Reproducing Kernel Hilbert Spaces

Dr. Matthieu R Bloch

Wednesday October 20, 2021


  • Drop date: October 30, 2021

  • More office hours

    • Tuesdays 8am-9am on BlueJeans (
    • Come prepared!
  • Midterm 2: initially scheduled for Wednesday November 3, 2021

    • Moved to Monday November 8, 2021 (gives you weekend to prepare)
    • Coverage: everything since Midterm 1 (dont’ forget the fundamentals though), emphasis on regression

What’s on the agenda for today?

  • Last time:
    • Motivation for RKHS
    • Functional on Hilbert spaces
  • Today:
    • Reproducing Kernel Hilbert Spaces
  • Reading: Romberg, lecture notes 10

Linear functionals on Hilbert spaces

  • In what follows, \(\calF\) is a Hilbert space with scalar field \(\bbR\)

  • A functional \(F:\calF\to\bbR\) associates real-valued number to an element of a Hilbert space \(\calF\)

  • Notation can be tricky when the Hilbert space is a space of functions: \(F\) can act on a function \(f\in\calF\)

  • Examples

  • A functional \(F:\calF\to\bbR\) is continuous at \(x\in\calF\) if \[ \forall \epsilon>0\exists\delta>0\textsf{ such that } \norm[\calF]{x-y}\leq \delta\Rightarrow \abs{F(x)-F(y)}\leq\epsilon\qquad \forall x,y\in\calF \] If this is true for every \(x\in\calF\), \(F\) is continuous.

  • Warning: I wasn’t careful enough last time in the definition of continuity

    1. All norms are continuous functionals
    2. \(F:\calF\to\bbR:x\mapsto\dotp{x}{c}\) for some \(c\in\calF\) is continuous

Continuous linear functionals on Hilbert spaces

  • A functional \(F\) is linear if \(\forall a,b\in\bbR\) \(\forall x,y\in\calF\) \(F(ax+by) = aF(x)+bF(y)\).
  • Continuous linear functions are much more constrained than one would imagine

  • A linear functional \(F:\calF\to\bbR\) is bounded if there exists \(M>0\) such that \[ \forall x\in\calF\quad\abs{F(x)}\leq M\norm[\calF]{x} \]
  • A linear functional on a Hilbert space that is countinuous at \(0\) is bounded.
  • For a linear functional \(F:\calF\to\bbR\), the following statements are equivalent:

    1. \(F\) is continuous at 0
    2. \(F\) is continuous at some point \(x\in\calF\)
    3. \(F\) is continuous everywhere on \(\calF\)
    4. \(F\) is uniformly continuous everywhere on \(\calF\)

Representation of (continuous) linear functionals

  • Let \(F:\calF\to\bbR\) be a linear functional on an \(n\)-dimensional Hilbert space \(\calF\).

    Then there exists \(c\in\calF\) such that \(F(x)=\dotp{x}{c}\) for every \(x\in\calF\)

  • Linear functional over finite dimensional Hilbert spaces are continuous!

  • This is not true in infinite dimension

  • Let \(F:\calF\to\bbR\) be a continuous linear functional on a (possible infinite dimensional) separable Hilbert space \(\calF\).

    Then there exists \(c\in\calF\) such that \(F(x)=\dotp{x}{c}\) for every \(x\in\calF\)
  • If \(\set{\psi_n}_{n\geq 1}\) is an orthobasis for \(\calH\), then we can construct \(c\) above as \[ c\eqdef \sum_{n=1}^\infty F(\psi_n)\psi_n \]

Reproducing Kernel Hilbert Spaces

  • An RKHS is a Hilbert space \(\calH\) of real-valued functions \(f:\bbR^d\to\bbR\) in which the sampling operation \(\calS_\bftau:\calH\to\bbR:f\mapsto f(\bftau)\) is continuous for every \(\bftau\in\bbR^d\).

    In other words, for each \(\bftau\in\bbR^d\), there exists \(k_\bftau\in\calH\) s.t. \[ f(\bftau) = {\dotp{f}{k_\bftau}}_\calH\text{ for all } f\in\calH \]

  • The kernel of an RKHS is \[ k:\bbR^d\times\bbR^d\to\bbR:(\bft,\bftau)\mapsto k_{\bftau}(\bft) \] where \(k_\bftau\) is the element of \(\calH\) that defines the sampling at \(\bftau\).

  • A (separable) Hilbert space with orthobasis \(\set{\psi_n}_{n\geq 1}\) is an RKHS iff \(\forall \bftau\in\bbR^d\) \(\sum_{n=1}^\infty\abs{\psi_{n}(\tau)}^2<\infty\)

RKHS and non orthogonal basis

  • If \(\set{\phi_n}_{n\geq 1}\) is a Riesz basis for \(\calH\), we know that every \(x\in\calH\) can be written \[ x = \sum_{n\geq 1}\alpha_n\phi_n\textsf{ with } \alpha_n\eqdef\dotp{x}{\smash{\widetilde{\phi}_n}} \] where \(\set{\widetilde{\phi}_n}_{n\geq 1}\) is the dual basis.

  • A (separable) Hilbert space with Riesz basis \(\set{\phi_n}_{n\geq 1}\) is an RKHS with kernel \[ k(\bft,\bftau) =\sum_{n=1}^\infty \phi_n(\bftau)\widetilde{\phi}_n(\bft) \] iff \(\forall \bftau\in\bbR^d\) \(\sum_{n=1}^\infty\abs{\phi_{n}(\tau)}^2<\infty\)