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Summary Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors.
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Minimal pole representation and analytic continuation of matrix-valued correlation functions
We present a minimal pole method for analytically continuing matrix-valued imaginary frequency correlation functions to the real axis, enabling precise access to off-diagonal elements and thus improving the interpretation of self-energies and susceptibilities in quantum simulations. Traditional methods for matrix-valued analytic continuation tend to be either noise sensitive or make ad hoc positivity assumptions. Our approach avoids these issues via the construction of a compact pole representation with shared poles through exponential fits, expanding upon prior work focused on scalar functions. We test our method across various scenarios, including fermionic and bosonic response functions, with and without noise, and for both continuous and discrete spectra of real materials and model systems. Our findings demonstrate that this technique addresses the shortcomings of existing methodologies, such as artificial broadening and positivity violations. The paper is supplemented with a sample implementation in PYTHON.
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- Award ID(s):
- 2310182
- PAR ID:
- 10616939
- Publisher / Repository:
- American Physical Society
- Date Published:
- Journal Name:
- Physical Review B
- Volume:
- 110
- Issue:
- 23
- ISSN:
- 2469-9950
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevB.110.235131
