# ℓ-norm and its Application to Learning Theory

ℓ-norm and its Application to Learning Theory We investigate connections between an important parameter in the theory of Banach spaces called the ℓ-norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the ℓ-norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the ℓ-norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# ℓ-norm and its Application to Learning Theory

, Volume 5 (2) – Oct 3, 2004
15 pages

/lp/springer_journal/norm-and-its-application-to-learning-theory-EI2qYR7mUk
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1011430329313
Publisher site
See Article on Publisher Site

### Abstract

We investigate connections between an important parameter in the theory of Banach spaces called the ℓ-norm, and two properties of classes of functions which are essential in Learning Theory – the uniform law of large numbers and the Vapnik–Chervonenkis (VC) dimension. We show that if the ℓ-norm of a set of functions is bounded in some sense, then the set satisfies the uniform law of large numbers. Applying this result, we show that if X is a Banach space which has a nontrivial type, then the unit ball of its dual satisfies the uniform law of large numbers. Next, we estimate the ℓ-norm of a set of {0,1}-functions in terms of its VC dimension. Finally, we present a `Gelfand number' like estimate of certain classes of functions. We use this estimate to formulate a learning rule, which may be used to approximate functions from the unit balls of several Banach spaces.

### Journal

PositivitySpringer Journals

Published: Oct 3, 2004

### References

• Scale sensitive dimensions, uniform convergence and learnability
Alon, N.; Ben-David, S.; Cesa-Bainchi, N.; Haussler, D.

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