Loosen Up! An introduction to frames.faculty.nps.edu/rgera/organized...

transcript

Frames

Vectors and VectorSpaces

OrthonormalBases

Frames

4 (Interrelated)Research AreasApplied Math

Linear Algebra

Geometry

Operator Theory

Frames forUndergraduates

Loosen Up! An introduction to frames.

Keri A. Kornelson

University of Oklahoma - Normankkornelson@math.ou.edu

Joint AMS/MAA MeetingsPanel: This could be YOUR graduate research!

New Orleans, LAJanuary 7, 2011

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Quick review of vector spaces, Rn:

Vectors in Rn are sometimes represented as columns:

x(1)x(2)

...x(n)

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Quick review of vector spaces, Rn:

Vectors in Rn are sometimes represented as columns:

x(1)x(2)

...x(n)

and sometimes as arrows:

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Norms and Dot Products2 vectors

x(1)x(2)

...x(n)

y(1)y(2)

...y(n)

DefinitionThe norm or length of x is

‖x‖ =

A vector with norm 1 is called a unit vector .

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Norms and Dot Products2 vectors

x(1)x(2)

...x(n)

y(1)y(2)

...y(n)

DefinitionThe norm or length of x is

‖x‖ =

A vector with norm 1 is called a unit vector .

DefinitionThe dot product or inner product of x and y is

〈x , y〉 =n∑

x(i)y(i).

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Orthogonal Vectors

DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Orthogonal Vectors

DefinitionTwo vectors are orthogonal if their inner (dot) product is zero,i.e. if their “arrows” are perpendicular.

DefinitionTwo vectors are orthonormal if they are orthogonal and areboth unit vectors.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Orthonormal Bases

DefinitionA collection of vectors {bi}

ni=1 is an orthonormal basis (ONB)

for Rn if the vectors are pairwise orthonormal and form a basis.

Some handy facts about ONBs:

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Orthonormal Bases

I The unique expansion coefficients are found by the dotproduct.

cibi =

〈x , bi〉bi

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Orthonormal Bases

I The unique expansion coefficients are found by the dotproduct.

cibi =

〈x , bi〉bi

I Parseval’s Identity:

‖x‖2 =

|〈x , bi〉|2

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

A Silly Example

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

A Silly Example

= 4b1 + 5b2 + 6b3

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

A Silly Example

= 4b1 + 5b2 + 6b3

〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

A Silly Example

= 4b1 + 5b2 + 6b3

〈x , b1〉 = 4, 〈x , b2〉 = 5, 〈x , b3〉 = 6.

‖x‖2 =√

42 + 52 + 62

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Orthonormal Bases, cont.

ONB’s are pretty restrictive...they all look alike somehow.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

I All vectors have norm 1.

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Linear Algebra

Geometry

Operator Theory

I The number of vectors n equals the dimension of thespace: Rn.

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Linear Algebra

Geometry

Operator Theory

I All the vectors are pairwise orthogonal.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

I There’s not much flexibility to tailor an ONB to a particularapplication, and there is no resilience to losses or errors indata reconstruction.

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OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Signal transmission

If we agree on an ONB {bi}ni=1, then I can just send you the

coefficients of x and you can find x .

In our silly example, I send you 4, 5, 6 and you can compute

x = 4b1 + 5b2 + 6b3 =

Same idea works for voice on a cell phone or pictures sentover the internet.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Signal transmission

If we agree on an ONB {bi}ni=1, then I can just send you the

coefficients of x and you can find x .

In our silly example, I send you 4, 5, 6 and you can compute

x = 4b1 + 5b2 + 6b3 =

Same idea works for voice on a cell phone or pictures sentover the internet.

If one data point gets lost using an ONB, there is noinformation about what it was.

4, ?, 6 −→ 4b1 + ?b2 + 6b3 =

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Loosen up!The concept of a frame for a vector space allows for morewiggle room than ONBs. The vectors are allowed to

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Stretch out

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Stretch out

Move around a bit

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Stretch out

Move around a bit

Even invite a few friends over!

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Frames

Recall Parseval’s Identity for ONB {bi}: ‖x‖2 =n∑

|〈x , bi〉|2

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Linear Algebra

Geometry

Operator Theory

Frames

|〈x , bi〉|2

DefinitionA frame for Rn is a collection of vectors {fi}k

i=1 that satisfy alooser condition than Parseval’s identity. There are constantsA,B > 0 (called frame bounds ) such that

A‖x‖2 ≤

|〈x , fi〉|2 ≤ B‖x‖2.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Frames

|〈x , bi〉|2

A‖x‖2 ≤

|〈x , fi〉|2 ≤ B‖x‖2.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Frames

|〈x , bi〉|2

A‖x‖2 ≤

|〈x , fi〉|2 ≤ B‖x‖2.

A frame is tight if A = B and Parseval if A = B = 1.

‖x‖2 =

|〈x , fi〉|2 (look familiar?)

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Handy facts about frames:

I In finite-dimensional spaces, the frames are exactly thespanning sets for the space.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Every ONB is a Parseval frame, but there are more!

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Every ONB is a Parseval frame, but there are more!

I Parseval frames also satisfy the reconstruction property ofONBs:

ci fi =k∑

〈x , fi〉fi

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.

Frames

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Linear Algebra

Geometry

Operator Theory

ProjectionsTheoremEvery frame is the projection of a basis for a larger space.Every Parseval frame is the projection of a orthonormal basisfor a larger space.

ExampleR

3 orthonormal basis projected onto the plane.

yields the Parseval frame with 3 equal-norm vectors for R2.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

4 frame-related research areas

1. Applied Math

2. Linear Algebra

3. Geometry

4. Operator Theory

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Sample Applied Math Research Problems

I Which frames have the best resilience to 1, 2, or moreerasures?

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Build tight frames which are tailored to a particularapplication.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Build the sparsest possible tight frame of givensize/redundancy.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

I Find an algorithm like Gram-Schmidt that generates tightframes from a given frame sequence.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Find an algorithm that numerically converges to a tightframe under given constraints (same norms, for example).

I Wavelet frames.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Synthesis operator, frame potential

Let {fi}ki=1 be a frame for Rn. We can create the n × k matrix S

which has the frame vectors as columns.

↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

I The frame {fi}ki=1 is Parseval if and only if the rows of S

are orthonormal.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

↑ ↑ ↑f1 f2 · · · fk↓ ↓ ↓

Theorem

I The frame {fi}ki=1 is Parseval if and only if the rows of S

are orthonormal.

I {fi}ki=1 is Parseval iff SS∗ is the identity matrix on R

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Sample Linear Algebra Research Problems

I Find a tight frame with k vectors for Rn, where the vectorsare all the same length.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Find a tight frame with k vectors for Rn, where the vectorsare all lie on an ellipsoid.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Find a tight frame with k vectors for Rn, where the vectorshave a given sequence of norms.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I These all relate to classical problems about writingoperators as sums of projections!

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

I These all relate to classical problems about writingoperators as sums of projections!

I Frame potential - a real quantity that is minimized at tightframes, simulating electromagnetic potential.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

R2 frames

Another way to think about a vector in R2:[

a cos θa sin θ

Theorem

A frame{[

ai cos θi

ai sin θi

i=1is a tight frame for R2 if and only if

a2i cos 2θi

a2i sin 2θi

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

R2 frames

Another way to think about a vector in R2:[

a cos θa sin θ

Theorem

A frame{[

ai cos θi

ai sin θi

i=1is a tight frame for R2 if and only if

a2i cos 2θi

a2i sin 2θi

Proof.Recall S is the 2 × k matrix with the frame vectors as columns,and the frame is tight iff SS∗ is a scalar multiple of the identity.Computing S and using some trigonometric identities gives theresult.

Frames

OrthonormalBases

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Linear Algebra

Geometry

Operator Theory

Sample Geometric Research Problems

I The theorem yields lots of facts about R2 tight frames —for example All 4-vector unit frames consist of 2 ONBs.

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Is there a similar kind of characterization for tight framesin 3 or 4 dimensions?

I Find/characterize equiangular equal-norm tight frames(related to packing problems).

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Kadison-Singer Problem (1951)

The Kadison-Singer problem in operator theory has been opensince 1951.

It has recently been shown equivalent to a variety of problemshaving to do with finite frames and finite matrices.

ExampleDoes there exist an ε > 0 and a natural number r such that forall equal-norm Parseval frames {fi}2n

i=1 for Rn , there is apartition {Aj}

rj=1 of {1, 2, . . . , 2n} such that {fi}i∈Aj has Bessel

bound ≤ 1 − ε for all j = 1, 2, . . . , r .

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

For further reading...

Frames for UndergraduatesDeguang Han, University of Central Florida, Orlando, FL, Keri Kornelson, Grinnell College, IA, David Larson, Texas A&M University, College Station, TX, and Eric Weber, Iowa State University, Ames, IA

Student Mathematical Library

2007; 295 pp; softcover

Volume: 40

ISBN: 978-0-8218-4212-6

List Price: US$49

Member Price: US$39

Order Code: STML/40

Frames are a generalization of bases.

�eir study has a powerful impact in both

abstract and applied settings. �is book

provides an undergraduate-level introduc-

tion to the theory of frames, primarily in

finite-dimensional Hilbert spaces.

Instructional Venues:

• A special topics course about

frames and bases.

• A second linear algebra course.

• A resource for an undergraduate

research activity.

AMERICAN MATHEMATICAL SOCIETY

For many more publications of interest,

visit the AMS Bookstore

www.ams.org/bookstore

1-800-321-4AMS (4267), in the U. S. and Canada, or 1-401-455-4000 (worldwide); fax:1-401-455-4046; email:

cust-serv@ams.org. American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

Where you might go to study frames.... . . in no particular order:

I University of Oklahoma

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Maryland - also check out the NorbertWeiner Center

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Texas A& M University

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Missouri

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Cincinnati

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Colorado

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Iowa

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Iowa State University

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Houston

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OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I University of Oregon

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Georgia Institute of Technology

Frames

OrthonormalBases

Frames

Linear Algebra

Geometry

Operator Theory

I Georgia Institute of Technology

I Vanderbilt University

Loosen Up! An introduction to frames.faculty.nps.edu/rgera/organized...

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