## 5 Conclusions  
This study proposes a new Partial Least Squares regression model, FPLS. This model integrates filter learning and traditional PLS models into a single model. It can adaptively learn the parameters of the filter and PLS, improving the effectiveness of the filter. At the same time, we briefly analyze the convergence of the model, the bounds of the objective function, and the variance of the prediction distribution. Experiments on four datasets, Corn, Octane, Mango and Soil, verify the effectiveness, convergence, and bounds of the objective function of the model. Future work will primarily encompass the following aspects:  
* Further exploration of the relationship between filter length and \(\frac{w^{T}(X_{h})^{T}Y_{C}}{w^{T}X^{T}Y_{C}}\). In Figure 2, we pointed out its connection with local smoothness, and consequently chose filter lengths of 2, 3, and 4. Actually, as the length increases, \(\frac{w^{T}(X_{h})^{T}Y_{C}}{w^{T}X^{T}Y_{C}}\) exhibits a distinct regularity as shown in Figure 12a. Further in-depth research will be conducted on this issue.
* Utilizing trigonometric functions to transform the constraint condition \(\|w\|_{2}=1\). Define \(w_{1}=\cos\theta_{1}\), \(w_{2}=\sin\theta_{1}\cos\theta_{2}\), \(w_{3}=\sin\theta_{1}\sin\theta_{2}\cos\theta_{3},\cdots,w_{n-1}=\sin\theta_{1} \sin\theta_{2}\ldots\sin\theta_{n-2}\cos\theta_{n-1}\), \(w_{n}=\sin\theta_{1}\sin\theta_{2}\ldots\sin\theta_{n-2}\sin\theta_{n-1}\). In this way, we obtain a framework that utilizes neural networks to compute with orthogonal constraints, as shown in Figure 12b. Whether this network is equivalent to the BP network with 2-norm constraint requires further research.
* Utilizing a dilated filter to achieve multi-scale feature extraction. The filter can be generalized to \(h=\left[h_{1},\underbrace{0,\cdots,0}_{m},h_{2},\underbrace{0,\cdots,0}_{m}, \cdots,h_{n}\right]\). What is the relationship between this network and existing neural networks? This is an open question.  
**Author Contributions:** Y.M.: Conceptualization, writing--original draft, and methodology. L.Z.: writing--draft correction. W.C.: Investigation. J.L.: Validation. T.L.: Validation. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by Hubei Provincial Department of Education under grant number B2020061.  
**Data Availability Statement:** Data openly available in a public repository. [[http://www.eigenvector.com/data/Corn/index.html](http://www.eigenvector.com/data/Corn/index.html)]([http://www.eigenvector.com/data/Corn/index.html](http://www.eigenvector.com/data/Corn/index.html)) (accessed on 13 January 2023). [[https://github.com/xylocadenza/whpu](https://github.com/xylocadenza/whpu)]([https://github.com/xylocadenza/whpu](https://github.com/xylocadenza/whpu)) (accessed on 1 July 2025). [[https://data.mendeley.com/datasets/ph57](https://data.mendeley.com/datasets/ph57) ynng46/1]([https://data.mendeley.com/datasets/ph57](https://data.mendeley.com/datasets/ph57) ynng46/1) (accessed on 11 November 2023). [[https://www.scidb.cn/en/detail?dataSetId=93899](https://www.scidb.cn/en/detail?dataSetId=93899) ccb63054b5 fbf663c010d88892c8]([https://www.scidb.cn/en/detail?dataSetId=93899](https://www.scidb.cn/en/detail?dataSetId=93899) ccb63054b5 fbf663c010d88892c8) &version=V1 (accessed on 2 December 2023).  
Figure 12: Issues that require further research. (a) Relationship between \(\frac{w^{T}(X_{h})^{T}Y_{C}}{w^{T}X^{T}Y_{C}}\) and filter length; (b) a potential network structure.