Classification in the sense of similarity is an important issue. In this paper, we study unitary similarity classification in Topological Data Analysis. To consider similarity from a topological perspective, we define a pseudometric \(d_{S}^{(p)}\) to measure the distance between barcodes generated by persistent homology groups of topological spaces, and we provide that our pseudometric \(d_{S}^{(2)}\) is unitary similarity invariant. Thereby, we establish a connection between Operator Theory and Topological Data Analysis. We give the calculation formula of the pseudometric \(d_{S}^{(2)}\) \((d_{S}^{(1)})\) by arranging all eigenvalues of matrices determined by barcodes in descending order to get the infimum over all matchings. We construct comparative experiments on both synthetic datasets and waves from an online platform, the results demonstrate that our pseudometric \(d_{S}^{(2)}\) \((d_{S}^{(1)})\) is consistent under conformal linear transformations and nonlinear transformations, whereas the bottleneck and Wasserstein distances are not. In particular, our pseudometric on waves is only related to the waveform but is independent on the frequency and amplitude. Furthermore, the computation time for \(d_{S}^{(2)}\) \((d_{S}^{(1)})\) is significantly less than the computation time for bottleneck distance and is comparable to the computation time for accelerated Wasserstein distance between barcodes.
