Continuous analogues of matrix factorizations

Continuous analogues of matrixfactorizationsNASC seminar, 9th May 2014

Alex TownsendDPhil student

Mathematical InstituteUniversity of Oxford

(joint work with Nick Trefethen)

Many thanks to Gil Strang, MIT. Work supported by supported by EPSRC grant EP/P505666/1.

IntroductionDiscrete vs. continuous

v = column vector

A = tall skinny matrix

A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]

f(x , y)∫f(s, y)v(s) ds

?

chebfun

quasimatrix

chebfun2

chebop

cmatrix

[Battles & Trefethen, 04]

[Stewart, 98]

[T. & Trefethen, 13]

[Driscoll, Bornemann, &Trefethen, 08]


Interested in continuous analogues rather than infinite analogues.

Aside: Infinite analogues are Schmidt, Wiener–Hopf, infinite-dimensional QR, etc.

Alex Townsend @ Oxford 2/24


v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]


?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]








v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]

f(x , y)

∫f(s, y)v(s) ds

?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]








v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]


?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]








v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]


?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]








v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]


?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]








v = column vector


A = square matrix

Av

SVD, QR, LU, Chol

f(x)

[ f1(x) | · · · | fn(x) ]


?

chebfun

quasimatrix

chebfun2

chebop

cmatrix


[Stewart, 98]







IntroductionMatrices, quasimatrices, cmatrices

..

matrix

.

quasimatrix

.

cmatrix

.

m × n

.

[a, b] × n

.

[a, b] × [c,d]

..................

A cmatrix is a continuous function of (y , x) ∈ [a, b] × [c,d].


IntroductionMatrices vs. cmatrices

An m × n matrix: entries indexed by {1, . . . ,m} × {1, . . . , n}An [a,b] × [c, d] cmatrix: entries indexed by [a, b] × [c,d]

{1, . . . ,m} subset of R Question

Well-ordered Not well-ordered by < What is the 1st column?

Successor No successor What is the next column?

A null set Null subsets What sparsity makes sense?

Finite Infinite Convergence?

Three heroes: Smoothness pivoting εmach









Three heroes:

Smoothness pivoting εmach









Three heroes: Smoothness

pivoting εmach









Three heroes: Smoothness pivoting

εmach









Three heroes: Smoothness pivoting εmach


Singular value decompositionMatrix factorization

A = UΣVT , Σ = diagonal, U,V = orthonormal columns

..................................................A

.U

.Σ

.

VT

Exists: SVD exists and is (almost) uniqueApplication: A best rank r approx. is Ar = 1st r terms (in 2- & F-norm)Separable model: A =

∑nj=1 σjujvT

j is a sum of outer products

Computation: Bidiagonalize then iterate [Golub & Kahan (1965)]




..................................................A

.U

.Σ

.

VT

Exists: SVD exists and is (almost) unique

Application: A best rank r approx. is Ar = 1st r terms (in 2- & F-norm)Separable model: A =

∑nj=1 σjujvT






..................................................A

.U

.Σ

.

VT

Exists: SVD exists and is (almost) uniqueApplication: A best rank r approx. is Ar = 1st r terms (in 2- & F-norm)

Separable model: A =∑n

j=1 σjujvTj is a sum of outer products





..................................................A

.U

.Σ

.

VT


∑nj=1 σjujvT






..................................................A

.U

.Σ

.

VT


∑nj=1 σjujvT




Singular value decompositionContinuous analogue


..

u1

.

u2

....

σ1

.

σ2

....

vT2

.

vT1

....A

.U

.Σ

.

VT

.

At leastformally

Exists: SVD exists if A is continuous and is (almost) unique [Schmidt 1907]Application: A best rank r approx. is fr = 1st r terms (L2-norm) [Weyl 1912]Separable model: A =

∑∞j=1 σjujvT

j is a sum of “outer products”Computation: Avoid bidiagonalization




..

u1

.

u2

....

σ1

.

σ2

....

vT2

.

vT1

....A

.U

.Σ

.

VT

.

At leastformally

Exists: SVD exists if A is continuous and is (almost) unique [Schmidt 1907]

Application: A best rank r approx. is fr = 1st r terms (L2-norm) [Weyl 1912]Separable model: A =

∑∞j=1 σjujvT





..

u1

.

u2

....

σ1

.

σ2

....

vT2

.

vT1

....A

.U

.Σ

.

VT

.

At leastformally

Exists: SVD exists if A is continuous and is (almost) unique [Schmidt 1907]Application: A best rank r approx. is fr = 1st r terms (L2-norm) [Weyl 1912]

Separable model: A =∑∞

j=1 σjujvTj is a sum of “outer products”

Computation: Avoid bidiagonalization




..

u1

.

u2

....

σ1

.

σ2

....

vT2

.

vT1

....A

.U

.Σ

.

VT

.

At leastformally


∑∞j=1 σjujvT

j is a sum of “outer products”

Computation: Avoid bidiagonalization




..

u1

.

u2

....

σ1

.

σ2

....

vT2

.

vT1

....A

.U

.Σ

.

VT

.

At leastformally


∑∞j=1 σjujvT



Singular value decompositionAbsolute and uniform convergence of the SVD

TheoremLet A be an [a,b] × [c, d] cmatrix that is (uniformly) Lipschitz continuous in bothvariables. Then the SVD of A exists, the singular values are unique with σj → 0as j →∞, and

A =

∞∑j=1

σjujvTj ,

where the series is uniformly and absolutely convergent to A.

Proof.See [Schmidt 1907], [Hammerstein 1923], and [Smithies 1937]. �

If A satisfies the assumptions of the theorem, then A = UΣVT .


Singular value decompositionAlgorithm

1. Compute A = QARA

2. Compute quasimatrix QR,RT

A = QRRR

(Householder triangularization of a quasimatrix [Trefethen 08])

3. Compute SVD

..

≈.

A.

QA.

RA

..=

.......RT

A

.QR

.RR

........=

......................RR

.U

.Σ

.VT

A = (QAV)Σ(QRU)T

This is a continuous analogue of a discrete algorithm [Ipsen 90].


Singular value decompositionRelated work

Erhard Schmidt

Autonne, Bateman,Hammerstein, Kellogg,Picard, Smithies, Weyl

James Mercer

Aizerman, Braverman,Konig, Rozonoer

Carl Eckart & Gail Young

Golub, Hestenes, Kahan,Kogbetliantz, Reinsch


LU decompositionMatrix factorization

A = P−1LU, P = permutation, L = unit lower-triangular, U = upper-triangular.

..A

.P−1L

.U

.........................................................................................................

P−1L = “psychologically” lower-triangular

Exists: It (almost) exists and with extra conditions is (almost) uniqueApplication: Used to solve dense linear systems Ax = bSeparable model: A =

∑nj=1 `juT

j is a sum of outer products [Pan 2000]

Computation: Gaussian elimination with pivoting


LU decompositionContinuous analogue

A = LU, L = unit lower-triangular, U = upper-triangular.

..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U

Exists: It (usually) exists and with extra conditions is (almost) uniqueApplication: Can be used to “solve” integral equationsSeparable model: A =

∑∞j=1 `juT

j is a sum of outer productsComputation: Continuous analogue of GECP (GE with complete pivoting)


LU decompositionComputation

The standard point of view:

A different point of view:

..A

.P−1L

.U

.........................................................................................................

A ←− A − A(j, :)A(:, k )/A(j, k ) (GE step for matrices)

A ←− A − A(y0, :)A(:, x0)/A(y0, x0) (GE step for functions)

Each step of GE is a rank-1 update. We use complete pivoting.

Pivoting orders the columns and rows.




A different point of view:..

A.

P−1L.

U.........................................................................................................








A different point of view:..

A.

P−1L.

U.........................................................................................................






LU decompositionWhat is a triangular quasimatrix?

..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U

L = unit lower-triangularU = upper-triangular

What is a lower-triangular quasimatrix?

Red dots = 0’s, blue squares = 1’sPosition of 0’s is determined by pivoting strategyForward substitution has a continuous analogueMore precisely, L is lower-triangular wrt y1, y2, . . .



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1

y1

y2



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1

y1

y2

y1

y2

y3



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1

y1

y2

y1

y2

y3

y1

y2

y3

y4



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1

y1

y2

y1

y2

y3

y1

y2

y3

y4

y1

y2

y3

y4

y5



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U




y1

y1

y2

y1

y2

y3

y1

y2

y3

y4

y1

y2

y3

y4

y5

y1

y2

y3

y4

y5



..

`1

.

`2

....

uT1

.

uT2

....A

.L

.U


What is a lower-triangular quasimatrix?Red dots = 0’s, blue squares = 1’sPosition of 0’s is determined by pivoting strategyForward substitution has a continuous analogueMore precisely, L is lower-triangular wrt y1, y2, . . .

y1

y1

y2

y1

y2

y3

y1

y2

y3

y4

y1

y2

y3

y4

y5

y1

y2

y3

y4

y5


LU decompositionAbsolute and uniform convergence of LU

TheoremLet A be an [a,b] × [c, d] continuous cmatrix. Suppose A(·, x) is analytic in the“stadium” of radius 2ρ(b − a) about [a, b] for some ρ > 1 where it is bounded inabsolute value by M (uniformly in x). Then

A =

∞∑j=1

`juTj ,

where the series is uniformly and absolutelyconvergent to A. Moreover,∥∥∥∥∥∥∥A −

k∑j=1

`juTj

∥∥∥∥∥∥∥∞

≤ Mρ−k .

a b

2ρ(b−a)

stadium


LU decompositionA Chebfun2 application

Low rank function approximation

A = chebfun2(@(x,y) cos(10*(x.ˆ2+y))+sin(10*(x+y.ˆ2)));

contour(A, ’.’)

• = pivot locationRank = 125 Rank = 65 Rank = 28

Rank = 5 Rank = 33 Rank = 2

A(y , x) ≈k∑

j=1

`j(y)uj(x),∫ d

c

∫ b

aA(y , x)dydx ≈

k∑j=1

∫ b

a`j(y)dy

∫ d

cuj(x)dx .


LU decompositionA Chebfun2 application

SVD is optimal, but GE can be faster

2D Runge function:

A(y , x) =1

1 + γ(x2 + y2).

0 5 10 15 20 25 3010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Rank of approximant

Rela

tive e

rror

in L

2

SVD

GE

γ=1

γ=10

γ=100

Wendland’s CSRBFs:

As(y , x) = φ3,s(‖x − y‖2) ∈ C2s .

0 50 100 150 20010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Rank of approximant

Re

lative

err

or

in L

2

SVD

GE

φ3,1

∈ C2

φ3,3

∈ C6

φ3,0

∈ C0


LU decompositionRelated work

Eugene Tyrtyshnikov

Goreinov, Oseledets,Savostyanov,Zamarashkin

Mario Bebendorf

Gesenhues, Griebel,Hackbusch, Rjasanow

Keith Geddes

Carvajal, Chapman

Petros Drineas

Candes, Greengard,Mahoney, Martinsson,

Rokhlin

Moral of the story: Iterative GE is everywhere, under different guises

Many others: Halko, Liberty, Martinsson, O’Neil, Tropp, Tygert, Woolfe, etc.


Cholesky factorizationMatrix factorization

A = RTR, R = upper-triangular

..A

.RT

.R

.....................

Exists: Exists and is unique if A is a positive-definite matrixApplication: A numerical test for a positive-definite matrixSeparable model: A =

∑nj=1 rjrT


Computation: Cholesky algorithm, i.e., GECP on a positive definite matrix


Cholesky factorizationContinuous analogue

A = RTR, R = upper-triangular quasimatrix

..

r1

.

r2

.......

rT1

.

rT2

.

At leastformally

Pivoting: Essential. Continuous analogue of pivoted CholeskyExists: Exists and is essentially unique for nonnegative definite functions

DefinitionAn [a,b] × [a,b] continuous symmetric cmatrix A is nonnegative definite if

vTAv =

∫ b

a

∫ b

av(y)A(y , x)v(x)dxdy ≥ 0, ∀v ∈ C[a,b].


Cholesky factorizationConvergence

TheoremLet A be an [a,b] × [a,b] continuous, symmetric, and nonnegative definitecmatrix. Suppose that A(·, x) is analytic in the closed Bernstein ellipse E2ρ(b−a)with foci a and b with ρ > 1 and bounded in absolute value by M, uniformly in y.Then

A =

∞∑j=1

rjrTj ,

where the series is uniformly and absolutelyconvergent to A. Moreover,∥∥∥∥∥∥∥A −

k∑j=1

rjrTj

∥∥∥∥∥∥∥∞

≤32Mkρ−k

4ρ − 1.

a b

stadium

E2ρ(b−a)


Cholesky factorizationComputation

Pivoted Cholesky = GECP on nonnegative definite function1

−1 0 1−1

0

1

−1 0 1−1

0

1

−1 0 1−1

0

1

−1 0 1−1

0

1

−1 0 1−1

0

1

−1 0 1−1

0

1

0 2 4 6 8 10 12 1410

−15

10−10

10−5

100

Step

Piv

ot s

ize

Pivots in Cholesky

Each step is a rank 1 update: A ←− A − A(:, x0)A(x0, :)/A(x0, x0)1Always take the absolute maximum on the diagonal even if there is a tie with an off-diagonal entry.


Cholesky factorizationA Chebfun2 application

A test for symmetric nonnegative definite functions

A = chebfun2(@(x,y) cos(10*x.*y) + y + x.ˆ2 + sin(10*x.*y));

B = A.’ * A;

chol(B)

Inverse multiquadric

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

All the pivots are nonnegative and on the y = x line⇒ nonnegative definite.Alex Townsend @ Oxford 22/24

Demo

Demo


References

Z. Battles & L. N. Trefethen, An extension of MATLAB to continuous functions and operators, SISC, 25 (2004),pp. 1743–1770.

T. A. Driscoll, F. Bornemann, & L. N. Trefethen, The chebop system for automatic solution of differentialequations, BIT, 48 (2008), pp. 701–723.

C. Eckart & G. Young, The approximation of one matrix by another of lower rank, Psychometrika, 1 (1936),pp. 211–218.

N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd edition, SIAM, 2002.

E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkurlichenFunktionen nach System vorgeschriebener, Math. Ann., 63 (1907), pp. 433–476.

G. W. Stewart, Afternotes Goes to Graduate School, Philadelphia, SIAM, 1998.

T. & L. N. Trefethen, Gaussian elimination as an iterative algorithm, SIAM News, March 2013.

T. & L. N. Trefethen, An extension of Chebfun to two dimensions, to appear in SISC, 2013.


Continuous analogues of matrix factorizations

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