Abstract
Let \(\mathbf d=(d_{j})_{j\in \mathbb {I}_{m}}\in \mathbb {N}^{m}\) be a finite sequence (of dimensions) and \(\alpha =(\alpha _{i})_{i\in \mathbb {I}_{n}}\) be a sequence of positive numbers (of weights), where \(\mathbb {I}_{k}=\{1,\ldots ,k\}\) for \(k\in \mathbb {N}\). We introduce the (α, d)-designs, i.e., m-tuples \({\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}\) such that \(\mathcal F_{j}=\{f_{ij}\}_{i\in \mathbb {I}_{n}}\) is a finite sequence in \(\mathbb {C}^{d_{j}}\), \(j\in \mathbb {I}_{m}\), and such that the sequence of non-negative numbers \((\|f_{ij}\|^{2})_{j\in \mathbb {I}_{m}}\) forms a partition of αi, \(i\in \mathbb {I}_{n}\). We characterize the existence of (α, d)-designs with prescribed properties in terms of majorization relations. We show, by means of a finite step algorithm, that there exist (α, d)-designs \({\Phi }^{\text {op}}=(\mathcal {F}_{j}^{\text {op}})_{j\in \mathbb {I}_{m}}\) that are universally optimal; that is, for every convex function \(\varphi :[0,\infty )\rightarrow [0,\infty )\), then Φop minimizes the joint convex potential induced by φ among (α, d)-designs, namely
for every (α, d)-design \({\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}\), where \(\text {P}_{\varphi }(\mathcal F)=\text {tr}(\varphi (S_{\mathcal {F}}))\); in particular, Φop minimizes both the joint frame potential and the joint mean square error among (α, d)-designs. We show that in this case, \(\mathcal {F}_{j}^{\text {op}}\) is a frame for \(\mathbb {C}^{d_{j}}\), for \(j\in \mathbb {I}_{m}\). This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.
Similar content being viewed by others
References
Antezana, J., Massey, P., Ruiz, M., Stojanoff, D.: The Schur-Horn theorem for operators and frames with prescribed norms and frame operator. Illinois J. Math. 51, 537–560 (2007)
Benac, M.J., Massey, P., Stojanoff, D.: Convex potentials and optimal shift generated oblique duals in shift invariant spaces. J. Fourier Anal. Appl. 23(2), 401–441 (2017)
Benac, M.J., Massey, P., Stojanoff, D.: Frames of translates with prescribed fine structure in shift invariant spaces. J. Funct. Anal. 271(9), 2631–2671 (2016)
Benedetto, J.J., Fickus, M.: Finite normalized tight frames. Frames. Adv. Comput. Math. 18(2-4), 357–385 (2003)
Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)
Bodmann, B.G., Paulsen, V.I.: Frames, graphs and erasures. Linear Algebra Appl. 404, 118–146 (2005)
Casazza, P.G.: The art of frame theory. Taiwanese J. Math. 4(2), 129–201 (2000)
Casazza, P.G.: Custom building finite frames. In: Wavelets, Frames and Operator Theory, Volume 345 of Contemp, pp 61–86. Math., Amer. Math. Soc., Providence (2004)
Casazza, P.G., Fickus, M., Kovacevic, J., Leon, M.T., Tremain, J.C.: A Physical Interpretation of Tight Frames. Harmonic Analysis and Applications. Appl. Numer. Harmon. Anal., pp 51–76. Birkhäuser, Boston (2006)
Cahill, J., Fickus, M., Mixon, D.G., Poteet, M.J., Strawn, N.: Constructing finite frames of a given spectrum and set of lengths. Appl. Comput. Harmon. Anal. 35, 52–73 (2013)
Casazza, P.G., Leon, M.T.: Existence and construction of finite frames with a given frame operator. Int. J. Pure Appl. Math. 63(2), 149–157 (2010)
Casazza, P.G., Kutyniok, G. (eds.): Finite Frames: Theory and Applications. Birkhauser, Boston (2012). xii + 483 pp
Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003). xxii+ 440 pp
Dhillon, I.S., Heath, R.W. Jr., Sustik, M.A., Tropp, J.A.: Generalized finite algorithms for constructing Hermitian matrices with prescribed diagonal and spectrum. SIAM J. Matrix Anal. Appl. 27(1), 61–71 (2005)
Dykema, K., Freeman, D., Kornelson, K., Larson, D., Ordower, M., Weber, E.: Ellipsoidal tight frames and projection decomposition of operators. Illinois J. Math. 48, 477–489 (2004)
Feng, D.J., Wang, L., Wang, Y.: Generation of finite tight frames by Householder transformations. Adv. Comput. Math. 24, 297–309 (2006)
Fickus, M., Marks, J., Poteet, M.: A generalized Schur-Horn theorem and optimal frame completions. Appl. Comput. Harmon. Anal. 40(3), 505–528 (2016)
Fickus, M., Mixon, D.G., Poteet, M.J.: Frame completions for optimally robust reconstruction. In: Proceedings of SPIE 8138: 81380Q/1-8 (2011)
Hassibi, B., Sharif, M.: Fundamental limits in MIMO broadcast channels. IEEE J. Selected Areas Commun. 25(7), 1333–1344 (2007)
Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377, 31–51 (2004)
Kornelson, K.A., Larson, D.R.: Rank-One Decomposition of Operators and Construction of Frames. Wavelets, Frames and Operator Theory, Contemp. Math., vol. 345, pp 203–214. Amer. Math. Soc., Providence (2004)
Massey, P., Rios, N., Stojanoff, D.: Frame completions with prescribed norms: Local minimizers and applications. Adv. Comput. Math. (in press)
Massey, P., Ruiz, M.A.: Tight frame completions with prescribed norms. Sampl. Theory Signal Image Process. 7(1), 1–13 (2008)
Massey, P., Ruiz, M.: Minimization of convex functionals over frame operators. Adv. Comput. Math. 32(2), 131–153 (2010)
Massey, P., Ruiz, M., Stojanoff, D.: Optimal dual frames and frame completions for majorization. Appl. Comput Harmon. Anal. 34(2), 201–223 (2013)
Massey, P.G., Ruiz, M.A., Stojanoff, D.: Optimal frame completions. Adv. Comput. Math. 40, 1011–1042 (2014)
Massey, P., Ruiz, M., Stojanoff, D.: Optimal frame completions with prescribed norms for majorization. J. Fourier Anal. Appl. 20(5), 1111–1140 (2014)
Acknowledgments
We would like to thank the reviewers for several comments and suggestions that helped us to improve the contents of this manuscript.
Funding
This study was partially supported by CONICET (PICT ANPCyT 1505/15) and Universidad Nacional de La Plata (UNLP 11X829).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Holger Rauhut
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Benac, M.J., Massey, P., Ruiz, M. et al. Optimal frame designs for multitasking devices with weight restrictions. Adv Comput Math 46, 22 (2020). https://doi.org/10.1007/s10444-020-09762-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-020-09762-6