# Research

### Information geometry and applications

Information geometry consists in studying the intrinsic geometry of probability distribution families—regarded as statistical manifolds—using the tools from differential geometry. For instance, it is knwon that the metric given by the Fisher information matrix is essentially the unique Riemannian metric invariant under sufficient statistics on such manifolds; this metric induces a natural notion of distance between probability distributions from the same family. In this project, we aim to apply information-geometric methods to problems in signal processing and machine learning.

This is a master's project under supervision of Prof. Sueli I. R. Costa at the Laboratory of Discrete Mathematics and Codes (LMDC), Institute of Mathematics, Statistics and Scientific Computing (IMECC), Unicamp. This project is supported by FAPESP (grant no. 21/04516-8).

### Connections between data compression and Bayesian inference

We study the connections between data compression and Bayesian inference, particularly through the so-called context-tree algorithms (CTW and CTM). For different purposes in communications, such as feedback and storage, it is necessary to represent the channel state information (CSI) in an ‘economical’ way. We propose a context-tree-based approach for compressing time-varying CSI, which combines lossy vector quantisation, by means of data-adapted companders, with lossless compression, based on symbol probabilities estimated by a context-tree model.

This project was developped under supervision of Prof. Sheng Yang, at the Laboratory of Signals and Systems (L2S), CentraleSupélec.

##### Journal paper
• H.K. Miyamoto and S. Yang, “Context-Tree-Based Lossy Compression and Its Application to CSI Representation”, IEEE Transactions on Communications, early access, 2022, doi: 10.1109/TCOMM.2022.3173002 [IEEEXplore] [arXiv] [BibTeX]
##### Conference paper
• H.K. Miyamoto and S. Yang, “A CSI Compression Scheme Using Context Trees”, International Zurich Seminar on Information and Communication (IZS), Zurich, 2022, pp. 24-28. doi: 10.3929/ethz-b-000535273. [Link] [BibTeX]

### Spherical codes by Hopf foliations

Spherical codes are discrete sets of points on the surface of an Euclidean sphere, and have several applications in sicences and engineering. Problems with spherical codes involve finding optimal distributions of points relative to some parameter of interest, such as the minimum distance between two points. We address the spherical packing problem by exploiting the Hopf foliations in dimensions $$2^k$$, which yield a method for constructing spherical codes in such dimensions, for a given minimum distance. This procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between coding rate and effective constructiveness with low encoding complexity.

This was a scientific initiation project developped from 2016 to 2018 at the Institute of Mathematics, Statistics and Scientific Computing (IMECC), Unicamp, under supervision of Prof. Henrique N. Sá Earp and Prof. Sueli I. R. Costa. The project was supported by FAPESP (grant no. 16/05126-0) as part of the thematic project Security and reliability of information: theory and practice (grant no. 13/25977-7), of which Prof. Costa was one of the main researchers.

##### Journal paper
• H.K. Miyamoto, S.I.R. Costa and H.N. Sá Earp, “Constructive spherical codes by Hopf foliations”, IEEE Transactions on Information Theory, vol. 67, no. 12, pp. 7925-7939, dec. 2021, doi: 10.1109/TIT.2021.3114094. [IEEEXplore] [arXiv] [BibTeX]
##### Conference papers
• H.K. Miyamoto, H.N. Sá Earp and S.I.R. Costa, “Constructive spherical codes in $$2^k$$ dimensions” IEEE International Symposium on Information Theory (ISIT), Paris, 2019, pp. 1612-1616. doi: 10.1109/ISIT.2019.8849464. [IEEEXplore] [BibTeX]
• H.K. Miyamoto, H.N. Sá Earp and S.I.R. Costa, “Construção de códigos esféricos usando a fibração de Hopf”, Jornada Nacional de Iniciação Científica (JNIC), 70ª Reunião Anual da SBPC, UFAL, Maceió, 2018. Anais: resumos da 70ª Reunião Anual da SBPC. São Paulo: SBPC, 2018. ISSN: 2176-1221. (in Portuguese) [Link]
• H.K. Miyamoto, H.N. Sá Earp and S.I.R. Costa, “Construction of spherical codes using the Hopf fibration”, XXV Congresso de Iniciação Científica da Unicamp, Campinas, 2017. doi: 10.19146/pibic-2017-78809. [Link]
##### Presentations
• Construction of spherical codes using the Hopf fibration. International Congress of Mathematicians (ICM) (short communication). Rio de Janeiro, 2018. [Abstract]
• A recursive algorithmic construction for spherical codes in dimensions $$\mathbb{R}^{2^k}$$. Latin American Week on Coding and Information (LAWCI). IMECC/Unicamp, Campinas, 2018. [Poster]
• Construção de códigos esféricos usando a fibração de Hopf. Seminários DivulgaMat, IMECC/Unicamp, Campinas, 2018. [Link]
• Construção de códigos esféricos usando a fibração de Hopf. Jornada de Matemática, Matemática Aplicada e Educação Matemática (J3M), UFPR, Curitiba, 2017. [Link]
• Construção de códigos esféricos usando a fibração de Hopf. Mostra Científica IEEE Day, FEEC/Unicamp, Campinas, 2017. [Link]