@article{Multiresolution:2172,
      recid = {2172},
      author = {Eskreis-Winkler, Jonathan Mark},
      title = {Multiresolution Analysis on Discrete Spaces},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2020-03},
      pages = {193},
      abstract = {It continues to be much cheaper to store data than to  analyze it. This state of data analysis motivates methods  that make minimal assumptions in order to reduce the  complexity of data. In order to address scalability in  certain applied, network-based problems, we bring together  three distinct fields: multiresolution analysis (MRA),  hyperbolic embeddings, and community detection. Together,  the work in these fields provides a path forward for  certain difficult problems in large-scale data analysis. We  provide a theoretical background, a description of our  contributions, and a number of applications to show the  viability of our ideas in a real-world setting.

The  primary contributions of this dissertation are threefold.  First, we draw a connection between a class of  currently-used methods in computational MRA to hyperbolic  representations of data. This connection allows us to  suggest a rationale for when different MRA methods are  appropriate, and to define related tree-based kernels for  wavelet construction.

Second, we broaden the existing work  describing the mechanics of how the multiresolution matrix  factorization (MMF) summarizes data. We look at this MRA  framework from two perspectives. First, we consider how MMF  may be viewed as a hyperbolic embedding. Second, we  consider the way MMF may be seen as a regularization  operator. We show that for certain graphs, the  regularization imposed by MMF enables methods to perform  more accurate inference than when no regularization is  used. This work fits within the existing field of high  dimensional statistics where regularization may turn a  massive problem into a manageable one.

Third, we show the  way that MMF, and other similar multiresolution methods,  may be used efficiently in unsupervised and semisupervised  settings. By taking advantage of the localization of  wavelets in time and frequency space of graphs, we are able  to detect complex, multiscale, overlapping community  structures, as well as combine network structure with the  small amounts of labeling for semisupervised learning.},
      url = {http://knowledge.uchicago.edu/record/2172},
      doi = {https://doi.org/10.6082/uchicago.2172},
}