2023-03-01 Singular Value Decomposition
Contents
2023-03-01 Singular Value Decomposition#
Last time#
Householder QR
Composition of reflectors
Today#
Comparison of interfaces
Profiling
Cholesky QR
Matrix norms and conditioning
Geometry of the SVD
using LinearAlgebra
using Plots
default(linewidth=4, legendfontsize=12)
function vander(x, k=nothing)
if isnothing(k)
k = length(x)
end
m = length(x)
V = ones(m, k)
for j in 2:k
V[:, j] = V[:, j-1] .* x
end
V
end
function gram_schmidt_classical(A)
m, n = size(A)
Q = zeros(m, n)
R = zeros(n, n)
for j in 1:n
v = A[:,j]
R[1:j-1,j] = Q[:,1:j-1]' * v
v -= Q[:,1:j-1] * R[1:j-1,j]
R[j,j] = norm(v)
Q[:,j] = v / R[j,j]
end
Q, R
end
function qr_householder(A)
m, n = size(A)
R = copy(A)
V = [] # list of reflectors
for j in 1:n
v = copy(R[j:end, j])
v[1] += sign(v[1]) * norm(v) # <---
v = normalize(v)
R[j:end,j:end] -= 2 * v * v' * R[j:end,j:end]
push!(V, v)
end
V, R
end
qr_householder (generic function with 1 method)
Composition of reflectors#
This turns applying reflectors from a sequence of vector operations to a sequence of (smallish) matrix operations. It’s the key to high performance and the native format (QRCompactWY) returned by Julia qr().
x = LinRange(-1, 1, 20)
A = vander(x)
Q, R = qr(A)
#norm(Q[:,1])
LinearAlgebra.QRCompactWY{Float64, Matrix{Float64}}
Q factor:
20×20 LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}}:
-0.223607 -0.368394 -0.430192 0.437609 … -3.23545e-5 -5.31905e-6
-0.223607 -0.329616 -0.294342 0.161225 0.000550027 0.000101062
-0.223607 -0.290838 -0.173586 -0.0383868 -0.00436786 -0.000909558
-0.223607 -0.252059 -0.067925 -0.170257 0.0214511 0.00515416
-0.223607 -0.213281 0.0226417 -0.243417 -0.0726036 -0.0206167
-0.223607 -0.174503 0.0981139 -0.266901 … 0.178209 0.06185
-0.223607 -0.135724 0.158492 -0.24974 -0.323416 -0.144317
-0.223607 -0.0969458 0.203775 -0.200966 0.429021 0.268016
-0.223607 -0.0581675 0.233964 -0.129612 -0.386119 -0.402025
-0.223607 -0.0193892 0.249058 -0.0447093 0.157308 0.491364
-0.223607 0.0193892 0.249058 0.0447093 … 0.157308 -0.491364
-0.223607 0.0581675 0.233964 0.129612 -0.386119 0.402025
-0.223607 0.0969458 0.203775 0.200966 0.429021 -0.268016
-0.223607 0.135724 0.158492 0.24974 -0.323416 0.144317
-0.223607 0.174503 0.0981139 0.266901 0.178209 -0.06185
-0.223607 0.213281 0.0226417 0.243417 … -0.0726036 0.0206167
-0.223607 0.252059 -0.067925 0.170257 0.0214511 -0.00515416
-0.223607 0.290838 -0.173586 0.0383868 -0.00436786 0.000909558
-0.223607 0.329616 -0.294342 -0.161225 0.000550027 -0.000101062
-0.223607 0.368394 -0.430192 -0.437609 -3.23545e-5 5.31905e-6
R factor:
20×20 Matrix{Float64}:
-4.47214 0.0 -1.64763 0.0 … -0.514468 2.22045e-16
0.0 2.71448 1.11022e-16 1.79412 -2.498e-16 0.823354
0.0 0.0 -1.46813 5.55112e-17 -0.944961 -2.23779e-16
0.0 0.0 0.0 -0.774796 3.83808e-17 -0.913056
0.0 0.0 0.0 0.0 0.797217 -4.06264e-16
0.0 0.0 0.0 0.0 … -3.59496e-16 0.637796
0.0 0.0 0.0 0.0 -0.455484 -1.3936e-15
0.0 0.0 0.0 0.0 4.40958e-16 -0.313652
0.0 0.0 0.0 0.0 -0.183132 1.64685e-15
0.0 0.0 0.0 0.0 4.82253e-16 0.109523
0.0 0.0 0.0 0.0 … 0.0510878 5.9848e-16
0.0 0.0 0.0 0.0 -2.68709e-15 0.0264553
0.0 0.0 0.0 0.0 -0.0094344 -2.94383e-15
0.0 0.0 0.0 0.0 2.08514e-15 0.00417208
0.0 0.0 0.0 0.0 0.0010525 -2.24994e-15
0.0 0.0 0.0 0.0 … -1.64363e-15 -0.000385264
0.0 0.0 0.0 0.0 -5.9057e-5 7.69025e-16
0.0 0.0 0.0 0.0 1.76642e-16 -1.66202e-5
0.0 0.0 0.0 0.0 -1.04299e-6 -1.68771e-16
0.0 0.0 0.0 0.0 0.0 1.71467e-7
This works even for very nonsquare matrices#
A = rand(1000000, 5)
Q, R = qr(A)
@show size(Q)
@show norm(Q*R - A)
R
Q * vcat(zeros(1000000 - 3), [1,0,0])
size(Q) = (1000000, 1000000)
norm(Q * R - A) = 1.5627435002979355e-12
1000000-element Vector{Float64}:
-0.000710716962490326
-0.0006221486003295463
-0.001481395515703071
0.0006986145984357538
0.00030719398438148966
-3.7070273376922167e-6
-1.846993064721867e-6
5.892273227768894e-7
-3.4792383435730715e-6
2.2249702781554393e-6
-1.8148213625435382e-6
-1.3842243836198498e-8
-2.943731575863994e-6
⋮
5.4302791596705405e-8
-1.2863812388719472e-6
-3.1178852037441024e-6
-1.454678366309665e-6
1.129578508676748e-6
-1.0958261567418687e-6
-1.4438579316234545e-7
4.451050785148942e-7
-1.2256371019533118e-6
0.9999963354732083
4.843751131958308e-7
2.761092851686931e-7
This is known as a “full” (or “complete”) QR factorization, in contrast to a reduced QR factorization in which \(Q\) has the same shape as \(A\).
How much memory does \(Q\) use?
Compare to numpy.linalg.qr#
Need to decide up-front whether you want full or reduced QR.
Full QR is expensive to represent.
Cholesky QR#
so we should be able to use \(L L^T = A^T A\) and then \(Q = A L^{-T}\).
function qr_chol(A)
R = cholesky(A' * A).U
Q = A / R
Q, R
end
A = rand(20,20)
Q, R = qr_chol(A)
@show norm(Q' * Q - I)
@show norm(Q * R - A)
norm(Q' * Q - I) = 1.3547676815713508e-13
norm(Q * R - A) = 1.289504394296693e-15
1.289504394296693e-15
x = LinRange(-1, 1, 20)
A = vander(x)
Q, R = qr_chol(A)
@show norm(Q' * Q - I)
@show norm(Q * R - A);
PosDefException: matrix is not positive definite; Cholesky factorization failed.
Stacktrace:
[1] checkpositivedefinite
@ /usr/share/julia/stdlib/v1.7/LinearAlgebra/src/factorization.jl:18 [inlined]
[2] cholesky!(A::Hermitian{Float64, Matrix{Float64}}, ::Val{false}; check::Bool)
@ LinearAlgebra /usr/share/julia/stdlib/v1.7/LinearAlgebra/src/cholesky.jl:266
[3] cholesky!(A::Matrix{Float64}, ::Val{false}; check::Bool)
@ LinearAlgebra /usr/share/julia/stdlib/v1.7/LinearAlgebra/src/cholesky.jl:298
[4] #cholesky#143
@ /usr/share/julia/stdlib/v1.7/LinearAlgebra/src/cholesky.jl:394 [inlined]
[5] cholesky (repeats 2 times)
@ /usr/share/julia/stdlib/v1.7/LinearAlgebra/src/cholesky.jl:394 [inlined]
[6] qr_chol(A::Matrix{Float64})
@ Main ./In[9]:2
[7] top-level scope
@ In[14]:3
[8] eval
@ ./boot.jl:373 [inlined]
[9] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)
@ Base ./loading.jl:1196
Can we fix this?#
Note that the product of two triangular matrices is triangular.
R = triu(rand(5,5))
R * R
5×5 Matrix{Float64}:
0.937018 0.166589 1.28394 0.856513 1.91001
0.0 0.147171 0.422197 0.410025 1.03557
0.0 0.0 0.466456 0.381229 0.947622
0.0 0.0 0.0 0.690276 0.506928
0.0 0.0 0.0 0.0 0.0989786
function qr_chol2(A)
Q, R = qr_chol(A)
Q, R1 = qr_chol(Q)
Q, R1 * R
end
x = LinRange(-1, 1, 18)
A = vander(x)
Q, R = qr_chol2(A)
@show norm(Q' * Q - I)
@show norm(Q * R - A);
norm(Q' * Q - I) = 1.1333031064724956e-15
norm(Q * R - A) = 1.1896922215066695e-15
How fast are these methods?#
m, n = 5000, 2000
A = randn(m, n)
@time qr(A);
1.507001 seconds (7 allocations: 77.393 MiB, 0.34% gc time)
A = randn(m, n)
@time qr_chol2(A);
1.067684 seconds (21 allocations: 305.176 MiB, 0.90% gc time)
Profiling#
using ProfileSVG
@profview qr(A)
Condition number of a matrix#
We may have informally referred to a matrix as “ill-conditioned” when the columns are nearly linearly dependent, but let’s make this concept for precise. Recall the definition of (relative) condition number:
We understood this definition for scalar problems, but it also makes sense when the inputs and/or outputs are vectors (or matrices, etc.) and absolute value is replaced by vector (or matrix) norms. Consider matrix-vector multiplication, for which \(f(x) = A x\).
There are two problems here:
I wrote \(\kappa(A)\) but my formula depends on \(x\).
What is that \(\lVert A \rVert\) beastie?
Matrix norms induced by vector norms#
Vector norms are built into the linear space (and defined in term of the inner product). Matrix norms are induced by vector norms, according to
This equation makes sense for non-square matrices – the vector norms of the input and output spaces may differ.
Due to linearity, all that matters is direction of \(x\), so it could equivalently be written
The formula makes sense, but still depends on \(x\).#
Consider the matrix
What is the norm of this matrix?
What is the condition number when \(x = [1,0]^T\)?
What is the condition number when \(x = [0,1]^T\)?
Condition number of the matrix#
The condition number of matrix-vector multiplication depends on the vector. The condition number of the matrix is the worst case (maximum) of the condition number for any vector, i.e.,
If \(A\) is invertible, then we can rephrase as
Evidently multiplying by a matrix is just as ill-conditioned of an operation as solving a linear system using that matrix.
Least squares and the normal equations#
A least squares problem takes the form: given an \(m\times n\) matrix \(A\) (\(m \ge n\)), find \(x\) such that
Is this the same as saying \(A^T (A x - b) = 0\)?
If \(QR = A\), is it the same as \(Q^T (A x - b) = 0\)?
In the quiz, we showed that \(QQ^T\) is an orthogonal projector onto the range of \(Q\). If \(QR = A\),
Solution by QR (Householder)#
Solve \(R x = Q^T b\).
QR factorization costs \(2 m n^2 - \frac 2 3 n^3\) operations and is done once per matrix \(A\).
Computing \(Q^T b\) costs \(4 (m-n)n + 2 n^2 = 4 mn - 2n^2\) (using the elementary reflectors, which are stable and lower storage than naive storage of \(Q\)).
Solving with \(R\) costs \(n^2\) operations. Total cost per right hand side is thus \(4 m n - n^2\).
This method is stable and accurate.
Solution by Cholesky#
The mathematically equivalent form \((A^T A) x = A^T b\) are called the normal equations. The solution process involves factoring the symmetric and positive definite \(n\times n\) matrix \(A^T A\).
Computing \(A^T A\) costs \(m n^2\) flops, exploiting symmetry.
Factoring \(A^T A = R^T R\) costs \(\frac 1 3 n^3\) flops. The total factorization cost is thus \(m n^2 + \frac 1 3 n^3\).
Computing \(A^T b\) costs \(2 m n\).
Solving with \(R^T\) costs \(n^2\).
Solving with \(R\) costs \(n^2\). Total cost per right hand side is thus \(2 m n + 2 n^2\).
The product \(A^T A\) is ill-conditioned: \(\kappa(A^T A) = \kappa(A)^2\) and can reduce the accuracy of a least squares solution.
Solution by Singular Value Decomposition#
Next, we will discuss a factorization
the SVD exists for all matrices (including non-square and deficient matrices)
\(U,V\) have orthogonal columns (while \(X\) can be arbitrarily ill-conditioned). Indeed, if a matrix is symmetric and positive definite (all positive eigenvalues), then \(U=V\) and \(\Sigma = \Lambda\). Computing an SVD requires a somewhat complicated iterative algorithm, but a crude estimate of the cost is \(2 m n^2 + 11 n^3\). Note that this is similar to the cost of \(QR\) when \(m \gg n\), but much more expensive for square matrices. Solving with the SVD involves
Compute \(U^T b\) at a cost of \(2 m n\).
Solve with the diagonal \(n\times n\) matrix \(\Sigma\) at a cost of \(n\).
Apply \(V\) at a cost of \(2 n^2\). The total cost per right hand side is thus \(2 m n + 2n^2\).
Activity: Geometry of the Singular Value Decomposition#
default(aspect_ratio=:equal)
function peanut()
theta = LinRange(0, 2*pi, 50)
r = 1 .+ .4*sin.(3*theta) + .6*sin.(2*theta)
r' .* [cos.(theta) sin.(theta)]'
end
function circle()
theta = LinRange(0, 2*pi, 50)
[cos.(theta) sin.(theta)]'
end
function Aplot(A)
"Plot a transformation from X to Y"
X = peanut()
Y = A * X
p = scatter(X[1,:], X[2,:], label="in")
scatter!(p, Y[1,:], Y[2,:], label="out")
X = circle()
Y = A * X
q = scatter(X[1,:], X[2,:], label="in")
scatter!(q, Y[1,:], Y[2,:], label="out")
plot(p, q, layout=2)
end
Aplot (generic function with 1 method)
Aplot(1.1*I)
Diagonal matrices#
Perhaps the simplest transformation is a scalar multiple of the identity.
Aplot([2 0; 0 2])
The diagonal entries can be different sizes.
Aplot([2 0; 0 .5])
The circles becomes an ellipse that is aligned with the coordinate axes
Givens Rotation (as example of orthogonal matrix)#
We can rotate the input using a \(2\times 2\) matrix, parametrized by \(\theta\). Its transpose rotates in the opposite direction.
function givens(theta)
s = sin(theta)
c = cos(theta)
[c -s; s c]
end
G = givens(0.3)
Aplot(G)
Aplot(G')
Reflection#
We’ve previously seen that reflectors have the form \(F = I - 2 v v^T\) where \(v\) is a normalized vector. Reflectors satisfy \(F^T F = I\) and \(F = F^T\), thus \(F^2 = I\).
function reflect(theta)
v = [cos(theta), sin(theta)]
I - 2 * v * v'
end
Aplot(reflect(0.3))
Singular Value Decomposition#
The SVD is \(A = U \Sigma V^T\) where \(U\) and \(V\) have orthonormal columns and \(\Sigma\) is diagonal with nonnegative entries. It exists for any matrix (non-square, singular, etc.). If we think of orthogonal matrices as reflections/rotations, this says any matrix can be represented as reflect/rotate, diagonally scale, and reflect/rotate again.
Let’s try a random symmetric matrix.
A = randn(2, 2)
A += A' # make symmetric
@show det(A) # Positive means orientation is preserved
Aplot(A)
det(A) = -1.6820338013412035
U, S, V = svd(A)
@show norm(U * diagm(S) * V' - A) # Should be zero
Aplot(V') # Rotate/reflect in preparation for scaling
norm(U * diagm(S) * V' - A) = 4.002966042486721e-16
What visual features indicate that this is a symmetric matrix?
Is the orthogonal matrix a reflection or rotation?
Does this change when the determinant is positive versus negative (rerun the cell above as needed).
Parts of the SVD#
Aplot(diagm(S)) # scale along axes
Aplot(U) # rotate/reflect back
Putting it together#
Aplot(U * diagm(S) * V')
Observations#
The circle always maps to an ellipse
The \(U\) and \(V\) factors may reflect even when \(\det A > 0\)