Persistent homology is a robust data analysis tool, but it is fraught with limitations, including substantial computational intricacy. Deriving persistent homology is a demanding process, especially for voluminous or high-dimensional data collections. Since the size of simplicial complexes spirals upwards in tandem with data dimension and sample size, carrying out non-approximate or general algorithms computations is impractical (Otter et al., 2017). Furthermore, it is challenging to interpret abstract results using persistence diagrams and barcodes. Even though singular homology is easier to furnish with exact sequences, techniques to calculate persistent homology assert similar vectors may have distinct persistence diagrams (Hatcher, 2022).

Therefore, grasping domain-critical terms from these topographical markers is daunting. It may be challenging to apply persistent homology to diverse real-world issues (Gunnarssson et al., 2019). Although persistent homology is crafted to be robust to noise, in practicality, collecting inadequate or noisy data can lead to the concealment of authentic structures or the production of spurious topographical characteristics. While sometimes challenging, it is crucial to perform accurate pre-processing and parameter tuning (Chazal & Michel, 2017). Furthermore, the lack of a comprehensive statistical framework remains a stumbling block. It is challenging to conduct statistical inference and hypothesis testing when using persistent homology. Topological characteristics and confidence folds galore reclaiming rigorous probabilistic models is a perpetual ache in the research spectrum. Therefore, this limitation curtails extensive and practical utilization (Fasy et al., 2014).

Also, the results you get will depend on the size of the filters you choose, as well as the methodology you are using. You need to do some tweaking to get the right results Therefore, it can be a challenge to incorporate dimensional scrutiny seamlessly into traditional data analysis methods and ML workflows. Although persistent homology is a significant form of analysis, it is necessary to note its shortcomings and the amount of expertise required to understand and utilize the same in various disciplines.

References

Otter, N., Porter, M. A., Tillmann, U., Grindrod, P., & Harrington, H. A. (2017). A roadmap for the computation of persistent homology. EPJ Data Sci, 6(1), 17.

Hatcher, A. (2022). Algebraic Topology.

Chazal, F., & Michel, B. (2017). An introduction to topological data analysis: fundamental and practical aspects for data scientists.

Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., & Singh, A. (2014). Confidence sets for persistence diagrams. The Annals of Statistics, 42(6), 2301-2339.

Gunnarsson, B., Lecci, F., & Rinaldo, A. (2019). A tutorial on persistence diagrams and their use in data analysis. Wiley Interdisciplinary Reviews: Computational Statistics, 11(2)