Output truncation error for truncated decompositions

Project.toml

@@ -22,7 +22,7 @@ Aqua = "0.6, 0.7, 0.8"
 ChainRulesCore = "1"
 ChainRulesTestUtils = "1"
 CUDA = "5"
-JET = "0.9"
+JET = "0.9, 0.10"
 LinearAlgebra = "1"
 SafeTestsets = "0.1"
 StableRNGs = "1"

docs/src/user_interface/truncations.md

@@ -12,17 +12,15 @@ Truncation strategies allow you to control which eigenvalues or singular values
 Truncation strategies can be used with truncated decomposition functions in two ways, as illustrated below.
 For concreteness, we use the following matrix as an example:
 
-```jldoctest truncations
+```jldoctest truncations; output=false
 using MatrixAlgebraKit
 using MatrixAlgebraKit: diagview
 
 A = [2 1 0; 1 3 1; 0 1 4];
 D, V = eigh_full(A);
-
 diagview(D) ≈ [3 - √3, 3, 3 + √3]
 
 # output
-
 true
 ```
 
@@ -31,38 +29,35 @@ true
 The simplest approach is to pass a `NamedTuple` with the truncation parameters.
 For example, keeping only the largest 2 eigenvalues:
 
-```jldoctest truncations
-Dtrunc, Vtrunc = eigh_trunc(A; trunc = (maxrank = 2,));
+```jldoctest truncations; output=false
+Dtrunc, Vtrunc, ϵ = eigh_trunc(A; trunc = (maxrank = 2,));
 size(Dtrunc, 1) <= 2
 
 # output
-
 true
 ```
 
 Note however that there are no guarantees on the order of the output values:
 
-```jldoctest truncations
+```jldoctest truncations; output=false
 diagview(Dtrunc) ≈ diagview(D)[[3, 2]]
 
 # output
-
 true
 ```
 
 You can also use tolerance-based truncation or combine multiple criteria:
 
-```jldoctest truncations
-Dtrunc, Vtrunc = eigh_trunc(A; trunc = (atol = 2.9,));
+```jldoctest truncations; output=false
+Dtrunc, Vtrunc, ϵ = eigh_trunc(A; trunc = (atol = 2.9,));
 all(>(2.9), diagview(Dtrunc))
 
 # output
-
 true
 ```
 
-```jldoctest truncations
-Dtrunc, Vtrunc = eigh_trunc(A; trunc = (maxrank = 2, atol = 2.9));
+```jldoctest truncations; output=false
+Dtrunc, Vtrunc, ϵ = eigh_trunc(A; trunc = (maxrank = 2, atol = 2.9));
 size(Dtrunc, 1) <= 2 && all(>(2.9), diagview(Dtrunc))
 
 # output
@@ -72,7 +67,7 @@ true
 In general, the keyword arguments that are supported can be found in the `TruncationStrategy` docstring:
 
 ```@docs; canonical = false
-TruncationStrategy
+TruncationStrategy()
 ```
 
 
@@ -81,33 +76,22 @@ TruncationStrategy
 For more control, you can construct [`TruncationStrategy`](@ref) objects directly.
 This is also what the previous syntax will end up calling.
 
-```jldoctest truncations
+```jldoctest truncations; output=false
 Dtrunc, Vtrunc = eigh_trunc(A; trunc = truncrank(2))
 size(Dtrunc, 1) <= 2
 
 # output
-
 true
 ```
 
-```jldoctest truncations
-Dtrunc, Vtrunc = eigh_trunc(A; trunc = truncrank(2) & trunctol(; atol = 2.9))
+```jldoctest truncations; output=false
+Dtrunc, Vtrunc, ϵ = eigh_trunc(A; trunc = truncrank(2) & trunctol(; atol = 2.9))
 size(Dtrunc, 1) <= 2 && all(>(2.9), diagview(Dtrunc))
 
 # output
 true
 ```
 
-## Truncation with SVD vs Eigenvalue Decompositions
-
-When using truncations with different decomposition types, keep in mind:
-
-- **`svd_trunc`**: Singular values are always real and non-negative, sorted in descending order. Truncation by value typically keeps the largest singular values.
-
-- **`eigh_trunc`**: Eigenvalues are real but can be negative for symmetric matrices. By default, `truncrank` sorts by absolute value, so `truncrank(k)` keeps the `k` eigenvalues with largest magnitude (positive or negative).
-
-- **`eig_trunc`**: For general (non-symmetric) matrices, eigenvalues can be complex. Truncation by absolute value considers the complex magnitude.
-
 ## Truncation Strategies
 
 MatrixAlgebraKit provides several built-in truncation strategies:
@@ -127,3 +111,31 @@ When strategies are combined, only the values that satisfy all conditions are ke
 combined_trunc = truncrank(10) & trunctol(; atol = 1e-6);
 ```
 
+## Truncation Error
+
+When using truncated decompositions such as [`svd_trunc`](@ref), [`eig_trunc`](@ref), or [`eigh_trunc`](@ref), an additional truncation error value is returned.
+This error is defined as the 2-norm of the discarded  singular values or eigenvalues, providing a measure of the approximation quality.
+For `svd_trunc` and `eigh_trunc`, this corresponds to the 2-norm difference between the original and the truncated matrix.
+For the case of `eig_trunc`, this interpretation does not hold because the norm of the non-unitary matrix of eigenvectors and its inverse also influence the approximation quality.
+
+
+For example:
+```jldoctest truncations; output=false
+using LinearAlgebra: norm
+U, S, Vᴴ, ϵ = svd_trunc(A; trunc=truncrank(2))
+norm(A - U * S * Vᴴ) ≈ ϵ # ϵ is the 2-norm of the discarded singular values
+
+# output
+true
+```
+
+### Truncation with SVD vs Eigenvalue Decompositions
+
+When using truncations with different decomposition types, keep in mind:
+
+- **[`svd_trunc`](@ref)**: Singular values are always real and non-negative, sorted in descending order. Truncation by value typically keeps the largest singular values. The truncation error gives the 2-norm difference between the original and the truncated matrix.
+
+- **[`eigh_trunc`](@ref)**: Eigenvalues are real but can be negative for symmetric matrices. By default, eigenvalues are treated by absolute value, e.g. `truncrank(k)` keeps the `k` eigenvalues with largest magnitude (positive or negative). The truncation error gives the 2-norm difference between the original and the truncated matrix.
+
+- **[`eig_trunc`](@ref)**: For general (non-symmetric) matrices, eigenvalues can be complex. By default, eigenvalues are treated by absolute value. The truncation error gives an indication of the magnitude of discarded values, but is not directly related to the 2-norm difference between the original and the truncated matrix.
+

ext/MatrixAlgebraKitChainRulesCoreExt.jl

@@ -2,7 +2,7 @@ module MatrixAlgebraKitChainRulesCoreExt
 
 using MatrixAlgebraKit
 using MatrixAlgebraKit: copy_input, initialize_output, zero!, diagview,
-    TruncatedAlgorithm, findtruncated, findtruncated_svd
+    TruncatedAlgorithm, findtruncated, findtruncated_svd, truncation_error
 using ChainRulesCore
 using LinearAlgebra
 
@@ -113,15 +113,20 @@ for eig in (:eig, :eigh)
             Ac = copy_input($eig_f, A)
             DV = $(eig_f!)(Ac, DV, alg.alg)
             DV′, ind = MatrixAlgebraKit.truncate($eig_t!, DV, alg.trunc)
-            return DV′, $(_make_eig_t_pb)(A, DV, ind)
+            ϵ = truncation_error(diagview(DV[1]), ind)
+            return (DV′..., ϵ), $(_make_eig_t_pb)(A, DV, ind)
         end
         function $(_make_eig_t_pb)(A, DV, ind)
-            function $eig_t_pb(ΔDV)
+            function $eig_t_pb(ΔDVϵ)
                 ΔA = zero(A)
-                MatrixAlgebraKit.$eig_pb!(ΔA, A, DV, unthunk.(ΔDV), ind)
+                ΔD, ΔV, Δϵ = ΔDVϵ
+                if !MatrixAlgebraKit.iszerotangent(Δϵ) && !iszero(unthunk(Δϵ))
+                    throw(ArgumentError("Pullback for eig_trunc! does not yet support non-zero tangent for the truncation error"))
+                end
+                MatrixAlgebraKit.$eig_pb!(ΔA, A, DV, unthunk.((ΔD, ΔV)), ind)
                 return NoTangent(), ΔA, ZeroTangent(), NoTangent()
             end
-            function $eig_t_pb(::Tuple{ZeroTangent, ZeroTangent}) # is this extra definition useful?
+            function $eig_t_pb(::Tuple{ZeroTangent, ZeroTangent, ZeroTangent}) # is this extra definition useful?
                 return NoTangent(), ZeroTangent(), ZeroTangent(), NoTangent()
             end
             return $eig_t_pb
@@ -152,15 +157,20 @@ function ChainRulesCore.rrule(::typeof(svd_trunc!), A, USVᴴ, alg::TruncatedAlg
     Ac = copy_input(svd_compact, A)
     USVᴴ = svd_compact!(Ac, USVᴴ, alg.alg)
     USVᴴ′, ind = MatrixAlgebraKit.truncate(svd_trunc!, USVᴴ, alg.trunc)
-    return USVᴴ′, _make_svd_trunc_pullback(A, USVᴴ, ind)
+    ϵ = truncation_error(diagview(USVᴴ[2]), ind)
+    return (USVᴴ′..., ϵ), _make_svd_trunc_pullback(A, USVᴴ, ind)
 end
 function _make_svd_trunc_pullback(A, USVᴴ, ind)
-    function svd_trunc_pullback(ΔUSVᴴ)
+    function svd_trunc_pullback(ΔUSVᴴϵ)
         ΔA = zero(A)
-        MatrixAlgebraKit.svd_pullback!(ΔA, A, USVᴴ, unthunk.(ΔUSVᴴ), ind)
+        ΔU, ΔS, ΔVᴴ, Δϵ = ΔUSVᴴϵ
+        if !MatrixAlgebraKit.iszerotangent(Δϵ) && !iszero(unthunk(Δϵ))
+            throw(ArgumentError("Pullback for svd_trunc! does not yet support non-zero tangent for the truncation error"))
+        end
+        MatrixAlgebraKit.svd_pullback!(ΔA, A, USVᴴ, unthunk.((ΔU, ΔS, ΔVᴴ)), ind)
         return NoTangent(), ΔA, ZeroTangent(), NoTangent()
     end
-    function svd_trunc_pullback(::Tuple{ZeroTangent, ZeroTangent, ZeroTangent}) # is this extra definition useful?
+    function svd_trunc_pullback(::Tuple{ZeroTangent, ZeroTangent, ZeroTangent, ZeroTangent}) # is this extra definition useful?
         return NoTangent(), ZeroTangent(), ZeroTangent(), NoTangent()
     end
     return svd_trunc_pullback

src/implementations/eig.jl

@@ -108,7 +108,8 @@ end
 
 function eig_trunc!(A, DV, alg::TruncatedAlgorithm)
     D, V = eig_full!(A, DV, alg.alg)
-    return first(truncate(eig_trunc!, (D, V), alg.trunc))
+    DVtrunc, ind = truncate(eig_trunc!, (D, V), alg.trunc)
+    return DVtrunc..., truncation_error!(diagview(D), ind)
 end
 
 # Diagonal logic

src/implementations/eigh.jl

@@ -111,7 +111,8 @@ end
 
 function eigh_trunc!(A, DV, alg::TruncatedAlgorithm)
     D, V = eigh_full!(A, DV, alg.alg)
-    return first(truncate(eigh_trunc!, (D, V), alg.trunc))
+    DVtrunc, ind = truncate(eigh_trunc!, (D, V), alg.trunc)
+    return DVtrunc..., truncation_error!(diagview(D), ind)
 end
 
 # Diagonal logic

src/implementations/svd.jl

@@ -237,8 +237,9 @@ function svd_vals!(A::AbstractMatrix, S, alg::LAPACK_SVDAlgorithm)
 end
 
 function svd_trunc!(A, USVᴴ, alg::TruncatedAlgorithm)
-    USVᴴ′ = svd_compact!(A, USVᴴ, alg.alg)
-    return first(truncate(svd_trunc!, USVᴴ′, alg.trunc))
+    U, S, Vᴴ = svd_compact!(A, USVᴴ, alg.alg)
+    USVᴴtrunc, ind = truncate(svd_trunc!, (U, S, Vᴴ), alg.trunc)
+    return USVᴴtrunc..., truncation_error!(diagview(S), ind)
 end
 
 # Diagonal logic
@@ -381,7 +382,12 @@ function svd_trunc!(A::AbstractMatrix, USVᴴ, alg::TruncatedAlgorithm{<:GPU_Ran
     _gpu_Xgesvdr!(A, S.diag, U, Vᴴ; alg.alg.kwargs...)
     # TODO: make this controllable using a `gaugefix` keyword argument
     gaugefix!(svd_trunc!, U, S, Vᴴ, size(A)...)
-    return first(truncate(svd_trunc!, USVᴴ, alg.trunc))
+    # TODO: make sure that truncation is based on maxrank, otherwise this might be wrong
+    USVᴴtrunc, ind = truncate(svd_trunc!, (U, S, Vᴴ), alg.trunc)
+    Strunc = diagview(USVᴴtrunc[2])
+    # normal `truncation_error!` does not work here since `S` is not the full singular value spectrum
+    ϵ = sqrt(norm(A)^2 - norm(Strunc)^2) # is there a more accurate way to do this?
+    return USVᴴtrunc..., ϵ
 end
 
 function MatrixAlgebraKit.svd_compact!(A::AbstractMatrix, USVᴴ, alg::GPU_SVDAlgorithm)

src/implementations/truncation.jl

@@ -116,3 +116,13 @@ end
 _ind_intersect(A::AbstractVector, B::AbstractVector{Bool}) = _ind_intersect(B, A)
 _ind_intersect(A::AbstractVector{Bool}, B::AbstractVector{Bool}) = A .& B
 _ind_intersect(A, B) = intersect(A, B)
+
+# Truncation error
+# ----------------
+truncation_error(values::AbstractVector, ind) = truncation_error!(copy(values), ind)
+# destroys input in order to maximize accuracy:
+# sqrt(norm(values)^2 - norm(values[ind])^2) might suffer from floating point error
+function truncation_error!(values::AbstractVector, ind)
+    values[ind] .= zero(eltype(values))
+    return norm(values)
+end

src/interface/eig.jl

@@ -32,16 +32,19 @@ See also [`eig_vals(!)`](@ref eig_vals) and [`eig_trunc(!)`](@ref eig_trunc).
 @functiondef eig_full
 
 """
-    eig_trunc(A; [trunc], kwargs...) -> D, V
-    eig_trunc(A, alg::AbstractAlgorithm) -> D, V
-    eig_trunc!(A, [DV]; [trunc], kwargs...) -> D, V
-    eig_trunc!(A, [DV], alg::AbstractAlgorithm) -> D, V
+    eig_trunc(A; [trunc], kwargs...) -> D, V, ϵ
+    eig_trunc(A, alg::AbstractAlgorithm) -> D, V, ϵ
+    eig_trunc!(A, [DV]; [trunc], kwargs...) -> D, V, ϵ
+    eig_trunc!(A, [DV], alg::AbstractAlgorithm) -> D, V, ϵ
 
 Compute a partial or truncated eigenvalue decomposition of the matrix `A`,
 such that `A * V ≈ V * D`, where the (possibly rectangular) matrix `V` contains 
 a subset of eigenvectors and the diagonal matrix `D` contains the associated eigenvalues,
 selected according to a truncation strategy.
 
+The function also returns `ϵ`, the truncation error defined as the 2-norm of the 
+discarded eigenvalues.
+
 ## Keyword arguments
 The behavior of this function is controlled by the following keyword arguments:
 

src/interface/eigh.jl

@@ -12,15 +12,18 @@ docs_eigh_note = """
 """
 
 """
-    eigh_full(A; kwargs...) -> D, V
-    eigh_full(A, alg::AbstractAlgorithm) -> D, V
-    eigh_full!(A, [DV]; kwargs...) -> D, V
-    eigh_full!(A, [DV], alg::AbstractAlgorithm) -> D, V
+    eigh_full(A; kwargs...) -> D, V, ϵ
+    eigh_full(A, alg::AbstractAlgorithm) -> D, V, ϵ
+    eigh_full!(A, [DV]; kwargs...) -> D, V, ϵ
+    eigh_full!(A, [DV], alg::AbstractAlgorithm) -> D, V, ϵ
 
 Compute the full eigenvalue decomposition of the symmetric or hermitian matrix `A`,
 such that `A * V = V * D`, where the unitary matrix `V` contains the orthogonal eigenvectors
 and the real diagonal matrix `D` contains the associated eigenvalues.
 
+The function also returns `ϵ`, the truncation error defined as the 2-norm of the 
+discarded eigenvalues.
+
 !!! note
     The bang method `eigh_full!` optionally accepts the output structure and
     possibly destroys the input matrix `A`. Always use the return value of the function
@@ -34,16 +37,19 @@ See also [`eigh_vals(!)`](@ref eigh_vals) and [`eigh_trunc(!)`](@ref eigh_trunc)
 @functiondef eigh_full
 
 """
-    eigh_trunc(A; [trunc], kwargs...) -> D, V
-    eigh_trunc(A, alg::AbstractAlgorithm) -> D, V
-    eigh_trunc!(A, [DV]; [trunc], kwargs...) -> D, V
-    eigh_trunc!(A, [DV], alg::AbstractAlgorithm) -> D, V
+    eigh_trunc(A; [trunc], kwargs...) -> D, V, ϵ
+    eigh_trunc(A, alg::AbstractAlgorithm) -> D, V, ϵ
+    eigh_trunc!(A, [DV]; [trunc], kwargs...) -> D, V, ϵ
+    eigh_trunc!(A, [DV], alg::AbstractAlgorithm) -> D, V, ϵ
 
 Compute a partial or truncated eigenvalue decomposition of the symmetric or hermitian matrix
 `A`, such that `A * V ≈ V * D`, where the isometric matrix `V` contains a subset of the
 orthogonal eigenvectors and the real diagonal matrix `D` contains the associated eigenvalues,
 selected according to a truncation strategy.
 
+The function also returns `ϵ`, the truncation error defined as the 2-norm of the discarded
+eigenvalues.
+
 ## Keyword arguments
 The behavior of this function is controlled by the following keyword arguments:
 

src/interface/svd.jl

@@ -42,16 +42,19 @@ See also [`svd_full(!)`](@ref svd_full), [`svd_vals(!)`](@ref svd_vals) and
 @functiondef svd_compact
 
 """
-    svd_trunc(A; [trunc], kwargs...) -> U, S, Vᴴ
-    svd_trunc(A, alg::AbstractAlgorithm) -> U, S, Vᴴ
-    svd_trunc!(A, [USVᴴ]; [trunc], kwargs...) -> U, S, Vᴴ
-    svd_trunc!(A, [USVᴴ], alg::AbstractAlgorithm) -> U, S, Vᴴ
+    svd_trunc(A; [trunc], kwargs...) -> U, S, Vᴴ, ϵ
+    svd_trunc(A, alg::AbstractAlgorithm) -> U, S, Vᴴ, ϵ
+    svd_trunc!(A, [USVᴴ]; [trunc], kwargs...) -> U, S, Vᴴ, ϵ
+    svd_trunc!(A, [USVᴴ], alg::AbstractAlgorithm) -> U, S, Vᴴ, ϵ
 
 Compute a partial or truncated singular value decomposition (SVD) of `A`, such that
-`A * (Vᴴ)' =  U * S`. Here, `U` is an isometric matrix (orthonormal columns) of size
+`A * (Vᴴ)' ≈ U * S`. Here, `U` is an isometric matrix (orthonormal columns) of size
 `(m, k)`, whereas  `Vᴴ` is a matrix of size `(k, n)` with orthonormal rows and `S` is a
 square diagonal matrix of size `(k, k)`, with `k` is set by the truncation strategy.
 
+The function also returns `ϵ`, the truncation error defined as the 2-norm of the 
+discarded singular values.
+
 ## Keyword arguments
 The behavior of this function is controlled by the following keyword arguments:
 

src/interface/truncation.jl

@@ -181,3 +181,8 @@ Base.:&(::NoTruncation, ::NoTruncation) = notrunc()
 # disambiguate
 Base.:&(::NoTruncation, trunc::TruncationIntersection) = trunc
 Base.:&(trunc::TruncationIntersection, ::NoTruncation) = trunc
+
+@doc """
+    truncation_error(values, ind)
+Compute the truncation error as the 2-norm of the values that are not kept by `ind`.
+""" truncation_error, truncation_error!