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Learning Manifolds with K-Means and K-Flats
Guillermo D Canas · Tomaso Poggio · Lorenzo Rosasco

Thu Dec 06 02:00 PM -- 12:00 AM (PST) @ Harrah’s Special Events Center 2nd Floor #None

We study the problem of estimating a manifold from random samples. In particular, we consider piecewise constant and piecewise linear estimators induced by k-means and k-flats, and analyze their performance. We extend previous results for k-means in two separate directions. First, we provide new results for k-means reconstruction on manifolds and, secondly, we prove reconstruction bounds for higher-order approximation (k-flats), for which no known results were previously available. While the results for k-means are novel, some of the technical tools are well-established in the literature. In the case of k-flats, both the results and the mathematical tools are new.

Author Information

Guille D Canas (Massachusetts Institute of Technology)
Tomaso Poggio (MIT)

Tomaso A. Poggio, is the Eugene McDermott Professor in the Dept. of Brain & Cognitive Sciences at MIT and the director of the new NSF Center for Brains, Minds and Machines at MIT of which MIT and Harvard are the main member Institutions. He is a member of both the Computer Science and Artificial Intelligence Laboratory and of the McGovern Brain Institute. He is an honorary member of the Neuroscience Research Program, a member of the American Academy of Arts and Sciences, a Founding Fellow of AAAI and a founding member of the McGovern Institute for Brain Research. Among other honors he received the Laurea Honoris Causa from the University of Pavia for the Volta Bicentennial, the 2003 Gabor Award, the Okawa Prize 2009, the AAAS Fellowship and the 2014 Swartz Prize for Theoretical and Computational Neuroscience. He is one of the most cited computational scientists with contributions ranging from the biophysical and behavioral studies of the visual system to the computational analyses of vision and learning in humans and machines. With W. Reichardt he characterized quantitatively the visuo-motor control system in the fly. With D. Marr, he introduced the seminal idea of levels of analysis in computational neuroscience. He introduced regularization as a mathematical framework to approach the ill-posed problems of vision and the key problem of learning from data. In the last decade he has developed an influential hierarchical model of visual recognition in the visual cortex. The citation for the recent 2009 Okawa prize mentions his “…outstanding contributions to the establishment of computational neuroscience, and pioneering researches ranging from the biophysical and behavioral studies of the visual system to the computational analysis of vision and learning in humans and machines.” His research has always been interdisciplinary, between brains and computers. It is now focused on the mathematics of learning theory, the applications of learning techniques to computer vision and especially on computational neuroscience of the visual cortex. A former Corporate Fellow of Thinking Machines Corporation and a former director of PHZ Capital Partners, Inc., he is a director of Mobileye and was involved in starting, or investing in, several other high tech companies including Arris Pharmaceutical, nFX, Imagen, Digital Persona and Deep Mind. Tomaso Poggio Eugene McDermott Professor Director NSF Science & Technology Center for Brains, Minds and Machines(CBMM) http://cbmm.mit.edu/ Core founding scientific advisor, MIT Quest for Intelligence McGovern Institute CSAIL (Computer Science and Artificial Intelligence Lab) Brain Sciences Department M.I.T., 46-5177B see http://whereis.mit.edu/?selection=46&Buildings=go 43 Vassar Street Cambridge, MA 02142 E-mail: tp@ai.mit.edu Phone: 617-253-5230 Fax: 617-253-2964 Web: http://cbcl.mit.edu/people/poggio/poggio-new.htm PoggioLab Web page: http://cbcl.mit.edu/

Lorenzo Rosasco (University of Genova- MIT - IIT)

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