Daniel Kaslovsky

Ph.D., Applied Mathematics



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Research


Massive quantities of data are produced by a wide range of scientific disciplines. While typically complex, high-dimensional, and noisy, such data sets often have a concise underlying structure. I am interested in developing new theoretical analyses and computational methods that utilize this inherent structure for efficient representation and processing of complex data. This work combines techniques and concepts from probability and geometry in high dimension, linear algebra, and statistics.


Computational Data Analysis & Random Sampling

As a NSF Postdoctoral Fellow for Transformative Computational Science using Cyberinfrastructure, my research focuses on the development of computational algorithms based on random sampling for efficient processing of compressible data. This work draws on recent advances in randomized numerical linear algebra and compressive sensing, both of which use random sampling to achieve great efficiency with near optimal accuracy for sparse and low-rank data models.


Geometric Sparsity in High Dimension

I am investigating the low-dimensional geometric structure of manifold-valued data embedded in a high-dimensional space. The "geometric sparsity" afforded by this low-dimensional structure allows for efficient parameterization of such data. For my dissertation research, I developed results for optimal approximation of the local tangent space from manifold samples corrupted by high-dimensional noise. I am particularly interested in applications of these results to image processing in biology and medicine. My thesis advisor was Professor Francois Meyer, with whom I continue to collaborate.


Adaptive Methods for Time-Frequency Analysis

Empirical Mode Decomposition (EMD) is an adaptive method for analysis of nonstationary data. In contrast with Fourier analysis, EMD relaxes the requirement of linear projection onto a fixed basis and thus produces a sparse decomposition from which a well-localized, time-varying frequency representation may be constructed. I am interested in the application of such data-driven, adaptive decompositions for the analysis of biomedical signals. As part of a collaboration with Professor Kenneth Wright and the University of Colorado Sleep and Chronobiology Laboratory, I have developed EMD-based tools for analyzing sleep EEG signals. These tools allow for new insight into neuronal activity and brain physiology that is otherwise unavailable using standard Fourier-based methods.




Please also see information on the NSF-IGERT program: Computational Optical Sensing and Imaging (COSI), in which I was a fellow while at the University of Colorado, Boulder.