The current data-age is witnessing an unprecedented confluence of disciplines. A single data analysis task can demand expertise in computer science, statistics, functional analysis, optimization, or more. But what aspects of data are driving this rich interaction? We may single out at least two: size and form.
We hear a lot about "size" but less about "form" of data. Today, I will talk more about form, in particular about geometric form and structure. Our motivation lies in a large number of applications whose data are not merely vectors, but richer objects such as matrices, strings, functions, graphs, trees, etc. Dealing with such data in their "intrinsic representation" raises deep mathematical and algorithmic concerns replete with open problems.
I illustrate these ideas concretely by describing them for data analysis with positive definite matrices. These matrices enjoy great importance in machine learning, statistics, and numerous other areas; they lie at the heart of modern convex optimization; and their incredibly rich geometry enables our advances in "geometric data analysis", a rapidly growing area of research. Indeed, exploiting inexactness and geometry we can even solve some difficult nonconvex optimization problems in geometric data analysis.
Time permitting, I will also mention some fascinating connections of our work well-beyond machine learning to areas such as game-theory, nonlinear dynamics, signal processing, algebra, and quantum information theory.
Suvrit Sra is a Sr. Research Scientist at the Max Planck Institute for Intelligent Systems, in Tübingen, Germany. He obtained Ph.D. in Computer Science from the University of Texas at Austin in 2007. In Spring 2013 he was visiting faculty at UC Berkeley (EECS), and currently he is a visiting faculty member in the Machine Learning Department at Carnegie Mellon University. His research is dedicated to bridging a number of mathematical areas (such as geometry, analysis, noncommutative algebra, convex analysis, matrix analysis, statistics, optimization, etc.) with data-driven real-world applications. His work has won several awards; most notably the "SIAM 2011 Outstanding Paper Prize". He regularly organizes a workshops on "Optimization for Machine Learning" at the Neural Information Processing Systems (NIPS) conference, and has recently (co)-edited a book with the same title.
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