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Case Western Reserve University DEPARTMENT OF MATHEMATICS |
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DEPARTMENT OF MATHEMATICS COLLOQUIA |
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Colloquium/Analysis SeminarProfessor Monika LUDWIG
Special ColloquiumProfessor Boris KASHIN
ColloquiumProfessor V.N. TEMLYAKOV ABSTRACT The talk is devoted to theoretical aspects of sparse approximation. The main motivation for the study of sparse approximation is that many real world signals can be well approximated by sparse ones. Sparse approximation automatically implies a need for nonlinear approximation and, in particular, for greedy approximation. We will discuss greedy approximation in different settings: with respect to bases and redundant dictionaries, in Hilbert and in Banach spaces.
Special ColloquiumProfessor Jenny BRYNJARSDOTTIR ABSTRACT Science-based simulation models are widely used to predict the behaviour of complex physical systems. It is also common to use observations of the physical systems to learn about the values of parameters within the model, a process usually called calibration. The values of parameters in the model may be of intrinsic scientific interest, so that learning about them contributes to the underlying science. Another reason for calibration is to improve the predictive performance of the simulator. In order to make appropriate use of observations of the physical system, however, it is important to recognise model discrepancy, the difference between reality and the simulator output. We illustrate through a simple example that an analysis that does not account for model discrepancy will lead to biased and over-confident parameter estimates and predictions. The challenge with incorporating model discrepancy in a statistical analysis of computer models is the confounding with calibration parameters, which will only be resolved with meaningful priors. For our simple example, we model the model-discrepancy via a Gaussian Process and demonstrate that by accounting for model discrepancy our prediction within the range of data is correct. However, only with realistic priors on model discrepancy do we uncover true parameters.
Special ColloquiumProfessor Yan DONGHUI ABSTRACT The tissue microarray (TMA) technology provides a way to efficiently evaluate large numbers of immunohistochemically-stained tissue images and has been successfully used in many applications in clinical outcome analysis, biomarker validations, and cancer research. Central to the use of TMAs is their scoring, which currently relies mainly on manual evaluation due to difficulties in quantifying the staining patterns (highly heterogeneous and not "localized" in position, size and shape). In response to concerns about the subjectivity and variability of manual TMA evaluation, we develop an automatic scoring algorithm--TACOMA. TACOMA effectively captures the statistical regularity in TMA images by statistics about the transition of gray levels, and a few "representative" image patches serve as prior information that allows TACOMA to focus on biologically relevant features and score in a similar way as the pathologists. Experiments show that TACOMA rivals pathologists in terms of both accuracy and reproducibility. Moreover, it is easily interpretable in that it reveals salient pixels in an image that are most relevant to scoring. I will also discuss work towards two challenges in the training of TACOMA, namely, label noise in the scoring by pathologists and issues on small training sample. We establish a sharp bound on the impact of data contamination to classification and give insights on the success of a thinning-based co-training strategy for small training sample.
Special ColloquiumProfessor Ick Hoon JIN ABSTRACT
Case Western Reserve University |
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