Abstract
Given a finite-dimensional real inner product space V and a finite subgroup G of linear isometries, max filtering affords a bilipschitz Euclidean embedding of the orbit space V/G. We identify the max filtering maps of minimum distortion in the setting where G is a reflection group. Our analysis involves an interplay between Coxeter’s classification and semidefinite programming.
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References
Balan, R., Haghani, N., Singh, M.: Permutation invariant representations with applications to graph deep learning (2022). arXiv:2203.07546
Bandeira, A.S., Blum-Smith, B., Kileel, J., Perry, A., Weed, J., Wein, A.S.: Estimation under group actions: recovering orbits from invariants (2023). arXiv:1712.10163
Bendory, T., Edidin, D., Leeb, W., Sharon, N.: Dihedral multi-reference alignment. IEEE Trans. Inform. Theory 68, 3489–3499 (2022)
Boyd, S., Vandenberghe, L.: Convex optimization. Press, Cambridge U (2004)
Cahill, J., Chen, X.: A note on scalable frames, SampTA (2013). arXiv:1301.7292
Cahill, J., Contreras, A., Contreras-Hip, A.: Complete set of translation invariant measurements with Lipschitz bounds. Appl. Comput. Harmon. Anal. 49, 521–539 (2020)
Cahill, J., Iverson, J.W., Mixon, D.G., Packer, D.: Group-invariant max filtering (2022). arXiv:2205.14039
Carlsson, M.: von Neumann’s trace inequality for Hilbert-Schmidt operators. Expo. Math. 39, 149–157 (2021)
Coxeter H.S.M.: The complete enumeration of finite groups of the form \(R_i^2=(R_iR_j)^{k_{ij}}=1\). J. London Math. Soc. s1-10, 21–25 (1935)
Godsil C.: Spectra of adjacency matrices of bipartite graphs. Math Stack Exch (2019). math.stackexchange.com/questions/3220357/spectra-of-adjacency-matrices-of-bipartite-graphs
Kane, R.M.: Reflection groups and invariant theory. Springer (2001)
Maslouhi, M., Okoudjou, K.A.: On root frames in \(\mathbb{R} ^d\). Sampl. Theory Signal Process. Data Anal. 21, 16 (2023)
Mixon, D.G., Qaddura Y.: Injectivity, stability, and positive definiteness of max filtering (2022). arXiv:2212.11156
Perry, A., Weed, J., Bandeira, A.S., Rigollet, P., Singer, A.: The sample complexity of multireference alignment. SIAM J. Math. Data Science 1, 497–517 (2019)
Varga, R.S.: Iterative matrix analysis. Springer (1962)
Acknowledgements
The authors thank Jameson Cahill, Dan Edidin, Joey Iverson, John Jasper, and Yousef Qaddura for enlightening discussions, Ben Blum-Smith for helping us simplify our proof of Lemma 5, and the anonymous referee for various comments and suggestions that improved the presentation of our results.
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Communicated by: Felix Krahmer
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Mixon, D.G., Packer, D. Max filtering with reflection groups. Adv Comput Math 49, 82 (2023). https://doi.org/10.1007/s10444-023-10084-6
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DOI: https://doi.org/10.1007/s10444-023-10084-6