Medical Data Analytic Seminar

In this talk, I will introduce very brief and fundamental ideas that underlie deep neural network from an applied mathematics perspective. At the heart of neural network, the concepts are from calculus, approximation theory, linear algebra, and numerical methods. In particular, I will go through the followings in detail: What is an artificial neural network and how is the network trained for solving partial differential equations?

Dimensional reduction is useful in exploratory data analysis. In many applications, the interesting features of “target” dataset may be obscured by high variance component from “background” dataset. In this situation, we hope to find the subspace that contains variance enriched in target dataset relative to background dataset. Contrastive PCA (cPCA) (Abid et al., 2018) is proposed for this setting. Usually the cPCA is solved by eigenvalue decomposition. However, it becomes impractical for high dimensional data. To the best of our knowledge, there is no computationally feasible algorithm for high dimensional cPCA. In this talk, we propose a geometric line search algorithm for it. Convergence analysis is provided. Numerical experiments are conducted to show its empirical performance.

Abstract

Several applications in optimization, image, and signal processing deal with data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p matrices with orthonormal columns. Some applications, like finding the Riemannian center of mass, require evaluating the geodesic distance between two arbitrary points on St(n,p). Since no explicit formula is known for computing the distance on St(n,p), one has to resort to numerical methods. In this talk, we will see how to use the shooting method, a classical numerical algorithm for solving initial value problems, to compute the distance on St(n,p). We will showcase three example applications in the contexts of shape analysis, summary statistics, and model order reduction.

Abstract

Abstract

The world population is expected to increase to approximately 11 billion by 2100. The aging population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010 to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will expand our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information and optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long-term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to Dementia in the elderly makes the cerebral environment notably worthy of investigation through numerical and computational modelling.

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