Finite Tight Frames

1
Finite Tight Frames

• Dejun Feng
• Tsinghua University
• Long Wang
• Southern Polytechnic State Univ.
• Yang Wang
• Georgia Institute of Technology

2
Finite Tight Frames

• Motivation
• Introduction
• Questions and Results
• An Algorithm
• Examples

3
Motivation

• Orthonormal Bases
• Orthonormal bases have played important roles in
  both mathematical theory and real applications.
  But they are not robust to data loses.
• In many applications, coding for signal
  transmission often requires bases with certain
  properties, such as redundancy, different lengths
  of bases vectors which othonormal bases dont
  have.

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Motivation

• Make multiple copies of each vector in an
  orthonormal basis.
• Too expensive, inefficient
• Search for better redundant bases Tight Frames
• resilience to additive noise,
• robustness to data loses,
• numerical stability of reconstruction
• greater freedom to capture signal characteristics.

5
Introduction

• Frames and Tight Frames
• Let be a Hilbert space. A set of
  vectors
• in is called a frame if there
  exist
• such that for any , we have
• The constants are called
  lower frame bound and upper frame bound.

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Introduction

• A frame is called a tight frame if
  .
• Equal-norm Frames
• A frame where all elements have the same norm is
  called an equal-norm frame.
• We only consider finite dimensional Hilbert space
  

7
Examples

• The union of two orthonormal bases is a tight
  frame with frame bound 2
• is a tight frame in .
• ----- Mercedes-Benz frame

8

More definitions

• Frame matrix
• A matrix is
  called a frame matrix (FM) if rank ( )
  
• is called a tight frame matrix (TFM) if
• for some
• is
  a frame matrix (resp. TFM) if and only if the
  column vectors of form a frame (resp.
  tight frame) of .

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More Definitions

• Condition number
• Let be a
  frame matrix. Let
• be the maximal
  and minimal eigenvalues of , respectively.
  Then
• is called the
  condition number of
• .
• is a TFM if and only if

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Questions

• Given vectors , how
  many vectors do we need to add in order to obtain
  a tight frame?
• If only a fixed number of vectors are allowed to
  be added, how small can we make the condition
  number to be?

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Main Results

• Theorem 1 (D.J. Feng, W Y. Wang)
• For any ,
  let
  
• Suppose that are
  all the eigenvalues of
• . Then for any vectors
  , the matrix
• 
  satisfies
• where

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Theorem 1(Continued)

• Furthermore, the equality can be attained by some
  
• . In particular, at most
  vectors are needed to make
  a TF.

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Lemmas

• Lemma 1.1 Let
  positive semi-
• definite Hermitian matrices. Let
• be the eigenvalues of
• be the eigenvalues of . Suppose that
• is positive semi-definite with
• for some integer Then

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Proof

• Lemma 1.1 Let
  positive semi-
• definite Hermitian matrices. Let
• be the eigenvalues of
• be the eigenvalues of . Suppose that
• is positive semi-definite with
• for some integer Then

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Proof

• Lemma 1.2 Let positive
  semi-definite Hermitian matrix with
  . Then there exists a matrix

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Questions

• Given vectors with
  equal norm 1, how many vectors do we need to add
  in order to obtain an equal-norm tight frame?

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Main Results

• Theorem 2(D. J. Feng, W Y. Wang)
• For any with equal
  norm 1, let
• . Suppose that
• are all the eigenvalues of . Denote
  by q the smallest integer greater than or equal
  to Then we can find
  such that
• form an
  equal-norm TF.

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Proof

• Lemma 2.1 Let be two real numbers with
  
• and let
• Then for any
• such that the
  eigenvalues of the
• matrix
• are exactly

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Proof

• Lemma 2.2 Let positive
  semi-definite Hermitian matrix with eigenvalues
• Let
  integer with
• Then for any
• we may construct a
• such that the eigenvalues of the matrix
• are exactly

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Questions

• For a given sequence
  
• is it possible to find a TF
  such that
• 
  ?
• If so, how to construct such a TF?
• For example, is it possible to find a TF
• in such that
• No.

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More Known Results

• Theorem 3 (P. Casazza, M. Leon J. C. Tremain)
• Let
  Then there exists a TF
• 
  such that
• 
  
• if and only if
• This is called the fundamental inequality.

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More Definitions

• A Householder matrix is a matrix of the form
• Any Householder matrix is unitary.
• If A is a TFM and U is a unitary matrix, then AU
  is a TFM.

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Main Results

• Theorem 4 (D. J. Feng, W Y. Wang)
• For a given sequence
  satisfying the fundamental
  inequality
• Inductively, we can construct a sequence
  of matrices by using a sequence of Householder
  matrices
• such that
  ,

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Main Results (Cont.)

• with
  possibly some columns
• interchanged and the matrices
  satisfies the following properties
• If we denote
  then
• Furthermore, for any

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Proof

• Lemma 4.1
• Let
• Then for any
• we can find
• In fact, can be found
  explicitly as follows

26
Lemmas
27
Lemmas

• Lemma 4.2
• Let and
• For any
• we can construct a Householder matrix
• Such that the column vectors
  of
• satisfy

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Algorithm

• Let be two positive integers be
  given and Let
  be given and satisfy the fundamental
  inequality
• Let

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Algorithm

• Multiply and let
• Repeat the following for
• Calculate the norm of the
• Compare
• then search
  for a column with norm great than or equal to
  and then swap it with (k1)-th column.
• then skip.
• then search
  for a column with norm less than or equal to
  and then swap it with (k1)-th column.
• , where
• 
  

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Algorithm

• , the
  result will be the TFM
• with prescribed norms.

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Example

• For
• Our algorithm yields the following TFM

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The End

• Thank you!