Phd Thesis Defence by Mr. Shankhadeep Mondal on 26th March

Title: A study of optimal dual frames for erasures
Name : Mr. Shankhadeep Mondal
Date: 26-3-2024
Time:4:00 P.M
Venue: PSB 3201

Abstract

Frames have demonstrated their significant utility in data transmission by virtue of their redundant features, which facilitate the reconstruction of data with minimal errors even in the presence of erasures and distortions. These applications have naturally prompted inquiries into the identification of optimal dual frames or dual pairs that can provide superior approximations to the original signals. We analyse the erasure problem in the following two broad contexts:

1. The existence and characterization of a dual pair minimizes, among all dual pairs, the maximum probabilistic error operator's measure obtained by considering the various possible locations of a fixed number of erasures.
2. For a given frame, the existence and characterization of a dual frame minimizes, among all the dual frames of the given frame, the maximum probabilistic error operator's measure.

We identify optimal dual pairs and dual frames using diverse measures of the error operator, including the operator norm, Frobenius norm, spectral radius, and numerical radius. Our investigation also extends to probabilistic erasure models, exploring their behaviour under operator norm, spectral radius, and their averages. Our investigations reveal that equiangular tight frames and their canonical duals often exhibit optimality among the dual pairs.

References

1. P. Devaraj and Shankhadeep Mondal, Spectrally optimal dual frames for erasures Proceedings-Mathematical Sciences 133 (2023), 24.
2. S. Arati, P. Devaraj and Shankhadeep Mondal, Optimal dual frames and dual pairs for probability modelled erasures, Advances in Operator Theory, (2024), 1-30
3. S. Arati, P. Devaraj and Shankhadeep Mondal, Probabilistic spectral-operator-averaged optimal dual frames for erasures. (preprint)
4. S. Arati, P. Devaraj and Shankhadeep Mondal, Optimal dual frame pairs for erasures. (preprint)
5. S. Pehlivan, D. Han, and R. Mohapatra, Linearly connected sequences and spectrally optimal dual frames for erasures, Journal of Functional Analysis 265 (2013), 2855–2876
6. R. B. Holmes and V. I. Paulsen, Optimal frames for erasures, Linear Algebra and its Applications 377 (2004), 31–51.
7. J. Leng, D. Han, and T. Huang, Optimal dual frames for communication coding with probabilistic erasures, IEEE Transactions on Signal Processing 59 (2011), 5380–5389.