervised learning technique. Central to this model is the mapping of function data from an infinite-dimensional function space to a finite-dimensional parameter space. This is accomplished by concurrently approximating multiple functions with a common set of B-spline basis functions by any chosen order, with their knot distribution determined by the Iterative Local Placement Algorithm, a newly proposed free knot placement algorithm. In contrast to the conventional equidistant knot placement strategy that uniformly distributes knot locations based on a predefined number of knots, our proposed algorithms determine knot location according to the local complexity of the input or output functions. The performance of our knot placement algorithms is shown to be robust in both single-function approximation and multiple-function approximation contexts. Furthermore, the effectiveness and advantage of the proposed prediction model in handling both function-on-scalar regression and function-on-function regression problems are demonstrated through several real data applications, in comparison with four groups of state-of-the-art methods.

Topic modeling is admittedly a convenient way to monitor markets trend. Conventionally, Latent Dirichlet Allocation, LDA, is considered a must-do model to gain this type of information. By given the merit of deducing keyword with token conditional probability in LDA, we can know the most possible or essential topic. However, the results are not intuitive because the given topics cannot wholly fit human knowledge. LDA offers the first possible relevant keywords, which also brings out another problem of whether the connection is reliable based on the statistic possibility. It is also hard to decide the topic number manually in advance. As the booming trend of using fuzzy membership to cluster and using transformers to embed words, this work presents the fuzzy topic modeling based on soft clustering and document embedding from state-of-the-art transformer-based model. In our practical application in a press release monitoring, the fuzzy topic modeling gives a more natural result than the traditional output from LDA.

Extractive models usually formulate text summarization as extracting fixed top-$k$ salient sentences from the document as a summary. Few works exploited extracting finer-grained Elementary Discourse Unit (EDU) with little analysis and justification for the extractive unit selection. Further, the selection strategy of the fixed top-$k$ salient sentences fits the summarization need poorly, as the number of salient sentences in different documents varies and therefore a common or best $k$ does not exist in reality. To fill these gaps, this paper first conducts the comparison analysis of oracle summaries based on EDUs and sentences, which provides evidence from both theoretical and experimental perspectives to justify and quantify that EDUs make summaries with higher automatic evaluation scores than sentences. Then, considering this merit of EDUs, this paper further proposes an EDU-level extractive model with Varying summary Lengths and develops the corresponding learning algorithm. EDU-VL learns to encode and predict probabilities of EDUs in the document, generate multiple candidate summaries with varying lengths based on various $k$ values, and encode and score candidate summaries, in an end-to-end training manner. Finally, EDU-VL is experimented on single and multi-document benchmark datasets and shows improved performances on ROUGE scores in comparison with state-of-the-art extractive models, and further human evaluation suggests that EDU-constituent summaries maintain good grammaticality and readability.