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Learning in Hilbert vs. Banach Spaces: A Measure Embedding Viewpoint
Bharath Sriperumbudur · Kenji Fukumizu · Gert Lanckriet

Tue Dec 13 08:45 AM -- 02:59 PM (PST) @

The goal of this paper is to investigate the advantages and disadvantages of learning in Banach spaces over Hilbert spaces. While many works have been carried out in generalizing Hilbert methods to Banach spaces, in this paper, we consider the simple problem of learning a Parzen window classifier in a reproducing kernel Banach space (RKBS)---which is closely related to the notion of embedding probability measures into an RKBS---in order to carefully understand its pros and cons over the Hilbert space classifier. We show that while this generalization yields richer distance measures on probabilities compared to its Hilbert space counterpart, it however suffers from serious computational drawback limiting its practical applicability, which therefore demonstrates the need for developing efficient learning algorithms in Banach spaces.

Author Information

Bharath Sriperumbudur (The Pennsylvania State University)
Kenji Fukumizu (Institute of Statistical Mathematics / Preferred Networks / RIKEN AIP)
Gert Lanckriet (U.C. San Diego)

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